Borel subalgebras of Cartan Type Lie Algebras

TL;DR

This paper classifies certain subalgebras of the Jacobson-Witt algebra over fields with positive characteristic, focusing on trigonalizable subalgebras related to Borel subalgebras, and determines their conjugation classes and properties.

Contribution

It introduces a classification of trigonalizable subalgebras related to Borel subalgebras in Jacobson-Witt algebras, extending previous work on homogeneous Borel subalgebras.

Findings

Conjugation classes of these subalgebras are determined.

Properties such as filtration and dimension are analyzed.

Relations to previously studied homogeneous Borel subalgebras are established.

Abstract

Let $ W(n) $ be Jacobson-Witt algebra over algebraic closed field $ \mathbb{K} $ with positive characteristic $ p>2. $ It is difficult to classify all Borel subalgebras of $ W(n) $ or non-classical restricted simple Lie algebras. The present paper and \cite{S7} study two kinds of subalgebras which are easily to understand and highly related to Borel subalgebras. In \cite{S7}, the last author investigates a class of special Borel subalgebras of $W(n)$ which is called homogeneous Borel subalgebras. The present paper focuses on subalgebras of $ W(n) $ which are related to Borel subalgebras such that firstly, they could be trigonalizable; and secondly, they essentially belong to the ones investigated in \cite{S7}. In this paper, the conjugation classes of these subalgebras and representative for each class will be determined. Then some properties such as filtration and dimension will be investigated.

Equations

$$\mathsf{gdim}(\mathfrak{l},({\mathfrak{l}}_i)):=\sum _{i\in \mathbb{Z}}(\mathsf{dim}{\mathfrak{l}}_i)t^i\in \mathbb{N}[[t,t^{-1}]].$$

$$\mathsf{gdim}(A,(A_{(n)})):=\mathsf{gdim}(\mathsf{gr}(A),(\mathsf{gr}(A{)}_n)).$$

$$W(n)=\underset{i=1}{\overset{n}{⨁}}A(n){\partial }_i,$$

$$W(n)={\oplus }_{i=-1}^{s}W(n{)}_i,s=n(p-1)-1,\text{ where }$$

$$W(n{)}_i:={\mathsf{span}}_{\mathbb{K}}\{x^{\underline{a}}{\partial }_l\mid |\underline{a}|=i+1\}.$$

$$U=\{g\in G\big{|}(g-Id_{\mathfrak{g}})(\mathfrak{g}_{\geq i})\subset\mathfrak{g}_{\geq i+1}\}$$

$${\Psi }_{\mu }\left(\begin{array}{c}{\partial }_1\\ ⋮\\ {\partial }_n\end{array}\right)=\left(\begin{array}{c}{\Psi }_{\mu }({\partial }_1)\\ ⋮\\ {\Psi }_{\mu }({\partial }_n)\end{array}\right)=J(\mu {)}^{-1}\left(\begin{array}{c}{\partial }_1\\ ⋮\\ {\partial }_n\end{array}\right).$$

$$\dim (\mathfrak{t}\cap {\mathfrak{g}}_{\ge 0})=\dim ({\mathfrak{t}}^{\prime }\cap {\mathfrak{g}}_{\ge 0}).$$

$${\mathfrak{t}}_r=\sum _{i=1}^n\mathbb{K}{z}_i{\partial }_i,r=0,\cdots ,n,$$

$$\mathbb{K}[z_1,\cdots ,z_{n-r};z_{n-r+1},\cdots ,z_n]$$

$$[x_i,x_{i^{\prime }}]=[y_i,y_{i^{\prime }}]=[x_i,y_j]=x_i^p=y_j^p-1=0.$$

$$W(n)=\underset{s}{⨁}W(n{)}_s^{({\mathfrak{t}}_r)},\text{with}W(n{)}_s^{({\mathfrak{t}}_r)}=⟨Z^{\underline{a}(i)}{\partial }_i\mid |\underline{a}(i)|=s+1,i=1,\cdots ,n⟩.$$

$$\mathfrak{h}=\sum _{\alpha \in I^n}{\mathfrak{h}}_{\alpha }^{({\mathfrak{t}}_r)}.$$

$$S:=\{{\partial }_1,x_1{\partial }_1,{\partial }_2,x_1{\partial }_2,\cdots ,x_1^{p-1}{\partial }_2,x_2{\partial }_2,x_1{x}_2{\partial }_2\cdots ,x_1^{p-1}{x}_2{\partial }_2\}.$$

$${\mathfrak{b}}_q={\mathfrak{b}}_0(x_1,\cdots ,x_{n-q})\oplus Q_q\oplus {\mathfrak{b}}_q(x_{n-q+1},\cdots ,x_n)={\mathfrak{t}}_0\oplus C_q.$$

$$Q_q=⟨u^{\underline{a}(i)}{w}^{\underline{b}(i)}{\partial }_i\mid (\underline{a}(i),\underline{b}(i),i)\in {\Gamma }_1\cup {\Gamma }_2⟩,$$

$$C_q=⟨u^{\underline{a}(i)}{w}^{\underline{b}(i)}{\partial }_i\mid (\underline{a}(i),\underline{b}(i),i)\in {\Lambda }_1\cup {\Lambda }_2⟩,$$

$${\Gamma }_1=$$

$${\Gamma }_2=$$

$${\Lambda }_1=$$

$${\Lambda }_2=$$

$$C_n^{[1]}\subseteq$$

$$C_n^{[p]}\subseteq$$

$$C_n^{[p^s]}\subseteq$$

$$C_q^{[1]}\subseteq$$

$$C_q^{[p]}\subseteq$$

$$C_q^{[p^q]}\subseteq$$

$$C_q^{[s{p}^q]}\subseteq$$

$$C_q^{[t+s{p}^q]}\subseteq$$

$$C_q^{[t+s{p}^q+(n-q)p^q]}\subseteq$$

$$C_q^{[t+s{p}^q+p^{n-q}{p}^q]}=\subseteq$$

$$r(\mathfrak{h}):=\text{max}\{r\mid \text{there exists}\sigma \in \text{Aut}(W(n))\text{such that }\sigma (t_r)\subset \mathfrak{h}\},$$

$$r(\mathfrak{b})=\mathsf{dim}({\mathsf{pr}}_0(\mathfrak{b})),$$

$$\mathsf{gdim}(\mathfrak{b})=n(Q^{n-r}-1)Q^r{t}^{-1}+\frac{1-Q^r}{1-Q}{t}^{-1}-\frac{(n-r)(n+r+1)}{2}(Q^r-1)$$

$$+r-\frac{(n-r)(n-r-1)}{2},$$

Full text

In [15], the last author investigates a class of special Borel subalgebras of $W(n)$ which is called homogeneous Borel subalgebras. The present paper focuses on subalgebