On automorphisms of quantum Schubert cells

TL;DR

This paper investigates the automorphism groups of quantum Schubert cell algebras, providing rigidity results and confirming conjectures about their structure, especially for certain Lie algebra types and quantum matrix cases.

Contribution

It develops general rigidity results and determines automorphism groups for specific quantum Schubert cell algebras, advancing understanding of their symmetries and confirming related conjectures.

Findings

Automorphism group is a semidirect product of a torus and diagram symmetries in many cases.

Complete determination of automorphism groups for types F4 and G2.

Verification of conjectures for quantum symmetric matrices cases.

Abstract

Automorphisms of the quantum Schubert cell algebras ${\mathcal U}_q^\pm[w]$ of De Concini, Kac, Procesi and Lusztig and their restrictions to some key invariant subalgebras are studied. We develop some general rigidity results and apply them to completely determine the automorphism group in several cases. We focus primarily on those cases when the underlying Lie algebra $\mathfrak{g}$ is finite dimensional and simple with rank $r > 1$, and $w$ is a parabolic element of the Weyl group, say $w = w_o^Jw_o$, for some nonempty subset $J$ of simple roots. Here, ${\mathcal U}_q^\pm[w]$ is a deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of $\mathfrak{g}$. In this setting we conjecture that, with the exception of two specific low rank cases, the automorphism group of ${\mathcal U}_q^{\pm}[w]$ is the semidirect product of an algebraic torus of rank $r$ with the group of Dynkin diagram symmetries that preserve $J$. This conjecture is a more general form of the Launois-Lenagan and Andruskiewitsch-Dumas conjectures regarding the automorphism groups of the algebras of quantum matrices and the algebras ${\mathcal U}_q^+(\mathfrak{g})$, respectively. We completely determine the automorphism group in several instances, including all cases when $\mathfrak{g}$ is of type $F_4$ or $G_2$, as well as those cases when the quantum Schubert cell algebras are the algebras of quantum symmetric matrices.

Equations

$$\varphi (x_1)=a_1{x}_1,\varphi (x_2)=a_2{x}_2+b{x}_1{x}_3,\varphi (x_3)=a_3{x}_3,$$

$$\mathrm{Aut}\left({\mathcal{U}}_q^{+}\left(\mathfrak{g}\right)\right)\cong {\left({\mathbb{K}}^{\times }\right)}^{\mathrm{rank}\left(\mathfrak{g}\right)}⋊Dynkin-Aut\left(\mathfrak{g}\right),$$

$${Dynkin-Aut}_J(\mathfrak{g}):=\left\{\psi \in Dynkin-Aut(\mathfrak{g})\mid \psi (J)=J\right\}.$$

$$\mathrm{Aut}\left({\mathcal{U}}_q\left({\mathfrak{n}}_J\right)\right)\cong {\left({\mathbb{K}}^{\times }\right)}^{\mathrm{rank}\left(\mathfrak{g}\right)}⋊{Dynkin-Aut}_J(\mathfrak{g}).$$

$$\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+},$$

$${\mathfrak{n}}^{\pm }:=\underset{\alpha \in {\Delta }_{\pm }}{⨁}{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\alpha }:=\left\{x\in \mathfrak{g}\mid [h,x]=\alpha (h)x\text{ for all }h\in \mathfrak{h}\right\}.$$

$$c_{ij}:=\frac{2⟨{\alpha }_i,{\alpha }_j⟩}{⟨{\alpha }_i,{\alpha }_i⟩},(i,j\in \mathbf{I}),$$

$$s_i,(i\in \mathbf{I}).$$

$$T_i,(i\in \mathbf{I}).$$

$$\hat{q}:=q-q^{-1}.$$

$$[n{]}_{q_i}:=\frac{q_i^n-q_i^{-n}}{q_i-q_i^{-1}},[n{]}_{q_i}!:=[n{]}_{q_i}[n-1{]}_{q_i}\cdots [1{]}_{q_i}.$$

$$K_{\mu },E_i,F_i,(\mu \in Q,i\in \mathbf{I}).$$

$${\mathcal{U}}_q(\mathfrak{g})=\underset{\lambda \in Q}{⨁}{\mathcal{U}}_q(\mathfrak{g}{)}_{\lambda },{\mathcal{U}}_q(\mathfrak{g}{)}_{\mu }:=\left\{u\in {\mathcal{U}}_q(\mathfrak{g}):K_{\lambda }u=q^{⟨\lambda ,\mu ⟩}u{K}_{\lambda }\text{ for all }\lambda \in Q\right\}.$$

$$E_i\in {\mathcal{U}}_q(\mathfrak{g}{)}_{{\alpha }_i},F_i\in {\mathcal{U}}_q(\mathfrak{g}{)}_{-{\alpha }_i},K_{\mu }\in {\mathcal{U}}_q(\mathfrak{g}{)}_0,(i\in \mathbf{I},\mu \in Q).$$

$$[x,y]:=xy-q^{⟨\mu ,\eta ⟩}yx,x\in {\mathcal{U}}_q(\mathfrak{g}{)}_{\mu },y\in {\mathcal{U}}_q(\mathfrak{g}{)}_{\eta },$$

$$\left({\mathrm{ad}}_{q}x\right)(y)=[x,y],$$

$$\begin{array}{rl}& K_0=1,K_{\mu }{K}_{\lambda }=K_{\lambda +\mu },\\ & K_{\mu }{E}_i=q^{⟨\mu ,{\alpha }_i⟩}{E}_i{K}_{\mu },K_{\mu }{F}_i=q^{-⟨\mu ,{\alpha }_i⟩}{F}_i{K}_{\mu },\\ & E_i{F}_j-F_j{E}_i={\delta }_{ij}\frac{K_{{\alpha }_i}-K_{-{\alpha }_i}}{q_i-q_i^{-1}}\end{array}$$

$$\begin{array}{rl}& ({\mathrm{ad}}_q{E}_i{)}^{1-c_{ij}}(E_j)=0,(\text{for all i=j}),\\ & ({\mathrm{ad}}_q{F}_i{)}^{1-c_{ij}}(F_j)=0,(\text{for all i=j}).\end{array}$$

$${\mathcal{U}}_q(\mathfrak{g})\cong {\mathcal{U}}_q({\mathfrak{n}}^{-})\otimes {\mathcal{U}}_q(\mathfrak{h})\otimes {\mathcal{U}}_q({\mathfrak{n}}^{+}),$$

$${\mathrm{deg}}_Q(u)=\mu ,{\mathrm{deg}}_{\mathbb{Z}}(u)=n,$$

$$\begin{array}{rl}& T_i(K_{\mu })=K_{s_i(\mu )},\\ & T_i(E_j)=\{\begin{array}{ll}-F_i{K}_{{\alpha }_i},& (i=j),\\ {\left({\mathrm{ad}}_q{E}_i\right)}^{(-c_{ij})}(E_j),& (i\ne j),\end{array}\\ & T_i(F_j)=\{\begin{array}{ll}-K_{-{\alpha }_i}{E}_i,& (i=j),\\ {\left(-q_i\right)}^{-c_{ij}}{\left({\mathrm{ad}}_q{F}_i\right)}^{(-c_{ij})}(F_j),& (i\ne j),\end{array}\end{array}$$

$${\left({\mathrm{ad}}_q{E}_i\right)}^{(n)}:=\frac{{\left({\mathrm{ad}}_q{E}_i\right)}^n}{[n{]}_{q_i}!},{\left({\mathrm{ad}}_q{F}_i\right)}^{(n)}:=\frac{{\left({\mathrm{ad}}_q{F}_i\right)}^n}{[n{]}_{q_i}!}.$$

$$T_w=T_{i_1}{T}_{i_2}\cdots T_{i_N}.$$

$${\mathfrak{p}}_J={\mathfrak{l}}_J⋉{\mathfrak{n}}_J,$$

$$w_J:=w_o^J{w}_o\in W.$$

$$w_J=s_{i_1}\cdot s_{i_2}\cdots s_{i_N}\in W,$$

$${\beta }_1={\alpha }_{i_1},{\beta }_2=s_{i_1}{\alpha }_{i_2},...,{\beta }_N=s_{i_1}\cdots s_{i_{N-1}}{\alpha }_{i_N},$$

$$X_{{\beta }_1}=E_{i_1},X_{{\beta }_2}=T_{s_{i_1}}{E}_{i_2},...,X_{{\beta }_N}=T_{s_{i_1}}\cdots T_{s_{i_{N-1}}}{E}_{i_N}.$$

$${\Delta }_w:=\left\{{\beta }_1,\dots ,{\beta }_N\right\}.$$

$${\mathcal{U}}_q({\mathfrak{n}}_J):=⟨X_{{\beta }_1},\dots X_{{\beta }_N}⟩\subseteq {\mathcal{U}}_q({\mathfrak{n}}^{+}),$$

Full text

We focus primarily on those cases when the underlying Lie algebra $\mathfrak{g}$ is finite dimensional and simple with rank $r>1$, and $w$ is a parabolic element of the Weyl group, say $w=w_{o}^{J}w_{o}$, for some nonempty subset $J$ of simple roots. Here, ${\mathcal{U}}_{q}^{\pm}[w]$ is a deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of $\mathfrak{g}$. In this setting we conjecture that, with the exception of two specific low rank cases, the automorphism group of ${\mathcal{U}}_{q}^{\pm}[w]$ is the semidirect product of an algebraic torus of rank $r$ with the group of Dynkin diagram symmetries that preserve $J$. This conjecture is a more general form of the Launois-Lenagan [23] and Andruskiewitsch-Dumas [2] conjectures regarding the automorphism groups of the algebras of quantum matrices and the algebras ${\mathcal{U}}_{q}^{+}(\mathfrak{g})$, respectively. We completely determine the automorphism group in several instances, including all cases when $\mathfrak{g}$ is of type $F_{4}$ or $G_{2}$, as well as those cases when the quantum Schubert cell algebras are the algebras of quantum symmetric matrices.

The authors were supported by NSF grant DMS-1900823.

1. Introduction and summary of the results

Quantum Schubert cell algebras ${\mathcal{U}}_{q}^{\pm}[w]$ were introduced by De Concini, Kac, Procesi [7] and Lusztig [26]. They are a family of subalgebras of the Drinfeld-Jimbo quantized enveloping algebra ${\mathcal{U}}_{q}(\mathfrak{g})$ indexed by the elements $w$ of the Weyl group, and have appeared in several contexts, including ring theory [28, 32], crystal basis theory [22, 27], and cluster algebras [11, 13]. Several important cases arise when the Weyl group element $w$ is a parabolic element, say $w_{o}^{J}w_{o}$, for some nonempty subset $J$ of simple roots. Here, the corresponding algebra ${\mathcal{U}}_{q}^{\pm}[w]$ can be viewed as a deformation of the universal enveloping algebra of the nilradical $\mathfrak{n}_{J}$ of a parabolic subalgebra of $\mathfrak{g}$. In such a setting, we denote the quantum Schubert cell algebra by ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ and we refer to it as a quantized nilradical for short.

Throughout, the underlying base field for all algebras will be denoted by $\mathbb{K}$. We do not need to assume that $\mathbb{K}$ is algebraically closed or that it is of characteristic zero. The role of the Lie algebra $\mathfrak{g}$ in defining the $\mathbb{K}$-algebras ${\mathcal{U}}_{q}^{\pm}[w]$, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$, and ${\mathcal{U}}_{q}\left(\mathfrak{g}\right)$ can be viewed as purely symbolic. We will denote the multiplicative group of nonunits by $\mathbb{K}^{\times}$.

We will turn our attention towards studying the automorphisms of these algebras. We assume that the deformation parameter $q\in\mathbb{K}^{\times}$ is not a root of unity. Basically, there is a dichotomy in the structure of quantum algebras depending on whether or not $q$ is a root of unity. When $q$ is a root of unity, these algebras more closely resemble deformations of modular Lie algebras (Lie algebras over fields of positive characteristic). These algebras have large centers, and are therefore closer to being commutative. This, in effect, gives less control over automorphisms. In such situations, various types of noncommutative discriminants [4, 5, 6] have been developed as tools to study automorphisms (see e.g. [9, 10]).

Automorphism groups of quantized nilradicals (when $q$ is not a root of unity) have already been studied in several cases. For instance, when the underlying Lie algebra is $\mathfrak{sl}(n)$ and $J$ is a singleton, say $J=\left\{\alpha_{k}\right\}$, the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to the algebra of quantum $k\times(n-k)$ matrices. Launois and Lenagan prove in [23] that the automorphism group is $\left(\mathbb{K}^{\times}\right)^{n-1}$ whenever $k\neq n-k$ (and when $(n,k)\not\in\left\{(4,1),(4,3)\right\}$) by using certain properties of height one prime ideals. These techniques do not apply when $k=n-k$, yet they conjecture that the automorphism group in this remaining case is $\left(\mathbb{K}^{\times}\right)^{n-1}\rtimes\mathbb{Z}_{2}$. Their conjecture was already known to be true in the $2\times 2$ case by the work of Alev and Chamarie [1]. Launois and Lenagan later proved their conjecture for the $3\times 3$ case in [24]. Finally, the Launois-Lenagan conjecture was proved in the remaining cases by Yakimov in [35].

An interesting phenomenon regarding automorphisms arises when $k=1$ or $n-k=1$. In this setting ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ is isomorphic to $(n-1)$-dimensional quantum affine space ${\mathcal{A}}_{q}\left(\mathbb{K}^{n-1}\right)$. As an algebra, ${\mathcal{A}}_{q}\left(\mathbb{K}^{n-1}\right)$ is generated by elements $x_{1},\dots,x_{n-1}$ and has defining relations $x_{i}x_{j}=qx_{j}x_{i}$ whenever $i<j$. In [1], Alev and Chamarie studied automorphisms of several types of noncommutative algebras, including multiparameter and uniparameter quantum affine space. Their work predates the Launois-Lenagan conjecture. Interestingly, automorphisms of ${\mathcal{A}}_{q}\left(\mathbb{K}^{n-1}\right)$ send each generator $x_{i}$ to a scalar multiple of itself whenever $n\neq 4$. Alev and Chamarie proved in [1, Theorem 1.4.6] that every automorphism $\phi$ of ${\mathcal{A}}_{q}\left(\mathbb{K}^{3}\right)$ has the form

[TABLE]

where $a_{1},a_{2},a_{3}\in\mathbb{K}^{\times}$ and $b\in\mathbb{K}$. Hence, the automorphism group of ${\mathcal{A}}_{q}(\mathbb{K}^{3})$ is isomorphic to the semidirect product $\left(\mathbb{K}^{\times}\right)^{3}\rtimes\mathbb{K}$. On the other hand, if $n\neq 4$, they proved that the automorphism group of ${\mathcal{A}}_{q}(\mathbb{K}^{n-1})$ is isomorphic to $\left(\mathbb{K}^{\times}\right)^{n-1}$. Here, every automorphism sends $x_{i}$ to a nonzero multiple of itself.

Automorphism groups of quantized nilradicals have also been determined in all cases when $J$ is chosen to be the full set of simple roots. In particular, we assume now that $\mathfrak{g}$ is an arbitrarily chosen finite dimensional complex simple Lie algebra with $\operatorname{rank}(\mathfrak{g})=r>1$, and $J$ is the full set of simple roots. Here, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ is the entire positive part of ${\mathcal{U}}_{q}\left(\mathfrak{g}\right)$. The Chevalley generators $E_{1},\dots,E_{r}$ generate ${\mathcal{U}}_{q}^{+}\left(\mathfrak{g}\right)$ as an algebra and satisfy the $q$-Serre relations. With this, it is not too difficult to observe that for every $r$-tuple $(a_{1},\dots,a_{r})\in\left(\mathbb{K}^{\times}\right)^{r}$, there is an algebra automorphism $\phi$ of ${\mathcal{U}}_{q}^{+}\left(\mathfrak{g}\right)$ such that $\phi(E_{i})=a_{i}E_{i}$ ($i=1,\dots,r$). Furthermore, for every symmetry $\psi$ of the underlying Dynkin diagram, there is an algebra automorphism of ${\mathcal{U}}_{q}^{+}\left(\mathfrak{g}\right)$ given by the rule $E_{i}\mapsto E_{\psi(i)}$. Andruskiewitsch and Dumas [2] conjectured that the automorphism group of ${\mathcal{U}}_{q}^{+}\left(\mathfrak{g}\right)$ is generated by only these types of automorphisms. That is to say, they conjectured that

[TABLE]

where $\operatorname{Dynkin-Aut}\left(\mathfrak{g}\right)$ is the automorphism group of the Dynkin diagram of $\mathfrak{g}$. Yakimov proved this conjecture in [34] using a rigidity result involving quantum tori.

In describing the automorphism group ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ (for arbitrarily chosen $\mathfrak{g}$ and $J$), we need to introduce the subgroup of Dynkin diagram symmetries that fixes $J$,

[TABLE]

We conjecture the following result regarding the automorphism groups of quantized nilradicals.

Conjecture 1.1**.**

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra with $\operatorname{rank}(\mathfrak{g})>1$. Suppose $J$ is a nonempty subset of simple roots, and let ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ be the corresponding quantized nilradical. Then

[TABLE]

provided ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\not\cong\mathcal{A}_{q}(\mathbb{K}^{3})$.

We remark that ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ is isomorphic to ${\mathcal{A}}_{q}(\mathbb{K}^{3})$ in only two situations: (1) $\mathfrak{g}=\mathfrak{sl}(4)$ and $J=\left\{\alpha_{1}\right\}$, or (2) $\mathfrak{g}=\mathfrak{sl}(4)$ and $J=\left\{\alpha_{3}\right\}$.

Conjecture 1.1 above has been resolved in several cases. As mentioned above, the proof of the Launois-Lenagan conjecture covers the situation when $\mathfrak{g}$ is of type $A_{n}$ and $J$ is a singleton, whereas the Andruskiewitsch-Dumas conjecture handles the case when $\mathfrak{g}$ is arbitrary and $J$ is the full set of simple roots. We prove Conjecture 1.1 in some other situations, including when the underlying Lie algebra $\mathfrak{g}$ is of type $F_{4}$ or $G_{2}$.

Theorem 1.2**.**

If $\mathfrak{g}$ is the Lie algebra of type $F_{4}$ and $J$ is a nonempty subset of simple roots of $\mathfrak{g}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{4}$.

Theorem 1.3**.**

If $\mathfrak{g}$ is the Lie algebra of type $G_{2}$ and $J$ is a nonempty subset of simple roots of $\mathfrak{g}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{2}$.

We also develop some general theorems (Theorems 4.1, 4.2, 4.3, 4.4, 4.5) regarding automorphisms of quantum Schubert cell algebras that can be applied to help determine the automorphism groups of several other quantized nilradicals. More generally, quantum Schubert cell algebras belong to a larger family of algebras called Cauchon-Goodearl-Letzter (CGL) extensions, which originated in the works [3, 12]. Some general techniques have been developed in [14] to study automorphisms of CGL extensions. These techniques utilize properties of some key subalgebras of a CGL extension $R$, namely the normal subalgebra ${\mathcal{N}}(R)$ (the subalgebra generated by the normal elements), and the core ${\mathcal{C}}(R)$. Basically, the larger the core ${\mathcal{C}}(R)$, the more control one has over automorphisms [15, Theorem 4.2]. One has the most control over automorphisms when the core coincides with the entire algebra. Most quantized nilradicals appear to have this property.

Several other instances of quantized nilradicals appear in the literature, particularly when the nilradical $\mathfrak{n}_{J}$ is abelian. For example, when the underlying Lie algebra $\mathfrak{g}$ is of type $C_{n}$ and $J=\left\{\alpha_{n}\right\}$, the corresponding quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the algebra of quantum $n\times n$ symmetric matrices [21, 30]. When $\mathfrak{g}$ is the Lie algebra of type $D_{n}$ and $J=\left\{\alpha_{n-1}\right\}$ or $J=\left\{\alpha_{n}\right\}$, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the algebra of quantum antisymmetric matrices [31]. If $\mathfrak{g}$ is of type $B_{n}$ and $J=\left\{\alpha_{1}\right\}$, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the odd-dimensional quantum Euclidean space, which was introduced by Faddeev, Reshetikhin, and Takhtadzhyan [8, Definition 12]. Simplified relations for this algebra appear in [29, Sections 2.1-2.2]. If $\mathfrak{g}$ is of type $D_{n}$ and $J=\left\{\alpha_{1}\right\}$, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the even-dimensional quantum Euclidean space [8, 29]. The automorphism groups of even and odd-dimensional quantum Euclidean space are already known to satisfy Conjecture 1.1 [14, Example 4. 10]. We prove Conjecture 1.1 holds when ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the algebra of quantum symmetric matrices.

Theorem 1.4**.**

If $\mathfrak{g}$ is the Lie algebra of type $C_{n}$ with $n>1$ and $J=\left\{\alpha_{n}\right\}$ (i.e. ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the algebra of $n\times n$ quantum symmetric matrices), then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{n}$.

Other examples of quantized nilradicals ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ have been studied for cases when $\mathfrak{n}_{J}$ is non-abelian. For instance, the quantized nilradicals when $\mathfrak{g}$ is of type $A_{n}$ and $J$ is an arbitrary set of simple roots were studied in [20], where it was shown that ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to an algebra of coinvariants. With this, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ can be viewed as a deformation of the coordinate ring of a unipotent subgroup of a parabolic subgroup of $SL(n+1)$.

Each algebra ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ can be equipped with a $\mathbb{N}$-grading such that, with respect to this grading, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is connected and locally finite. We apply the results developed in Theorems 4.1, 4.2, 4.3, 4.4, 4.5 to illustrate that, for certain cases of $\mathfrak{g}$ and $J$, every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ that preserves the $\mathbb{N}$-grading acts diagonally on the graded component of degree one (see Proposition 4.7).

We choose the first case listed in Proposition 4.7, namely when $\mathfrak{g}$ is of type $B_{6}$ and $J=\left\{\alpha_{2},\alpha_{5}\right\}$, and completely determine the automorphism group of the corresponding quantized nilradical. We show here that every automorphism preserves the $\mathbb{N}$-grading by applying the results of [15, Theorem 4.2] involving the core of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. The same steps can be applied to other cases listed in Proposition 4.7.

Theorem 1.5**.**

If $\mathfrak{g}$ is the Lie algebra of type $B_{6}$ and $J=\left\{\alpha_{2},\alpha_{5}\right\}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{6}$.

Ideally, we would like to eventually develop a theory sufficient to completely determine the automorphism group of ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ in all cases. A more general endeavor is to develop a theory sufficient to describe the automorphism groups of quantum Schubert cell algebras ${\mathcal{U}}_{q}^{\pm}[w]$. Interestingly, Ceken, Palmieri, Wang, and Zhang [4] describe a family of algebras such that the automorphism group of each algebra in this family is isomorphic to the semidirect product of an algebraic torus and a finite group. While quantum Schubert cells don’t belong to this family of algebras, in many instances their automorphism groups seem to have this form. In a related work, one could attempt to find necessary and sufficient conditions on $w$ so that $\operatorname{Aut}\left({\mathcal{U}}_{q}^{\pm}[w]\right)$ is isomorphic to $\left(\mathbb{K}^{\times}\right)^{n}\rtimes G$ for some natural number $n\in\mathbb{N}$ and finite group $G$.

2. The algebra $\mathbf{{\mathcal{U}}_{q}(\mathfrak{g})}$

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra of rank $r$. Define the index set $\mathbf{I}:=\left\{1,2,\dots r\right\}$, and let $\Pi=\{\alpha_{i}\}_{i\in\mathbf{I}}$ be a set of simple roots of $\mathfrak{g}$ with respect to a fixed Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ such that the labelling of the simple roots agrees with the labelling in [17, Section 12.1].

The root system of $\mathfrak{g}$ will be denoted by $\Delta$, and the sets of positive and negative roots will be denoted by $\Delta_{+}$ and $\Delta_{-}$, respectively. The corresponding triangular decomposition of $\mathfrak{g}$ will be denoted by

[TABLE]

where

[TABLE]

As usual, let $Q=\oplus_{i\in\mathbf{I}}\mathbb{Z}\alpha_{i}$ be the root lattice of $\mathfrak{g}$, and let $\langle\,\,,\,\rangle:Q\times Q\to\mathbb{Z}$ be a symmetric nondegenerate ad-invariant $\mathbb{Z}$-bilinear form, normalized so that $\langle\alpha,\alpha\rangle=2$ for short roots $\alpha$. Define

[TABLE]

and let $(c_{ij})_{i,j\in\mathbf{I}}$ be the associated Cartan matrix of $\mathfrak{g}$. We will denote the simple reflections in the Weyl group $W$ of $\mathfrak{g}$ by

[TABLE]

The corresponding generators of the braid group ${\mathcal{B}}_{\mathfrak{g}}$ of $\mathfrak{g}$ will be denoted by

[TABLE]

Let $q\in\mathbb{K}$ be nonzero and not a root of unity, and let $q_{i}=q^{\langle\alpha_{i},\alpha_{i}\rangle/2}$ for $i\in\mathbf{I}$. As usual, define

[TABLE]

For a natural number $n\in\mathbb{N}$, define

[TABLE]

The quantized universal enveloping algebra ${\mathcal{U}}_{q}(\mathfrak{g})$ is an associative $\mathbb{K}$-algebra with standard Chevalley generators

[TABLE]

There is a standard $Q$-gradation on the algebra ${\mathcal{U}}_{q}(\mathfrak{g})$,

[TABLE]

With respect to this grading, the Chevalley generators are homogeneous elements. In particular,

[TABLE]

We will state the defining relations of ${\mathcal{U}}_{q}(\mathfrak{g})$, but first we find it convenient to introduce the abbreviation

[TABLE]

for $q$-commutators. We will adopt this notation throughout. Next, for every homogeneous $x\in{\mathcal{U}}_{q}(\mathfrak{g})_{\mu}$, we define the linear operator $\operatorname{ad}_{q}x:{\mathcal{U}}_{q}(\mathfrak{g})\to{\mathcal{U}}_{q}(\mathfrak{g})$ by the condition that

[TABLE]

for every homogeneous element $y\in{\mathcal{U}}_{q}(\mathfrak{g})$. With this, the defining relations of ${\mathcal{U}}_{q}(\mathfrak{g})$ are

[TABLE]

together with the $q$-Serre relations

[TABLE]

The algebra ${\mathcal{U}}_{q}(\mathfrak{g})$ has a triangular decomposition,

[TABLE]

where ${\mathcal{U}}_{q}(\mathfrak{n}^{-})$, ${\mathcal{U}}_{q}(\mathfrak{h})$, and ${\mathcal{U}}_{q}(\mathfrak{n}^{+})$ are the subalgebras of ${\mathcal{U}}_{q}(\mathfrak{g})$ generated by the $F$’s, $K$’s, and $E$’s respectively.

2.1. A $\mathbb{Z}$-grading on ${\mathcal{U}}_{q}(\mathfrak{g})$

The fundamental coweights of $\mathfrak{g}$ will be denoted by $\varpi_{i}^{\vee}$ ($i\in\mathbf{I}$). They are determined by the conditions $\langle\varpi_{i}^{\vee},\alpha_{j}\rangle=\delta_{ij}$ for all $i,j\in\mathbf{I}$. We will let $P^{\vee}=\oplus_{i\in\mathbf{I}}\mathbb{Z}\varpi_{i}^{\vee}$ be the coweight lattice of $\mathfrak{g}$. For every integral coweight $\lambda\in P^{\vee}$ (i.e. $\langle\lambda,\alpha_{i}\rangle\in\mathbb{Z}$ for every $i\in\mathbf{I}$), there is an associated $\mathbb{Z}$-grading on ${\mathcal{U}}_{q}(\mathfrak{g})$ given by assigning the degree $\langle\lambda,\mu\rangle$ to every $Q$-homogeneous element of degree $\mu$. To make the distinction between the $Q$-grading and the $\mathbb{Z}$-grading, we will write

[TABLE]

respectively, to mean that $u\in{\mathcal{U}}_{q}(\mathfrak{g})$ is a $Q$-homogeneous element of degree $\mu\in Q$ and a $\mathbb{Z}$-homogeneous element of degree $n$. Technically, $\operatorname{deg}_{\mathbb{Z}}$ depends on the coweight $\lambda$. However, we adopt the notation $\operatorname{deg}_{\mathbb{Z}}$ rather than $\operatorname{deg}_{\lambda}$ whenever the choice of coweight $\lambda$ is clear from the context.

2.2. Lusztig symmetries of $\mathbf{{\mathcal{U}}_{q}(\mathfrak{g})}$

In [26, Section 37.1.3], Lusztig defines an action of the braid group ${\mathcal{B}}_{\mathfrak{g}}$ via algebra automorphisms on ${\mathcal{U}}_{q}(\mathfrak{g})$. In fact, Lusztig defines the symmetries $T_{i,1}^{\prime}$, $T_{i,-1}^{\prime}$, $T_{i,1}^{\prime\prime}$, and $T_{i,-1}^{\prime\prime}$. By [26, Proposition 37.1.2], these are automorphisms of ${\mathcal{U}}_{q}(\mathfrak{g})$, while by [26, Theorem 39.4.3] they satisfy the braid relations. For short, we will adopt the abbreviation $T_{i}:=T^{\prime\prime}_{i,1}$. With this convention, Lusztig’s symmetries are given by the formulas

[TABLE]

where, for a nonnegative integer $n$,

[TABLE]

If $w\in W$ has a reduced expression $w=s_{i_{1}}\cdots s_{i_{N}}\in W$, we write

[TABLE]

A key property of the braid symmetries is given in the following proposition (see e.g. [19, Proposition 8.20]).

Proposition 2.1**.**

If $w\in W$ such that $w(\alpha_{i})=\alpha_{j}$, then $T_{w}(E_{i})=E_{j}$.

3. The quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$

For each nonempty set $J$ of simple roots, let $\mathfrak{p}_{J}$ be the parabolic subalgebra of $\mathfrak{g}$ obtained by deleting the roots in $J$. The Levi decomposition of $\mathfrak{p}_{J}$ will be denoted by

[TABLE]

where $\mathfrak{l}_{J}$ is the Levi subalgebra and $\mathfrak{n}_{J}$ is the nilradical.

Let $w_{o}$ denote the longest element of the Weyl group, and let $w_{o}^{J}\in W$ be the longest element in the subgroup $\langle s_{i}\mid i\not\in J\rangle\subseteq W$. Define

[TABLE]

For a reduced expression,

[TABLE]

where $N$ is the length of $w_{J}$, define the roots

[TABLE]

and root vectors

[TABLE]

We will denote the set of radical roots by

[TABLE]

These roots are precisely the positive roots that get sent to negative roots by the action of $w_{J}^{-1}$. An analogous construction can be applied to obtain a list of negative roots by replacing the $E$’s in (3.1) above with $F$’s. The subalgebra of ${\mathcal{U}}_{q}(\mathfrak{g})$ generated by the root vectors $X_{\beta_{1}},\dots,X_{\beta_{N}}$ is contained in the positive part ${\mathcal{U}}_{q}(\mathfrak{n}^{+})$ (see e.g. [19, Proposition 8.20]). This subalgebra will be denoted by ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$,

[TABLE]

and we refer to it as the quantized nilradical of $\mathfrak{p}_{J}$, or quantized nilradical for short. The subalgebra of ${\mathcal{U}}_{q}(\mathfrak{g})$ generated by the negative root vectors

[TABLE]

is isomorphic to ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$.

Quantized nilradicals belong to a larger class of algebras called quantum Schubert cell algebras, which are indexed by elements $w$ in the Weyl group. More generally, given a reduced expression of a Weyl group element $w$, the corresponding quantum Schubert cell algebra ${\mathcal{U}}_{q}^{+}[w]$ can be constructed in the same way as ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ by replacing a reduced expression for $w_{J}$ above with a reduced expression for $w$. De Concini, Kac, and Procesi [7, Proposition 2.2] proved that the algebra ${\mathcal{U}}_{q}^{+}[w]$ does not depend on the reduced expression for $w$. Furthermore, every quantum Schubert cell ${\mathcal{U}}_{q}^{+}[w]$ has a PBW basis

[TABLE]

of standard monomials, and they have presentations as iterated Ore extensions,

[TABLE]

For $1<i<j\leq N$, define the interval subalgebra

[TABLE]

as the subalgebra generated by $X_{i},X_{i+1},\dots,X_{j}$. Standard monomials

[TABLE]

form a basis of $\mathbf{U}_{[i,j]}$. The Levendorskii-Soibelmann straightening rule [25, Prop. 5.5.2] tells us that for all $1\leq i<j\leq N$,

[TABLE]

(recall (2.1)). As a consequence of the straightening rule, we have the following corollary.

Corollary 3.1**.**

If $1\leq i<j\leq N$ and there fails to exist a nonnegative integral combination of roots in $\left\{\beta_{i+1},\dots,\beta_{j-1}\right\}$ that sum to $\beta_{i}+\beta_{j}$, then $[X_{\beta_{i}},X_{\beta_{j}}]=0$.

Furthermore, every quantum Schubert cell algebra ${\mathcal{U}}_{q}^{+}[w]$ has a quantum cluster algebra structure (provided the deformation parameter $q$ satisfies some minor conditions) [13] with a set of frozen variables

[TABLE]

The normal subalgebra of ${\mathcal{U}}_{q}^{+}[w]$ (the subalgebra generated by the normal elements) is generated by the frozen variables [15, Proposition 2.7].

4. Automorphisms of quantum Schubert cells

In this section, we assume that $R$ is a quantum Schubert cell algebra, say $R={\mathcal{U}}_{q}^{+}[w]\subseteq{\mathcal{U}}_{q}(\mathfrak{g})$. Fix a reduced expression

[TABLE]

and let

[TABLE]

be the corresponding Lusztig root vectors. Recall that $R$ can be written as an iterated Ore extension

[TABLE]

Observe that the algebraic torus ${\mathcal{H}}=\left(\mathbb{K}^{\times}\right)^{r}$ of rank $r=\operatorname{rank}(\mathfrak{g})$ acts canonically on ${\mathcal{U}}_{q}(\mathfrak{g})$ via algebra automorphisms. An element $h=(h_{1},\dots,h_{r})\in{\mathcal{H}}$ acts by the rule

[TABLE]

for all $1\leq i\leq r$ and $\mu\in Q$. This action is preserved by $R$, and each Lusztig root vector $X_{\beta_{i}}$ is an ${\mathcal{H}}$-eigenvector. In fact, every $Q$-homogeneous element is an ${\mathcal{H}}$-eigenvector. Furthermore, the iterated Ore extension presentation in (4.1) is a symmetric Cauchon-Goodearl-Letzter (CGL) extension presentation for $R$ (see e.g. [13, Theorem 9.1.b]).

4.1. The function $\mathbf{\eta}$

Following [15], to every iterated Ore extension presentation $R$, as in (4.1) above, we define the rank of $R$

[TABLE]

Let $S$ be a set of cardinality $\operatorname{rank}(R)$, and let $\eta:\left\{1,\dots,N\right\}\to S$ be a function such that $\eta(\left\{k\in\left\{1,\dots,N\right\}:\delta_{k}=0\right\})=S$. That is, we assign to each trivial derivation $\delta_{k}$ a unique element in $S$. We require also that, for every $k\in\left\{1,\dots,N\right\}$ such that $\delta_{k}\neq 0$,

[TABLE]

The existence of such a function $\eta$ was proved in [15, Theorem 4.3], and it plays a key role in determining the homogeneous prime elements for any CGL extension $R$. When $R={\mathcal{U}}_{q}^{+}[w]$, the rank of $R$ agrees with the cardinality of the support of $w$,

[TABLE]

In this setting, the function $\eta:\left\{1,\dots,N\right\}\to\operatorname{supp}(w)$ can be defined by the rule

[TABLE]

(see e.g. [13, Theorem 9.5]).

4.2. The core of $\mathbf{R}$

Following [14, Section 4.1], define $P_{x}(R)$ to be the set of those $i\in\left\{1,\dots,N\right\}$ such that $X_{\beta_{i}}$ is prime. By [14, Proposition 2.6],

[TABLE]

For $1\leq j<k\leq N$, the element $Q_{jk}:=[X_{\beta_{j}},X_{\beta_{k}}]$ can be written uniquely as a linear combination of monomials $X_{\beta_{j+1}}^{m_{j+1}}\cdots X_{\beta_{k-1}}^{m_{k-1}}$. Let $F_{x}(R)$ be the set of those $i\in P_{x}(R)$ such that $X_{\beta_{i}}$ does not appear in any $Q_{jk}$. More precisely, no monomial in $Q_{jk}$ with a nonzero coefficient contains a positive power of $X_{\beta_{i}}$. Define $C_{x}(R):=\left\{1,\dots,N\right\}\backslash F_{x}(R)$. The core of $R$, denoted by ${\mathcal{C}}(R)$, is defined as the subalgebra generated by the $X_{\beta_{i}}$’s with $i\in C_{x}(R)$,

[TABLE]

4.3. Diagonal and graded automorphisms

An algebra automorphism $\phi:R\to R$ that sends every Lusztig root vector $X_{\beta_{i}}$ ($1\leq i\leq N$) to a scalar multiple of itself will be called a diagonal automorphism. This notion is dependent upon the choice of reduced expression for $w$. Hence, whenever we refer to automorphisms of this type, we have a fixed reduced expression for $w$ in mind. The set of diagonal automorphisms is a subgroup of the automorphism group $\operatorname{Aut}(R)$ of $R$. We will denote this subgroup by $\operatorname{Diag-Aut}(R)$. Thus, for an algebra automorphism $\phi:R\to R$,

[TABLE]

From Section 2.1, recall that every coweight $\lambda\in P^{\vee}$ induces a $\mathbb{Z}$-grading on ${\mathcal{U}}_{q}(\mathfrak{g})$. With this, the subalgebra $R={\mathcal{U}}_{q}^{+}[w]\subseteq{\mathcal{U}}_{q}(\mathfrak{g})$ inherits this grading,

[TABLE]

We assume throughout that $\lambda\in P^{\vee}$ is chosen so that the induced grading satisfies the following conditions:

1. (1)

$R=R_{0}\oplus R_{1}\oplus R_{2}\oplus\cdots$ (that is to say, $R_{d}=0$ whenever $d<0$), 2. (2)

$R_{d}$ is finite dimensional for every $d\geq 0$ (i.e. $R$ is locally finite), 3. (3)

$R_{0}=\mathbb{K}$ (i.e. $R$ is connected), and 4. (4)

$R$ is generated, as an algebra, by $R_{1}$.

These conditions mimic the standard grading on a commutative polynomial ring $\mathbb{K}[z_{1},\dots,z_{N}]$, where each variable $z_{i}$ is assigned degree $1$. It is always possible to choose $\lambda$ so that the first three conditions above are satisfied. For example, $\lambda=\sum_{i\in\mathbf{I}}\varpi_{i}^{\vee}$ is one such choice. However, it is not always possible to select $\lambda$ such that all four conditions are met. To give one example, it is not too difficult to verify that such a $\lambda$ fails to exist for the case when the underlying Lie algebra is of type $G_{2}$ and $w=s_{2}s_{1}s_{2}$.

An algebra automorphism $\phi:R\to R$ is a graded algebra automorphism if it respects the $\mathbb{Z}_{\geq 0}$-grading. That is to say,

[TABLE]

for all $d\geq 0$. The set of graded automorphisms is a subgroup of the automorphism group of $R$. We denote the subgroup of graded automorphisms by $\operatorname{Gr-Aut}(R)$. Observe we have a chain of subgroups,

[TABLE]

Using (4.2) and (4.3), one can easily determine the set $P_{x}(R)$ from the reduced expression $w=s_{1}\cdots s_{N}$. In many cases $P_{x}(R)$ is empty (and ${\mathcal{C}}(R)=R$), and in such situations we have the most control over the automorphisms of $R$ (see e.g. [14]). The following theorem describes sufficient conditions on $R$ to conclude that every automorphism of $R$ is graded.

Theorem 4.1**.**

Suppose $R={\mathcal{U}}_{q}^{+}[w]$ is a quantum Schubert cell algebra with Lusztig root vectors $X_{\beta_{1}},\dots,X_{\beta_{N}}$. Suppose ${\mathcal{C}}(R)=R$. Suppose also that $R$ is connected graded, locally finite, and generated by $R_{1}$. For every radical root $\beta_{i}\in\Delta_{w}$ with $X_{\beta_{i}}\in R_{1}$, suppose there exists $\beta_{j}\in\Delta_{w}$ such that $X_{\beta_{i}}X_{\beta_{j}}=\kappa X_{\beta_{j}}X_{\beta_{i}}$ for some scalar $\kappa\neq 1$. Then every algebra automorphism of $R$ is a graded automorphism. In other words,

[TABLE]

Proof.

It was shown in [14, Theorem 4.2] that if $R$ is a symmetric saturated CGL extension which is a connected graded algebra, then every unipotent automorphism restricted to ${\mathcal{C}}(R)$ is the identity. Since ${\mathcal{C}}(R)=R$, then the identity is the only unipotent automorphism of $R$. As a consequence of [14, Lemma 4.7] every automorphism $\phi$ is graded provided $\phi(R_{d})\subseteq\oplus_{j\geq d}R_{j}$ for all $d\geq 0$. However, this condition was established in [23, Proposition 4.2].

∎

4.4. The normal subalgebra $\mathbf{{\mathcal{N}}(R)}$ and the sets

$\mathbf{C_{d}^{m}}$ and $\mathbf{\gamma_{d,\ell}^{m}}$

Following [15], let ${\mathcal{N}}(R)$ be the normal subalgebra of $R$. It is the subalgebra generated by the normal elements of $R$. By [15, Theorem 4.3], ${\mathcal{N}}(R)$ is a generated by a finite set of $Q$-homogeneous prime elements

[TABLE]

We remark here that the element $\Theta_{i}\in R$ is written as $\Delta_{\varpi_{i},w\varpi_{i}}$ in [13, Section 9.4]. We have the following commutation relations,

[TABLE]

(see e.g. [33, Eq. 3.30] and [13, Eq. 9.23]).

For $d\in\mathbb{Z}_{\geq 0}$, $m\in\mathbb{Z}$, and $x\in R_{1}$, define

[TABLE]

For $d,\ell\in\mathbb{Z}_{\geq 0}$ and $m\in\mathbb{Z}$, define the set

[TABLE]

The following proposition is an easy observation.

Proposition 4.2**.**

Suppose $\phi\in\operatorname{Gr-Aut}(R)$. Let $C_{d}^{m}$ and $\gamma_{d,\ell}^{m}$ be as defined in (4.5) and (4.6). Then each $C_{d}^{m}$ is a $\phi$-invariant subspace of $R_{1}$, and each $\gamma_{d,\ell}^{m}$ is a $\phi$-invariant subset of $R_{1}$.

The following theorem gives us sufficient conditions to determine when a standard generator $\Theta_{j}$ of ${\mathcal{N}}(R)$ gets sent to a nonzero scalar multiple of itself by a graded algebra automorphism $\phi:R\to R$.

Theorem 4.3**.**

Suppose $\phi\in\operatorname{Gr-Aut}(R)$. For every $i\in\operatorname{supp}(w)$, let $d_{i}$ be the degree of $\Theta_{i}$. That is, $\Theta_{i}\in{\mathcal{N}}(R)_{d_{i}}$. If, for some $j\in\operatorname{supp}(w)$, there fails to exist a nonnegative integral combination of numbers in $\left\{d_{i}:i\in\operatorname{supp}(w)\text{ and }i\neq j\right\}$ that sum to $d_{j}$, then

[TABLE]

Proof.

Ordered monomials in the $\Theta_{i}$’s form a basis (over $\mathbb{K}$) of the normal subalgebra ${\mathcal{N}}(R)$ [15, Theorem 4.6]. As there fails to exist a nonnegative integral combination of numbers in $\left\{d_{i}:i\in\operatorname{supp}(w)\text{ and }i\neq j\right\}$ that sum to $d_{j}$, this implies that ${\mathcal{N}}(R)_{d_{j}}$ is a one-dimensional vector space over $\mathbb{K}$ spanned by the element $\Theta_{j}$. Since ${\mathcal{N}}(R)$ is an invariant subalgebra of $R$ under $\phi$, then we have $\phi({\mathcal{N}}(R)_{d_{j}})={\mathcal{N}}(R)_{d_{j}}$. Hence $\phi(\Theta_{j})\in\mathbb{K}^{\times}\Theta_{j}$.

∎

The following theorem gives sufficient conditions to determine when a Lusztig root vector $X_{\beta}$ gets sent to a nonzero scalar multiple of itself by a graded algebra automorphism $\phi:R\to R$.

Theorem 4.4**.**

Suppose $\phi\in\operatorname{Gr-Aut}(R)$ and $\phi(\Theta_{i})\in\mathbb{K}^{\times}\Theta_{i}$ for some $i\in\operatorname{supp}(w)$. Suppose also that there exists a radical root $\beta\in\Delta_{w}$ with $X_{\beta}\in R_{1}$ such that

[TABLE]

for every radical root $\beta^{\prime}\in\Delta_{w}\backslash\left\{\beta\right\}$ with $X_{\beta^{\prime}}\in R_{1}$, then

[TABLE]

Proof.

Suppose $x_{1},\dots,x_{n}$ is a list of the Lusztig root vectors in $R_{1}$. Without loss of generality, assume $x_{1}=X_{\beta}$. There are integers $d_{1},\dots,d_{n}$ which can be computed explicitly using (4.4) such that $x_{j}\Theta_{i}=q^{d_{j}}\Theta_{i}x_{j}$. The given hypotheses imply that $d_{1}$ is not equal to any number in $\left\{d_{2},\dots,d_{n}\right\}$.

The automorphism $\phi$ sends $x_{1}$ to a linear combination of $x_{1},\dots,x_{n}$, say $\phi(x_{1})=\sum c_{j}x_{j}$, ($c_{j}\in\mathbb{K}$). Applying $\phi$ to the relation $x_{1}\Theta_{i}=q^{d_{1}}\Theta_{i}x_{1}$ yields $\sum c_{j}x_{j}\Theta_{i}=\sum c_{j}q^{d_{1}}\Theta_{i}x_{j}=\sum c_{j}q^{d_{j}}\Theta_{i}x_{j}$. Thus, $\sum c_{j}(q^{d_{1}}-q^{d_{j}})\Theta_{i}x_{j}=0$. The elements $\Theta_{i}x_{1},\dots,\Theta_{i}x_{n}$ are $Q$-homogeneous and have distinct degrees with respect to the $Q$-gradation. Hence, each of the coefficients $c_{j}(q^{d_{1}}-q^{d_{j}})$ equals zero. Since $q$ is not a root of unity, $c_{2}=\cdots=c_{n}=0$.

∎

Using the same techniques in the proof of Theorem 4.4 above, we have a more general result.

Theorem 4.5**.**

Suppose $\phi\in\operatorname{Gr-Aut}(R)$. Define the set

[TABLE]

Suppose that $\beta\in\Delta_{w}$ is a radical root with $X_{\beta}\in R_{1}$ and satisfies the condition that for every radical root $\beta^{\prime}\in\Delta_{w}\backslash\left\{\beta\right\}$ with $X_{\beta^{\prime}}\in R_{1}$, there exists $i\in S(\phi,w)$ such that $\langle\beta-\beta^{\prime},\left(1+w\right)\varpi_{i}\rangle\neq 0$, then

[TABLE]

By applying the theorems above, we can prove that Conjecture 1.1 holds, for example, when the underlying Lie algebra $\mathfrak{g}$ is of type $G_{2}$.

Theorem 4.6**.**

If $\mathfrak{g}$ is the Lie algebra of type $G_{2}$ and $J$ is a nonempty subset of simple roots of $\mathfrak{g}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{2}$.

Proof.

We consider the reduced expression $w_{o}=s_{1}s_{2}s_{1}s_{2}s_{1}s_{2}$ for the longest element of the Weyl group of $\mathfrak{g}$. The corresponding radical roots and root vectors associated to this reduced expression will be denoted by $\beta_{1},\dots,\beta_{6}$ and $x_{1},\dots,x_{6}$, respectively. In other words, we define $x_{i}:=X_{\beta_{i}}$ ($i=1,\dots,6$). The positive part ${\mathcal{U}}_{q}^{+}(\mathfrak{g})$ of ${\mathcal{U}}_{q}(\mathfrak{g})$ is generated by $x_{1},\dots,x_{6}$. The defining relations appear in the work of Hu and Wang [16], where they index the root vectors by Lyndon words. The correspondence between our notation and their notation is $x_{1}\leftrightarrow E_{1}$, $[3]_{q}!x_{2}\leftrightarrow E_{1112}$, $[2]_{q}x_{3}\leftrightarrow E_{112}$, $[3]_{q}!x_{4}\leftrightarrow E_{11212}$, $x_{5}\leftrightarrow E_{12}$, $x_{6}\leftrightarrow E_{2}$. From [16, Eqns. 2.2 - 2.7 and Lemma 3.1], the defining relations in ${\mathcal{U}}_{q}^{+}(\mathfrak{g})$ are

2. $x_{1}x_{2}=q^{3}x_{2}x_{1}$,
3. $x_{1}x_{3}=qx_{3}x_{1}+[3]_{q}x_{2}$,
4. $x_{1}x_{4}=x_{4}x_{1}+q\widehat{q}x_{3}^{2}$,
5. $x_{1}x_{5}=q^{-1}x_{5}x_{1}+[2]_{q}x_{3}$,
6. $x_{1}x_{6}=q^{-3}x_{6}x_{1}+x_{5}$,
7. $x_{2}x_{3}=q^{3}x_{3}x_{2}$,
8. $x_{2}x_{4}=q^{3}x_{4}x_{2}+\eta x_{3}^{3}$,
9. $x_{2}x_{5}=x_{5}x_{2}+q\widehat{q}x_{3}^{2}$,
10. $x_{2}x_{6}=q^{-3}x_{6}x_{2}+\widehat{q}x_{3}x_{5}+\zeta x_{4}$,
11. $x_{3}x_{4}=q^{3}x_{4}x_{3}$,
12. $x_{3}x_{5}=qx_{5}x_{3}+[3]_{q}x_{4}$,
13. $x_{3}x_{6}=x_{6}x_{3}+q\widehat{q}x_{5}^{2}$,
14. $x_{4}x_{5}=q^{3}x_{5}x_{4}$,
15. $x_{4}x_{6}=q^{3}x_{6}x_{4}+\eta x_{5}^{3}$
16. $x_{5}x_{6}=q^{3}x_{6}x_{5}$,

where $\zeta:=q^{-3}-q^{-1}-q\in\mathbb{K}$ and $\eta=q^{3}\frac{\widehat{q}^{2}}{[3]_{q}}\in\mathbb{K}$.

Observe that when $J$ is a singleton, the parabolic element $w_{o}^{J}w_{o}$ has a unique reduced expression, and it appears as a substring of the reduced expression for the longest element $w_{o}$. Hence, each quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to an interval subalgebra of ${\mathcal{U}}_{q}^{+}(\mathfrak{g})$. In particular, when $J=\left\{\alpha_{1}\right\}$, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to the subalgebra generated by $x_{1},\dots,x_{5}$, whereas if $J=\left\{\alpha_{2}\right\}$, the corresponding quantized nilradical is isomorphic to the subalgebra generated by $x_{2},\dots,x_{6}$. For convenience, we will identify each quantized nilradical with an appropriate interval subalgebra.

First consider the case when $J=\left\{\alpha_{1}\right\}$. Here we choose the coweight $\lambda=\varpi_{1}\in P^{\vee}$ to equip ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ with a $\mathbb{N}$-gradation. With respect to this grading, the degree one generators are $x_{1}$ and $x_{5}$. The defining relations verify that ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is generated as an algebra by its degree one elements. By using (4.3), we have ${\mathcal{C}}\left({\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\right)$ is generated by $x_{1}$, $x_{3}$, and $x_{5}$. Thus, ${\mathcal{C}}\left({\mathcal{U}}_{q}(\mathfrak{n}_{J})\right)={\mathcal{U}}_{q}(\mathfrak{n}_{J})$ and Theorem 4.1 be can applied to conclude that every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ preserves the $\mathbb{N}$-grading. The elements $\Theta_{1}$ and $\Theta_{2}$ of the normal subalgebra have degrees $4$ and $6$, respectively. Hence, by Theorem 4.3, every automorphism sends $\Theta_{1}$ and $\Theta_{2}$ to nonzero multiples of themselves. Finally, one can verify that Theorem 4.5 can be applied to conclude that every automorphism sends $x_{1}$ and $x_{5}$ to nonzero multiples of themselves. Therefore, every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is a diagonal automorphism. As a CGL extension, the algebra ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ has rank 2. Thus, by [15, Theorems 5.3 and 5.5], $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{2}$.

The case when $J=\left\{\alpha_{2}\right\}$ is treated similarly, except now we choose the coweight $\varpi_{2}\in P^{\vee}$ to equip the corresponding quantized nilradical with a $\mathbb{N}$-gradation. Here, we identify ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ with the subalgebra of ${\mathcal{U}}_{q}^{+}(\mathfrak{g})$ generated by $x_{2},\dots,x_{6}$. With this identification, the degree one root vectors of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ are $x_{2}$, $x_{3}$, $x_{5}$, and $x_{6}$. Again, we see that ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is generated as an algebra by its elements of degree one, and the core ${\mathcal{C}}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))$ coincides with ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. All of the hypotheses of Theorem 4.1 apply. Hence, every automorphism preserves the $\mathbb{N}$-grading. The elements $\Theta_{1}$ and $\Theta_{2}$ of the normal subalgebra have degrees $2$ and $4$, respectively, in this setting. Thus, by Theorem 4.3, every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ sends $\Theta_{1}$ to a nonzero multiple of itself. Finally, by applying Theorem 4.4, we can conclude that every automorphism sends each degree one root vector to a nonzero multiple of itself. Thus, every automorphism is a diagonal automorphism. Finally, [15, Theorems 5.3 and 5.5] imply that $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{2}$.

The case when $J=\left\{\alpha_{1},\alpha_{2}\right\}$ has already been established in [34].

∎

Theorems 4.4 and 4.5 can be applied in several other cases to conclude that every graded automorphism of a quantized nilradical sends each degree one generator to a multiple of itself. If it can also be established that some particular quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ satisfies all of the hypotheses of Theorem 4.1, then the results of Goodearl and Yakimov in [15, Theorems 5.3 and 5.5] can be applied to conclude that $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{\operatorname{rank}(\mathfrak{g})}$.

Proposition 4.7**.**

As above, let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra, and let $J$ be a nonempty set of simple roots. Choose the coweight $\lambda:=\sum_{j\in J}\varpi_{j}\in P^{\vee}$ to equip the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ with a $\mathbb{N}$-gradation. The following list is an exhaustive list of all cases with $\operatorname{rank}(\mathfrak{g})\leq 9$ such that Theorems 4.4 and 4.5 can be applied to conclude that every graded automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ sends each degree one Lusztig root vector $X_{\beta}$ to a multiple of itself.

1. (1)

$\mathfrak{g}$* is of type $B_{6}$ and $J$ is $\left\{\alpha_{2},\alpha_{5}\right\}$ or $\left\{\alpha_{2},\alpha_{4},\alpha_{5}\right\}$.* 2. (2)

$\mathfrak{g}$* is of type $B_{7}$ or $B_{8}$ and $J=\left\{\alpha_{2},\alpha_{6},\alpha_{7}\right\}$.* 3. (3)

$\mathfrak{g}$* is of type $C_{5}$ and $J$ is one of $\left\{\alpha_{3},\alpha_{5}\right\}$, $\left\{\alpha_{1},\alpha_{3},\alpha_{5}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{5}\right\}$,*

$\left\{\alpha_{2},\alpha_{4},\alpha_{5}\right\}$. 4. (4)

$\mathfrak{g}$* is of type $C_{6}$ and $J$ is $\left\{\alpha_{2},\alpha_{4},\alpha_{6}\right\}$ or $\left\{\alpha_{1},\alpha_{3},\alpha_{5},\alpha_{6}\right\}$.* 5. (5)

$\mathfrak{g}$* is of type $D_{7}$ and $J$ is $\left\{\alpha_{2},\alpha_{4},\alpha_{6}\right\}$ or $\left\{\alpha_{2},\alpha_{4},\alpha_{7}\right\}$.* 6. (6)

$\mathfrak{g}$* is of type $D_{8}$ and $J$ is $\left\{\alpha_{2},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$ or $\left\{\alpha_{2},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$.* 7. (7)

$\mathfrak{g}$* is of type $E_{7}$ and $J$ is one of $\left\{\alpha_{3},\alpha_{5}\right\}$, $\left\{\alpha_{4},\alpha_{5}\right\}$, $\left\{\alpha_{4},\alpha_{7}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{5}\right\}$,*

$\left\{\alpha_{1},\alpha_{4},\alpha_{7}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{6}\right\}$, $\left\{\alpha_{2},\alpha_{4},\alpha_{6}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{5}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{7}\right\}$,

$\left\{\alpha_{3},\alpha_{5},\alpha_{6}\right\}$, $\left\{\alpha_{4},\alpha_{5},\alpha_{6}\right\}$, $\left\{\alpha_{4},\alpha_{6},\alpha_{7}\right\}$, $\left\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{6}\right\}$,

$\left\{\alpha_{1},\alpha_{4},\alpha_{5},\alpha_{6}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{6}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{7}\right\}$,

$\left\{\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$. 8. (8)

$\mathfrak{g}$* is of type $E_{8}$ and $J$ is one of $\left\{\alpha_{4},\alpha_{6}\right\}$, $\left\{\alpha_{4},\alpha_{7}\right\}$, $\left\{\alpha_{4},\alpha_{8}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{6}\right\}$,*

$\left\{\alpha_{1},\alpha_{4},\alpha_{7}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{8}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{7}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{6}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{7}\right\}$,

$\left\{\alpha_{3},\alpha_{4},\alpha_{8}\right\}$, $\left\{\alpha_{3},\alpha_{5},\alpha_{7}\right\}$, $\left\{\alpha_{4},\alpha_{5},\alpha_{7}\right\}$, $\left\{\alpha_{4},\alpha_{6},\alpha_{7}\right\}$, $\left\{\alpha_{4},\alpha_{6},\alpha_{8}\right\}$,

$\left\{\alpha_{4},\alpha_{7},\alpha_{8}\right\}$, $\left\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{6}\right\}$, $\left\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{8}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{5},\alpha_{7}\right\}$,

$\left\{\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$, $\left\{\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{7}\right\}$, $\left\{\alpha_{2},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$,

$\left\{\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{7}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$, $\left\{\alpha_{3},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$, $\left\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{6},\alpha_{7}\right\}$,

$\left\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$, $\left\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{6},\alpha_{8}\right\}$. 9. (9)

$\mathfrak{g}$* is of type $F_{4}$ and $J$ is one of $\left\{\alpha_{3}\right\}$, $\left\{\alpha_{1},\alpha_{3}\right\}$, $\left\{\alpha_{2},\alpha_{3}\right\}$, $\left\{\alpha_{2},\alpha_{4}\right\}$.* 10. (10)

$\mathfrak{g}$* is of type $G_{2}$ and $J=\left\{\alpha_{1}\right\}$ or $J=\left\{\alpha_{2}\right\}$*

We choose the first case listed above, namely when $\mathfrak{g}$ is of type $B_{6}$ and $J=\left\{\alpha_{2},\alpha_{5}\right\}$, and completely determine its automorphism group. It remains to show that every automorphism preserves the $\mathbb{N}$-grading in this case.

Theorem 4.8**.**

If $\mathfrak{g}$ is of type $B_{6}$ and $J=\left\{\alpha_{2},\alpha_{5}\right\}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{6}$.

Proof.

Consider the reduced expression

[TABLE]

for the parabolic element $w_{o}^{J}w_{o}$. The length of this Weyl group element is $31$. With this, let $\beta_{1},\dots,\beta_{31}$ and $x_{1},\dots,x_{31}$ denote the corresponding radical roots and root vectors, respectively. We will choose the coweight $\varpi_{2}+\varpi_{5}$ to equip the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ with a $\mathbb{N}$-gradation. By applying (4.3), we obtain that the core ${\mathcal{C}}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))$ coincides with ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. Secondly, for every root vector $x_{i}$ ($i=1,\dots,31$), there is another root vector $x_{j}$ ($j\neq i$) such that $x_{i}x_{j}=\kappa x_{j}x_{i}$ for some scalar $\kappa\neq 1$. One can always simply let $j=i+1$ or $j=i-1$. Observe that in the reduced word above, every letter $s_{k}$ is always adjacent to either $s_{k-1}$ or $s_{k+1}$. If the $i$-th letter is $s_{k}$ and the $(i+1)$-th letter is $s_{k\pm 1}$, then $x_{i}x_{i+1}=q^{\langle\alpha_{k},s_{k}(\alpha_{k\pm 1})\rangle}x_{i+1}x_{i}$. On the other hand, if the $i$-th letter is $s_{k}$ and the $(i-1)$-th letter is $s_{k\pm 1}$, then $x_{i-1}x_{i}=q^{\langle\alpha_{k\pm 1},s_{k\pm 1}(\alpha_{k})\rangle}x_{i}x_{i-1}$. In either case, the relevant hypothesis of Theorem 4.1 is satisfied.

Among the $31$ Lusztig root vectors exactly 15 of them are of degree one. We leave it to the reader to verify that each remaining root vector $x_{i}$ can be written, up to a scalar multiple, as a $q$-commutator $x_{j}x_{k}-q^{\langle\beta_{j},\beta_{k}\rangle}x_{k}x_{j}$ for some $j,k$. Thus ${\mathcal{U}}_{q}(\mathfrak{n})$ is generated by its elements of degree one. Hence all of the hypotheses of Theorem 4.1 are satisfied. Therefore every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ preserves the $\mathbb{N}$-grading.

The degrees of the normal elements $\Theta_{1},\dots,\Theta_{6}$ are $4,8,10,12,14$, and $7$, respectively. Hence, Theorem 4.3 tells us that every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ sends $\Theta_{1}$, $\Theta_{3}$, and $\Theta_{6}$ to multiples of themselves. Next, Theorem 4.5 can be applied to conclude that every automorphism sends each degree one generator $x_{i}$ to a multiple of itself. Hence, every automorphism is a diagonal automorphism. Finally, [15, Theorems 5.3 and 5.5] imply that $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{6}$.

∎

5. The automorphism group of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ for

$\mathfrak{g}=F_{4}$

We now consider the case when $\mathfrak{g}$ is the Lie algebra of type $F_{4}$. In this section we prove that, for every nonempty subset $J$ of simple roots of $\mathfrak{g}$, the automorphism group of the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to the algebraic torus $\left(\mathbb{K}^{\times}\right)^{4}$.

We consider the following reduced expressions for the longest element $w_{o}$ of the Weyl group of $\mathfrak{g}$:

[TABLE]

The main reason we consider these two particular reduced expressions for $w_{o}$ is that for every nonempty subset $J\subseteq\Pi$, a reduced expression for the corresponding parabolic element $w_{J}$ appears either as a substring of $\mathbf{R}[w_{o}]$ or as a substring of $\mathbf{R}^{\prime}[w_{o}]$. Thus, if we treat ${\mathcal{U}}_{q}(\mathfrak{n}^{+})$ as an iterated Ore extension over $\mathbb{K}$, then it follows that each quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ can be viewed as an interval subalgebra (defined in (3.2)) of ${\mathcal{U}}_{q}(\mathfrak{n}^{+})$. This is advantageous for finding explicit presentations of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ more efficiently. To make this more explicit, we will denote the Lusztig root vectors corresponding to the reduced expressions, $\mathbf{R}[w_{o}]$ and $\mathbf{R}^{\prime}[w_{o}]$, by

[TABLE]

respectively. Since every parabolic element $w_{J}$ appears as a substring of $\mathbf{R}[w_{o}]$ or $\mathbf{R}^{\prime}[w_{o}]$, then every quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic, as an algebra, to a subalgebra of ${\mathcal{U}}_{q}(\mathfrak{n}^{+})$ generated by either a contiguous sequence of $x_{i}$’s or $y_{i}$’s. The following example illustrates this when $J=\left\{\alpha_{2},\alpha_{4}\right\}$.

Example 5.1**.**

Suppose $J=\left\{\alpha_{2},\alpha_{4}\right\}$. The parabolic element $w_{J}\in W$ has a reduced expression

[TABLE]

This expression is obtained by removing the first and last letters from $\mathbf{R}[w_{o}]$. Thus ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is isomorphic to the interval subalgebra $\mathbf{U}_{[2,23]}\subseteq{\mathcal{U}}_{q}(\mathfrak{n}^{+})$,

[TABLE]

With this identification, $x_{2},x_{3},x_{4},x_{6},x_{7},x_{8},x_{22},x_{23}$ is a list of Lusztig root vectors of degree $1$.

Theorem 5.2**.**

If $\mathfrak{g}$ is the Lie algebra of type $F_{4}$ and $J$ is a nonempty subset of simple roots of $\mathfrak{g}$, then every automorphism of the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is a diagonal automorphism.

Proof.

The case when $J$ is the full set of simple roots was handled in [34, Theorem 5.1]. Thus, we suppose $J$ is a nonempty proper subset of the set of simple roots. Throughout this proof we will let $\phi$ be an arbitrary algebra automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. The algebra ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ satisfies the hypotheses of Theorem 4.1. Hence, $\phi$ is a graded automorphism. Thus, the sets $C_{d}^{m}$ and $\gamma_{d,\ell}^{m}$ are $\phi$-invariant.

Our objective is to prove that $\phi$ is a diagonal automorphism. The commutation relations given in Lemmas 6.1 and 6.2 show that ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is generated by the degree one component ${\mathcal{U}}_{q}(\mathfrak{n}_{J})_{1}$. Hence it suffices to show that each of the degree one generators of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ gets sent to a scalar multiple of itself under the map $\phi$. We handle this on a case by case basis for each subset $J$ of simple roots. The strategy is the same in each case. One key step is to observe that the sets $C_{d}^{m}$ and $\gamma_{d,\ell}^{m}$ (defined in (4.5) and (4.6)) are $\phi$-invariant. In several instances we will be able to characterize the $C_{d}^{m}$’s and $\gamma_{d,\ell}^{m}$’s, or intersections of them, as either the set of all scalar multiples or nonzero scalar multiples of a generator of $\mathcal{U}_{q}(\mathfrak{n}_{J})$. In these cases, we can immediately conclude that $\phi$ sends that particular generator to a multiple of itself. In other cases, we can show that the $C_{d}^{m}$’s (or intersections of $C_{d}^{m}$’s) are vector subspaces of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})_{1}$ spanned by either two or three generators of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. In these cases we will need to appeal to the defining relations of Lemmas 6.1 and 6.2 as well as Corollary 3.1 in order to conclude that $\phi$ indeed sends every degree one generator of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ to itself. Recall that Corollary 3.1 gives us sufficient conditions to conclude that certain $q$-commutators, say $[x_{i},x_{j}]$ or $[y_{i},y_{j}]$, equal [math]. Throughout this proof, we are tacitly applying this result whenever we state that a $q$-commutator equals [math].

$J=\left\{\alpha_{1}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[2,16]}$. The degree one generators are $y_{2}$, $y_{3}$, $y_{4}$, $y_{5}$, $y_{6}$, $y_{7}$, $y_{8}$, $y_{10}$, $y_{11}$, $y_{12}$, $y_{13}$, $y_{14}$, $y_{15}$, and $y_{16}$. We have $\mathbb{K}^{\times}y_{2}=\gamma_{6,3}^{6}$, $\mathbb{K}^{\times}y_{3}=\gamma_{6,2}^{6}$, $\mathbb{K}^{\times}y_{4}=\gamma_{4,1}^{4}\cap\gamma_{4,1}^{2}$, $\mathbb{K}^{\times}y_{5}=\gamma_{4,1}^{2}\cap\gamma_{4,2}^{0}$, $\mathbb{K}^{\times}y_{6}=\gamma_{6,3}^{2}$, $\mathbb{K}^{\times}y_{7}=\gamma_{6,2}^{2}\cap\gamma_{6,1}^{-2}$, $\mathbb{K}^{\times}y_{8}=\gamma_{6,3}^{0}\cap\gamma_{6,1}^{2}$, $\mathbb{K}^{\times}y_{10}=\gamma_{6,3}^{0}\cap\gamma_{6,1}^{-2}$, $\mathbb{K}^{\times}y_{11}=\gamma_{6,2}^{-2}\cap\gamma_{6,1}^{2}$, $\mathbb{K}^{\times}y_{12}=\gamma_{6,3}^{-2}$, $\mathbb{K}^{\times}y_{13}=\gamma_{4,1}^{-2}\cap\gamma_{4,2}^{0}$, $\mathbb{K}^{\times}y_{14}=\gamma_{4,1}^{-2}\cap\gamma_{4,1}^{-4}$, $\mathbb{K}^{\times}y_{15}=\gamma_{6,2}^{-6}$, $\mathbb{K}^{\times}y_{16}=\gamma_{6,3}^{-6}$. Hence $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{2}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[2,21]}$. The degree one generators are $x_{2}$, $x_{3}$, $x_{4}$, $x_{6}$, $x_{7}$, $x_{8}$, $x_{13}$, $x_{15}$, $x_{17}$, $x_{19}$, $x_{20}$, and $x_{21}$. We have $\mathbb{K}x_{2}=C_{4}^{2}\cap C_{6}^{2}$, $\mathbb{K}x_{3}=C_{4}^{2}\cap C_{6}^{-2}$, $\mathbb{K}x_{4}=C_{4}^{1}\cap C_{6}^{2}$, $\mathbb{K}x_{6}=C_{4}^{1}\cap C_{6}^{-2}$, $\mathbb{K}x_{17}=C_{4}^{-1}\cap C_{6}^{2}$, $\mathbb{K}x_{19}=C_{4}^{-1}\cap C_{6}^{-2}$, $\mathbb{K}x_{20}=C_{4}^{-2}\cap C_{6}^{2}$, $\mathbb{K}x_{21}=C_{4}^{-2}\cap C_{6}^{-2}$, $\mathbb{K}x_{7}\oplus\mathbb{K}x_{13}=C_{4}^{0}\cap C_{6}^{2}$, and $\mathbb{K}x_{8}\oplus\mathbb{K}x_{15}=C_{4}^{0}\cap C_{6}^{-2}$.

Thus, $\phi(x_{7})=ax_{7}+bx_{13}$ for some $a,b\in\mathbb{K}$. By applying $\phi$ to the relation $[x_{7},x_{17}]=0$ and using $[x_{13},x_{17}]=0$, we conclude $b=0$. Hence, $\phi(x_{7})\in\mathbb{K}x_{7}$. Similarly, by applying $\phi$ to the relation $[x_{13},x_{17}]=0$ we conclude $\phi(x_{13})\in\mathbb{K}x_{13}$. Analogously, we can conclude that $\phi(x_{8})\in\mathbb{K}x_{8}$ and $\phi(x_{15})\in\mathbb{K}x_{15}$ by applying $\phi$ to the relations $[x_{8},x_{19}]=0$ and $[x_{15},x_{19}]=0$. Hence $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{3}\right\}$ Here, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[4,23]}$. The degree one generators are $y_{4}$, $y_{5}$, $y_{17}$, $y_{19}$, $y_{21}$, and $y_{23}$. We have $\mathbb{K}y_{4}=C_{6}^{1}\cap C_{8}^{2}$, $\mathbb{K}y_{5}=C_{6}^{-1}\cap C_{8}^{2}$, $\mathbb{K}y_{17}=C_{6}^{1}\cap C_{8}^{0}$, $\mathbb{K}y_{19}=C_{6}^{1}\cap C_{8}^{-2}$, $\mathbb{K}y_{21}=C_{6}^{-1}\cap C_{8}^{0}$, and $\mathbb{K}y_{23}=C_{6}^{-1}\cap C_{8}^{-2}$. Therefore, $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{4}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[9,23]}$. The degree one generators are $x_{9}$, $x_{10}$, $x_{13}$, $x_{15}$, $x_{17}$, $x_{19}$, $x_{22}$, and $x_{23}$. We have $\mathbb{K}x_{9}=C_{6}^{3}$, $\mathbb{K}^{\times}x_{10}=C_{6}^{1}\cap\gamma_{8,2}^{4}$, $\mathbb{K}^{\times}x_{13}=C_{6}^{1}\cap\gamma_{8,1}^{4}$, $\mathbb{K}^{\times}x_{15}=C_{6}^{1}\cap\gamma_{8,0}^{4}$, $\mathbb{K}^{\times}x_{17}=C_{6}^{-1}\cap\gamma_{8,0}^{-4}$, $\mathbb{K}^{\times}x_{19}=C_{6}^{-1}\cap\gamma_{8,1}^{-4}$, $\mathbb{K}^{\times}x_{22}=C_{6}^{-1}\cap\gamma_{8,2}^{-4}$, and $\mathbb{K}x_{23}=C_{6}^{-3}$. Therefore, $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{2}\right\}$ Here, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[1,21]}$. The degree one generators are $x_{1}$, $x_{3}$, $x_{6}$, $x_{8}$, $x_{15}$, $x_{19}$, and $x_{21}$. We have $\mathbb{K}x_{3}=C_{6}^{2}$, $\mathbb{K}x_{6}=C_{6}^{1}$, $\mathbb{K}x_{19}=C_{6}^{-1}$, $\mathbb{K}x_{21}=C_{6}^{-2}$, and $\mathbb{K}x_{1}\oplus\mathbb{K}x_{8}\oplus\mathbb{K}x_{15}=C_{6}^{0}$. Hence $\phi(x_{3})\in\mathbb{K}x_{3}$, $\phi(x_{6})\in\mathbb{K}x_{6}$, $\phi(x_{19})\in\mathbb{K}x_{19}$, $\phi(x_{21})\in\mathbb{K}x_{21}$, and there exist scalars $a_{ij}\in\mathbb{K}$, ($1\leq i,j\leq 3$) such that

[TABLE]

By applying $\phi$ to the relation $[x_{6},x_{15}]=0$ and using the relations $[x_{6},x_{8}]=0$ and $[x_{1},x_{6}]=x_{4}$ to straighten unordered monomials, we obtain

[TABLE]

Hence $a_{31}=a_{32}=0$. Therefore $\phi(x_{15})\in\mathbb{K}x_{15}$.

Next we apply $\phi$ to the relation $[x_{6},x_{8}]=0$ and use the relations $[x_{1},x_{6}]=x_{4}$ and $[x_{6},x_{15}]=0$ to straighten unordered monomials to conclude that $-q^{2}a_{21}x_{4}+a_{23}\left(1-q\right)x_{6}x_{15}=0$. Hence $a_{21}=a_{23}=0$. Thus, $\phi(x_{8})\in\mathbb{K}x_{8}$.

The relations $[x_{1},x_{19}]=x_{17}$ and $[x_{8},x_{17}]=0$ imply $[x_{8},[x_{1},x_{19}]]=0$. By applying $\phi$ to this relation and using the relations $[x_{8},x_{15}]=[x_{8},x_{19}]=[x_{15},x_{19}]=0$ to straighten any unordered monomials, we get

[TABLE]

Hence $a_{12}=a_{13}=0$. Thus, $\phi(x_{1})\in\mathbb{K}x_{1}$ and we conclude that $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{3}\right\}$ In this situation, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[2,23]}$. The degree one generators are $y_{2}$, $y_{3}$, $y_{17}$, $y_{19}$, $y_{21}$, and $y_{23}$. We have $\mathbb{K}y_{2}=C_{8}^{0}\cap C_{22}^{2}$, $\mathbb{K}y_{3}=C_{8}^{0}\cap C_{22}^{-2}$, $\mathbb{K}y_{17}=C_{8}^{1}\cap C_{22}^{2}$, $\mathbb{K}y_{19}=C_{8}^{1}\cap C_{22}^{-2}$, $\mathbb{K}y_{21}=C_{8}^{-1}\cap C_{22}^{2}$, and $\mathbb{K}y_{23}=C_{8}^{-1}\cap C_{22}^{-2}$. Hence $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{4}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[1,20]}$. The degree one generators are $y_{1}$, $y_{2}$, $y_{3}$, $y_{5}$, $y_{11}$, $y_{12}$, $y_{17}$, $y_{19}$, and $y_{20}$. We have $\mathbb{K}^{\times}y_{1}=C_{10}^{2}\cap\gamma_{14,1}^{2}$, $\mathbb{K}^{\times}y_{2}=C_{10}^{2}\cap\gamma_{14,1}^{4}$, $\mathbb{K}y_{3}=C_{10}^{2}\cap C_{14}^{0}$, $\mathbb{K}y_{5}=C_{10}^{0}\cap C_{14}^{0}$, $\mathbb{K}y_{11}=C_{10}^{-2}\cap C_{14}^{0}$, $\mathbb{K}^{\times}y_{12}=C_{10}^{-2}\cap\gamma_{14,1}^{-4}$, $\mathbb{K}^{\times}y_{17}=C_{10}^{0}\cap\gamma_{14,1}^{2}$, $\mathbb{K}^{\times}y_{19}=C_{10}^{0}\cap\gamma_{14,1}^{-2}$, and $\mathbb{K}^{\times}y_{20}=C_{10}^{-2}\cap\gamma_{14,1}^{-2}$. Hence $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{2},\alpha_{3}\right\}$ Here, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[3,24]}$. The degree one generators are $y_{3}$, $y_{17}$, $y_{21}$, and $y_{24}$. We have $\mathbb{K}y_{3}=C_{14}^{2}$, $\mathbb{K}y_{17}=C_{10}^{1}$, $\mathbb{K}y_{21}=C_{10}^{-1}$, and $\mathbb{K}y_{24}=C_{14}^{-2}$. Hence, $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{2},\alpha_{4}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[2,23]}$. The degree one generators are $x_{2}$, $x_{3}$, $x_{4}$, $x_{6}$, $x_{7}$, $x_{8}$, $x_{22}$, and $x_{23}$. We have $\mathbb{K}x_{2}=C_{10}^{2}\cap C_{14}^{2}$, $\mathbb{K}x_{3}=C_{10}^{-2}\cap C_{14}^{2}$, $\mathbb{K}x_{4}=C_{10}^{2}\cap C_{14}^{0}$, $\mathbb{K}x_{6}=C_{10}^{-2}\cap C_{14}^{0}$, $\mathbb{K}x_{7}=C_{10}^{2}\cap C_{14}^{-2}$, $\mathbb{K}x_{8}=C_{10}^{-2}\cap C_{14}^{-2}$, $\mathbb{K}x_{22}=C_{14}^{1}$, and $\mathbb{K}x_{23}=C_{14}^{-1}$. Hence, $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{3},\alpha_{4}\right\}$ We have ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[4,24]}$. The degree one generators are $x_{4}$, $x_{6}$, $x_{22}$, and $x_{24}$. We have $\mathbb{K}x_{4}=C_{12}^{2}$, $\mathbb{K}x_{24}=C_{12}^{-2}$, and $\mathbb{K}x_{6}\oplus\mathbb{K}x_{22}=C_{12}^{0}$. Thus, there exist scalars $a_{11},a_{12},a_{21},a_{22},\gamma,\delta\in\mathbb{K}$ such that $\phi(x_{6})=a_{11}x_{6}+a_{12}x_{22}$, $\phi(x_{22})=a_{21}x_{6}+a_{22}x_{22}$, $\phi(x_{4})=\gamma x_{4}$, and $\phi(x_{24})=\delta x_{24}$.

Observe first that the relation $[x_{4},x_{24}]=(q+q^{-1})x_{7}$ implies $\phi(x_{7})=\gamma\delta x_{7}$. Next apply $\phi$ to the relation $[x_{6},x_{7}]=0$ and use the relation $[x_{7},x_{22}]=x_{17}$ to straighten unordered monomials to obtain

[TABLE]

Hence $a_{12}=0$. Thus $\phi(x_{6})\in\mathbb{K}x_{6}$.

The relations $[x_{4},x_{22}]=x_{13}$ and $[x_{4},x_{13}]=0$ imply $[x_{4},[x_{4},x_{22}]]=0$. Applying $\phi$ to this relation and using the relations $[x_{4},x_{6}]=\left(q+q^{-1}\right)x_{5}$ and $[x_{4},x_{5}]=0$ to straighten unordered monomials we get

[TABLE]

Hence $a_{21}=0$. Therefore $\phi(x_{22})\in\mathbb{K}x_{22}$. Thus $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{2},\alpha_{3}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[2,24]}$. The degree one generators are $y_{2}$, $y_{17}$, $y_{21}$, and $y_{24}$. We have $\mathbb{K}y_{17}=C_{12}^{1}$, $\mathbb{K}y_{21}=C_{12}^{-1}$, and $\mathbb{K}y_{2}\oplus\mathbb{K}y_{24}=C_{12}^{0}$. Thus, there exist $b_{2},b_{24},c_{2},c_{24}\in\mathbb{K}$ such that $\phi(y_{2})=b_{2}y_{2}+b_{24}y_{24}$ and $\phi(y_{24})=c_{2}y_{2}+c_{24}y_{24}$. Applying $\phi$ to the relation $[y_{2},y_{17}]=0$ and using the identity $[y_{17},y_{24}]=y_{19}$ gives us

[TABLE]

Hence $b_{24}=0$. Next we observe that $[y_{2},y_{24}]=y_{3}$ and $[y_{2},y_{3}]=0$. Hence we have the relation $[y_{2},[y_{2},y_{24}]]=0$. Applying $\phi$ to this relation gives us $c_{2}\widehat{q}^{2}y_{2}^{3}=0$. Therefore $c_{2}=0$ and we conclude that $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{2},\alpha_{4}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[1,23]}$. The degree one generators are $x_{1}$, $x_{3}$, $x_{6}$, $x_{8}$, $x_{22}$, and $x_{23}$. We have $\mathbb{K}x_{3},=C_{18}^{2}$, $\mathbb{K}x_{8}=C_{18}^{-2}$, $\mathbb{K}x_{22}=C_{18}^{1}$, $\mathbb{K}x_{23}=C_{18}^{-1}$, and $\mathbb{K}x_{1}\oplus\mathbb{K}x_{6}=C_{18}^{0}$. Hence, there exist $b_{1},b_{6},c_{1},c_{6}\in\mathbb{K}$ such that $\phi(x_{1})=b_{1}x_{1}+b_{6}x_{6}$ and $\phi(x_{6})=c_{1}x_{1}+c_{6}x_{6}$. Applying $\phi$ to the relation $[x_{1},x_{22}]=0$ gives us

[TABLE]

because $[x_{6},x_{22}]=x_{15}$. Hence $b_{6}=0$. Similarly, by applying $\phi$ to the relation $[x_{3},x_{6}]=0$ we obtain

[TABLE]

because $[x_{1},x_{3}]=x_{2}$. Hence $c_{1}=0$. Therefore $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{1},\alpha_{3},\alpha_{4}\right\}$ In this situation, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}^{\prime}_{[1,23]}$. The degree one generators are $y_{1}$, $y_{2}$, $y_{3}$, $y_{21}$, and $y_{23}$. We have $\mathbb{K}y_{1}=C_{30}^{0}$, $\mathbb{K}y_{2}\oplus\mathbb{K}y_{21}=C_{30}^{2}$, and $\mathbb{K}y_{3}\oplus\mathbb{K}y_{23}=C_{30}^{-2}$. Hence there exist $a_{2},a_{21},b_{2},b_{21},c_{3},c_{23},d_{3},d_{23}\in\mathbb{K}$ such that $\phi(y_{2})=a_{2}y_{2}+a_{21}y_{21}$, $\phi(y_{21})=b_{2}y_{2}+b_{21}y_{21}$, $\phi(y_{3})=c_{3}y_{3}+c_{23}y_{23}$, and $\phi(y_{23})=d_{3}y_{3}+d_{23}y_{23}$. The relations $[y_{1},y_{17}]=0$ and $[y_{1},y_{21}]=y_{17}$ give us $[y_{1},[y_{1},y_{21}]]=0$. Applying $\phi$ to this relation and using the commutation relation $y_{1}y_{2}=y_{2}y_{1}$ gives us $b_{2}(2-q-q^{-1})y_{1}^{2}y_{2}=0$. Hence $b_{2}=0$. Next, since $[y_{1},y_{21}]=y_{17}$ and $[y_{2},y_{17}]=0$, we have the relation $[y_{2},[y_{1},y_{21}]]=0$. Applying $\phi$ to this relation and using the identity $[y_{17},y_{21}]=0$ gives us $a_{21}\left(q^{-1}-1\right)y_{17}y_{21}=0$. Therefore $a_{21}=0$. Observe next that $[y_{21},y_{23}]=[2]_{q}y_{22}$ and $[y_{21},y_{22}]=0$. Therefore $[y_{21},[y_{21},y_{23}]]=0$. Applying $\phi$ to this relation and using the identities $[y_{3},y_{21}]=y_{5}$ and $[y_{5},y_{21}]=[2]_{q}y_{11}$ give us $d_{3}[2]_{q}q^{2}y_{11}=0$. Thus $d_{3}=0$. Finally, applying $\phi$ to the relation $[y_{1},y_{3}]=0$ and using the identity $[y_{1},y_{23}]=y_{19}$ gives us $c_{23}\left((1-q)y_{1}y_{23}+qy_{19}\right)=0$. Hence $c_{23}=0$. Therefore $\phi$ is a diagonal automorphism.

$J=\left\{\alpha_{2},\alpha_{3},\alpha_{4}\right\}$ In this case, ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\cong\mathbf{U}_{[2,24]}$. The degree one generators are $x_{2}$, $x_{3}$, $x_{22}$, and $x_{24}$. We have $\mathbb{K}x_{2}=C_{18}^{2}$, $\mathbb{K}x_{3}=C_{18}^{-2}$, and $\mathbb{K}x_{22}\oplus\mathbb{K}x_{24}=C_{18}^{0}$. By applying $\phi$ to the relation $[x_{2},x_{22}]=0$ and using the relation $[x_{2},x_{24}]=x_{4}$ to straighten unordered monomials, we can conclude that $\phi(x_{22})\in\mathbb{K}x_{22}$.

Since $\mathbb{K}x_{22}\oplus\mathbb{K}x_{24}$ is a $\phi$-invariant subspace, there exist scalars $\gamma,\delta\in\mathbb{K}$ such that $\phi(x_{24})=\gamma x_{22}+\delta x_{24}$. The relations $[x_{2},x_{24}]=x_{4}$ and $[x_{3},x_{4}]=0$ give us the relation $[x_{3},[x_{2},x_{24}]]=0$. Applying $\phi$ to this relation and using the relations $[x_{2},x_{22}]=[x_{2},x_{3}]=[x_{3},x_{22}]=0$ to straighten any unordered monomials gives us $\gamma\left(q^{-2}-1\right)x_{2}x_{3}x_{22}=0$. Hence $\gamma=0$. Therefore $\phi(x_{24})\in\mathbb{K}x_{24}$ and $\phi$ is a diagonal automorphism.

∎

We are now able to prove the main result of this section. The following theorem proves Conjecture 1.1 when the underlying Lie algebra $\mathfrak{g}$ is of type $F_{4}$.

Theorem 5.3**.**

If $\mathfrak{g}$ is the Lie algebra of type $F_{4}$ and $J$ is a nonempty subset of simple roots of $\mathfrak{g}$, then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{4}$.

Proof.

By Theorem 5.2, every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is a diagonal automorphism. As a CGL extension, the algebra ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ has rank $4$. Thus, by [15, Theorems 5.3 and 5.5], $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{4}$.

∎

6. Two lemmas regarding ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ when

$\mathfrak{g}=F_{4}$

In this section we prove two lemmas regarding the quantized nilradicals ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ for the case when the underlying Lie algebra $\mathfrak{g}$ is of type $F_{4}$ and $J$ is any nonempty subset of simple roots.

Recall from Section 2.1 that for each coweight $\lambda\in P^{\vee}$, there is an induced $\mathbb{Z}$-grading on ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$. We will use the coweight $\lambda=\sum_{i\in J}\varpi_{i}^{\vee}\in P^{\vee}$. The two lemmas in this section explicitly show how every Lusztig root vector $X_{\beta}$ in ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ with $\operatorname{height}(\beta)>1$ can be written, up to a scalar multiple, as a $q$-commutator of other Lusztig root vectors. As a direct consequence of these lemmas, one can readily verify that each quantized nilradical ${\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)$ is generated, as an algebra, by the Lusztig root vectors of degree $1$. Hence, ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is a locally finite, connected, $\mathbb{N}$-graded algebra generated by its graded component of degree $1$.

Lemma 6.1**.**

Let $\mathfrak{g}$ be the Lie algebra of type $F_{4}$, and let $x_{1},\dots,x_{24}$ be the Lusztig root vectors (recall (5.3)) corresponding to the reduced expression $\mathbf{R}[w_{o}]$ (see (5.1)) of the longest element of the Weyl group of $\mathfrak{g}$. Then

2. $x_{2}=[x_{1},x_{3}]$,
3. $x_{4}=[x_{1},x_{6}]$,
4. $x_{4}=[x_{2},x_{24}]$,
5. $x_{5}=\frac{1}{[2]_{q}}[x_{4},x_{6}]$,
6. $x_{6}=[x_{3},x_{24}]$,
7. $x_{7}=[x_{1},x_{8}]$,
8. $x_{7}=\frac{1}{[2]_{q}}[x_{4},x_{24}]$,
9. $x_{8}=\frac{1}{[2]_{q}}[x_{6},x_{24}]$,
10. $x_{9}=[x_{6},x_{13}]$,
11. $x_{10}=[x_{8},x_{13}]$,
12. $x_{11}=\frac{1}{[2]_{q}}[x_{10},x_{13}]$,
13. $x_{12}=\frac{1}{[2]_{q}}[x_{10},x_{15}]$,
14. $x_{13}=[x_{1},x_{15}]$,
15. $x_{13}=[x_{4},x_{22}]$,
16. $x_{14}=\frac{1}{[2]_{q}}[x_{13},x_{15}]$,
17. $x_{15}=[x_{6},x_{22}]$,
18. $x_{16}=[x_{15},x_{17}]$,
19. $x_{17}=[x_{1},x_{19}]$,
20. $x_{17}=[x_{7},x_{22}]$,
21. $x_{18}=\frac{1}{[2]_{q}}[x_{17},x_{19}]$,
22. $x_{19}=[x_{8},x_{22}]$,
23. $x_{20}=[x_{1},x_{21}]$,
24. $x_{20}=\frac{1}{[2]_{q}}[x_{17},x_{22}]$,
25. $x_{21}=\frac{1}{[2]_{q}}[x_{19},x_{22}]$,
26. $x_{23}=[x_{22},x_{24}]$.

Proof.

Throughout the proof of this lemma, we adopt the abbreviation

[TABLE]

for a $q$-commutator and inductively define the nested $q$-commutator

[TABLE]

for $i_{1},i_{2},\dots,i_{n}\in\mathbf{I}$.

Since $s_{1}$ is the first simple reflection appearing the reduced expression $R[w_{o}]$, it is clear that $x_{1}=E_{1}$. As a direct consequence of Proposition 2.1, $x_{3}=E_{2}$, $x_{22}=E_{4}$, and $x_{24}=E_{3}$. At this point we have identified each Chevalley generator $E_{i}$ with a corresponding Lusztig root vector. Next we will show how the remaining Lusztig root vectors $x_{i}$ can be written, up to a scalar multiple, as $q$-commutators of other Lusztig root vectors. The next step will be to focus on those $x_{i}$ with $\langle\operatorname{deg}_{Q}(x_{i}),\varpi_{1}^{\vee}+\varpi_{2}^{\vee}+\varpi_{3}^{\vee}+\varpi_{4}^{\vee}\rangle=2$. In other words, we focus on the $x_{i}$’s such that $\operatorname{deg}_{Q}(x_{i})$ has height $2$. For short, we will say that the $\mathbb{N}$-degree of $x_{i}$ is $d$ whenever $\operatorname{deg}_{Q}(x_{i})$ has height $d$. We compute

[TABLE]

where, in the above computations, we adopt the underlining notation, as in $\mathbf{T}_{\underline{121}32}$ above, to highlight that braid relations in the Weyl group are being applied to the underlined part in moving from one step in the calculations to the next. We use the dot notation, as in $\mathbf{T}_{2\cdot 1232}$ above, to split a reduced word into two parts in order to indicate which Lusztig symmetries are being applied at that particular step. We continue to compute

[TABLE]

So far we have identified how each Lusztig root vector $x_{i}$ with $\mathbb{N}$-degree at most $3$ can be written as a $q$-commutator of Lusztig root vectors of smaller $\mathbb{N}$-degree. We continue in this manner focusing next on the Lusztig root vectors $x_{i}$ having $\mathbb{N}$-degree equal to $4$. We compute

[TABLE]

Next we show how each Lusztig root vector $x_{i}$ of $\mathbb{N}$-degree equal to $5$ (i.e. $x_{5}$, $x_{17}$, and $x_{21}$) can be written as a $q$-commutator of Lusztig root vectors of smaller $\mathbb{N}$-degree. We have

[TABLE]

Continuing in this manner, we get

[TABLE]

Finally, we can use $q$-associativity to prove the remaining identities,

[TABLE]

∎

Lemma 6.2**.**

Let $\mathfrak{g}$ be the Lie algebra of type $F_{4}$, and let $y_{1},\dots,y_{24}$ be the Lusztig root vectors (recall (5.3)) corresponding to the reduced expression $\mathbf{R}^{\prime}[w_{o}]$ (see (5.1)) of the longest element of the Weyl group of $\mathfrak{g}$. Then

2. $y_{3}=[y_{2},y_{24}]$,
3. $y_{4}=[y_{3},y_{17}]$,
4. $y_{5}=[y_{3},y_{21}]$,
5. $y_{6}=\frac{1}{[2]_{q}}[y_{4},y_{17}]$,
6. $y_{7}=\frac{1}{[2]_{q}}[y_{4},y_{19}]$,
7. $y_{8}=[y_{5},y_{17}]$,
8. $y_{9}=\frac{1}{[2]_{q}}[y_{8},y_{10}]$,
9. $y_{10}=[y_{5},y_{19}]$,
10. $y_{11}=\frac{1}{[2]_{q}}[y_{5},y_{21}]$,
11. $y_{12}=\frac{1}{[2]_{q}}[y_{5},y_{23}]$,
12. $y_{13}=[y_{10},y_{17}]$,
13. $y_{14}=[y_{12},y_{17}]$,
14. $y_{15}=\frac{1}{[2]_{q}}[y_{14},y_{17}]$,
15. $y_{16}=\frac{1}{[2]_{q}}[y_{14},y_{19}]$,
16. $y_{17}=[y_{1},y_{21}]$,
17. $y_{18}=\frac{1}{[2]_{q}}[y_{17},y_{19}]$,
18. $y_{19}=[y_{1},y_{23}]$,
19. $y_{19}=[y_{17},y_{24}]$,
20. $y_{20}=[y_{19},y_{21}]$,
21. $y_{22}=\frac{1}{[2]_{q}}[y_{21},y_{23}]$,
22. $y_{23}=[y_{21},y_{24}]$.

Proof.

In the proof of this lemma we adopt the same abbreviation for $q$-commutators as used in the proof of Lemma 6.1.

Observe first that the two reduced expressions for $w_{o}$ in (5.1) share a common substring, namely ${\mathbf{R}}^{\prime}[6,23]={\mathbf{R}}[2,19]$. Hence, there is an algebra isomorphism $\Phi:\mathbf{U}^{\prime}_{[6,23]}\to\mathbf{U}_{[2,19]}$ such that $\Phi(y_{i})=x_{i-4}$ for all $i\in[6,23]$. Thus, the commutation relations among the $x_{i}$’s given in Lemma 6.1 translate into commutation relations among the $y_{i}$’s. In particular, we have $y_{9}=\frac{1}{[2]_{q}}[y_{8},y_{10}]$, $y_{13}=[y_{10},y_{17}]$, $y_{14}=[y_{12},y_{17}]$, $y_{15}=\frac{1}{[2]_{q}}[y_{14},y_{17}]$, $y_{16}=\frac{1}{[2]_{q}}[y_{14},y_{19}]$, $y_{18}=\frac{1}{[2]_{q}}[y_{17},y_{19}]$, $y_{20}=[y_{19},y_{21}]$, and $y_{22}=\frac{1}{[2]_{q}}[y_{21},y_{23}]$.

Next we apply Proposition 2.1 to identify which of the $y_{i}$’s correspond to the standard Chevalley generators $E_{i}$. We get $y_{1}=E_{4}$, $y_{2}=E_{1}$, $y_{21}=E_{3}$, and $y_{24}=E_{2}$. We next identify the $y_{i}$’s that can be written as $q$-commutators of these $E_{i}$’s. For example, we have $y_{3}=\mathbf{T}_{\underline{41}}(E_{2})=\mathbf{T}_{1\cdot 4}(E_{2})=\mathbf{T}_{1}(E_{2})=E_{12}=[y_{2},y_{24}]$. We note here we have adopted the same underlining notation, as in $\mathbf{T}_{\underline{41}}$ above, as well as the dot notation, as in $\mathbf{T}_{1\cdot 4}$ above, used in the proof of Lemma 6.1. With this, we also have $y_{17}=\mathbf{T}_{4\cdot 123421323124321}(E_{3})=\mathbf{T}_{4}(E_{3})=E_{43}=[y_{1},y_{21}]$ and $y_{23}=\mathbf{T}_{\underline{4123421323124321323432}}(E_{3})=\mathbf{T}_{32\cdot 12321432341232143234}(E_{3})=\mathbf{T}_{32}(E_{3})=E_{32}=[y_{21},y_{24}]$.

Now that we have established some of the identities of this lemma, we can continue with this same strategy to establish further identities. We have

[TABLE]

Next we compute a few more identities that build off of the identities already established. We have

[TABLE]

Finally, we can use $q$-associativity to establish the remaining identities,

[TABLE]

∎

7. Quantum Symmetric Matrices

The algebra of $n\times n$ quantum symmetric matrices [21, 30] is a quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ for the case when the underlying Lie algebra $\mathfrak{g}$ is of type $C_{n}$ and $J=\left\{\alpha_{n}\right\}$. In this section, we prove that Conjecture 1.1 holds in this case.

We let $x_{ij}$ ($1\leq i\leq j\leq n$) denote the standard generators of the algebra of quantum symmetric matrices. The defining relations are given in [21, Proposition 5.2]. Let ${\mathcal{N}}_{J}$ be the normal subalgebra of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. It is generated by the normal elements $\Theta_{1},\dots,\Theta_{n}$. Here, the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is $\mathbb{N}$-graded with $\operatorname{deg}(x_{ij})=1$ for all $1\leq i\leq j\leq n$. In view of this, $\operatorname{deg}(\Theta_{i})=i$ (for $1\leq i\leq n$). The simple roots are $\alpha_{i}=e_{i}-e_{i+1}$ (for $1\leq i<n$) and $\alpha_{n}=2e_{n}$. The fundamental weights are $\varpi_{i}=e_{1}+\cdots+e_{i}$ (for $1\leq i\leq n$). Let $Q=\mathbb{Z}\alpha_{1}+\cdots+\mathbb{Z}\alpha_{n}$ and $P=\mathbb{Z}\varpi_{1}+\cdots+\mathbb{Z}\varpi_{n}$ be the root lattice and weight lattice respectively, and let $\langle-,-\rangle$ be the symmetric bilinear form on $P$ defined by the rule $\langle e_{i},e_{j}\rangle=\delta_{ij}$. The algebra ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is $Q$-graded with

[TABLE]

Consider the parabolic element $w_{J}:=w_{o}^{J}w_{o}\in W$. We have the reduced expression

[TABLE]

We recall (4.4) the commutation relations,

[TABLE]

for $1\leq i\leq j\leq n$ and $1\leq k\leq n$.

Proposition 7.1**.**

Suppose $\mathfrak{g}$ is the Lie algebra of type $C_{n}$ with $n>1$, and suppose $J=\left\{\alpha_{n}\right\}$. If $\psi$ is an automorphism of the quantized nilradical ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$, then $\psi(\Theta_{i})\in\mathbb{K}^{\times}\Theta_{i}$ for every $i\in\left\{1,\dots,n\right\}$.

Proof.

As in the proof of Theorem 4.8, the hypotheses in Theorem 4.1 involving the core and the existence of relations of the form $xy=\kappa yx$ can be seen to be satisfied by observing the relevant properties of the reduced expression (7.1). Hence, every automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ preserves the $\mathbb{N}$-grading.

The normal subalgebra ${\mathcal{N}}_{J}$ is invariant under any algebra automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. Since $\operatorname{dim}_{\mathbb{K}}({\mathcal{N}}_{J})_{1}=1$ (in fact, $({\mathcal{N}}_{J})_{1}=\mathbb{K}\Theta_{1}$), then $\psi(\Theta_{1})\in\mathbb{K}^{\times}\Theta_{1}$. Furthermore, since $\mathbb{K}x_{nn}=\left\{x\in({\mathcal{U}}_{q}(\mathfrak{n}_{J}))_{1}:x\Theta_{1}=q^{2}\Theta_{1}x\right\}$, $\psi(x_{nn})=\mathbb{K}^{\times}x_{nn}$.

For $k\in\left\{1,\dots,n\right\}$, $\operatorname{dim}_{\mathbb{K}}\left(\left({\mathcal{N}}_{J}\right)_{k}\right)={\mathcal{P}}(k)$, where ${\mathcal{P}}$ is the partition function. We also have $x_{nn}\Theta_{k}=q^{2}\Theta_{k}x_{nn}$ for every $k\in\left\{1,\dots,n-1\right\}$.

For a fixed natural number $k\in\mathbb{N}$, we represent a partition $\nu$ of $k\in\mathbb{N}$ (and write $\nu\vdash k$) by a weakly increasing sequence of natural numbers $\nu_{1}\leq\nu_{2}\leq\cdots$ such that $\sum_{i}\nu_{i}=k$. For a partition $\nu\vdash k$, we let $\operatorname{parts}(\nu)$ be the number of parts of $\nu$ and write $\nu=(\nu_{1},\nu_{2},\dots,\nu_{\operatorname{parts}(\nu)})$, and we define the monomial

[TABLE]

Suppose

[TABLE]

for scalars $c_{\nu,k}\in\mathbb{K}$. For $1\leq k<n$, we apply the automorphism $\psi$ to the relation $x_{nn}\Theta_{k}=q^{2}\Theta_{k}x_{nn}$ to conclude that

[TABLE]

However since $x_{nn}\Theta_{k}=q^{2}\Theta_{k}x_{nn}$ for every $k\in\left\{1,\dots.,n-1\right\}$, then in the above sum we can replace $x_{nn}\Theta^{\nu}$ with $q^{2\cdot\operatorname{parts}(\nu)}\Theta^{\nu}x_{nn}$. Thus,

[TABLE]

Since $q$ is not a root of unity, the only nonzero coefficients $c_{\nu,k}$ appearing in the above sum are those such that $\operatorname{parts}(\nu)=1$. In other words, there is at most one monomial $\Theta^{\nu}$ in the sum $\sum_{\nu\vdash k}c_{\nu,k}\Theta^{\nu}$ with a nonzero coefficient, namely $\Theta_{k}$. Hence $\psi(\Theta_{k})\in\mathbb{K}^{\times}\Theta_{k}$ for $k<n$.

Finally, consider $\Theta_{n}$. Since $\Theta_{n}$ generates the center of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ (see e.g. [18]), then $\psi(\Theta_{n})\in\mathbb{K}^{\times}\Theta_{n}$.

∎

Theorem 7.2**.**

If $\mathfrak{g}$ is the Lie algebra of type $C_{n}$ with $n>1$ and $J=\left\{\alpha_{n}\right\}$ (i.e. ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ is the algebra of $n\times n$ quantum symmetric matrices), then $\operatorname{Aut}({\mathcal{U}}_{q}(\mathfrak{n}_{J}))\cong\left(\mathbb{K}^{\times}\right)^{n}$.

Proof.

Suppose $\phi$ is an automorphism of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$. As established in Proposition 7.1, all hypotheses of Theorem 4.1 are satisfied. Hence, $\phi$ is a graded automorphism.

Next we will apply Theorem 4.5 to show that each $x_{ij}$ with $i+j\neq n+1$ gets sent to a multiple of itself by $\phi$. First, suppose $i$ and $j$ are chosen such that $1\leq i\leq j\leq n$ and $i+j<n+1$, and consider the corresponding element $x_{ij}$. We show that for every other Lusztig root vector $x_{k\ell}$, there is a normal element $\Theta_{p}$ such that $x_{ij}$ and $x_{k\ell}$ commute differently with $\Theta_{p}$. Equivalently, this means $\langle(e_{i}+e_{j})-(e_{k}+e_{\ell}),(1+w_{J})\varpi_{p}\rangle\neq 0$. With this, Theorem 4.5 implies $x_{ij}$ gets sent to a multiple of itself by the automorphism $\phi$. There are two cases to consider. If $i\neq k$, let $p=\operatorname{min}(i,k)$. On the other hand, if $i=k$, let $p=\operatorname{min}(j,\ell)$. Now suppose $i$ and $j$ are chosen so that $1\leq i\leq j\leq n$ and $i+j>n+1$, and consider the corresponding Lusztig root vector $x_{ij}$. As before, we will show that for every other Lusztig root vector $x_{k\ell}$, there is a normal element $\Theta_{p}$ such that $x_{ij}$ and $x_{k\ell}$ commute differently with $\Theta_{p}$. Here, if $j\neq\ell$, put $p=\operatorname{max}(j-1,\ell-1)$. However, if $j=\ell$, put $p=\operatorname{max}(i-1,k-1)$.

Now we will show that every Lusztig root vector $x_{ij}$ with $i+j=n+1$ gets sent to a multiple of itself by $\phi$. Theorem 4.5 does not apply here because these $x_{ij}$ all commute the same way with each normal element $\Theta_{p}$. In fact, each of these $x_{ij}$ commute with all of the elements in the normal subalgebra. None of the other $x_{ij}$’s behave this way. This means $S:=\operatorname{span}_{\mathbb{K}}\left\{x_{ij}\mid i+j=n+1\right\}$ is a $\phi$-invariant vector subspace of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})_{1}$ For every $y\in S$, define

[TABLE]

Each $C(y)$ is a vector space. Observe $\operatorname{dim}(C(y))=\operatorname{dim}(C(\phi(y))$. From the defining relations of ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$, we obtain that $C(x_{ij})$ is spanned by $\left\{x_{ik}:k<j\right\}\cup\left\{x_{kj}:k<i\right\}$. Hence $\operatorname{dim}(C(x_{ij}))=n-i$. The only elements $y\in S$ with $\operatorname{dim}(C(y))=n-i$ are the nonzero multiples of $x_{ij}$. Hence $\phi(x_{ij})\in\mathbb{K}^{\times}x_{ij}$.

We have shown now that $\phi$ is a diagonal automorphism. Since ${\mathcal{U}}_{q}(\mathfrak{n}_{J})$ has rank $n$ as a CGL extension, [15, Theorems 5.3 and 5.5] imply that $\operatorname{Aut}\left({\mathcal{U}}_{q}\left(\mathfrak{n}_{J}\right)\right)\cong\left(\mathbb{K}^{\times}\right)^{n}$.

∎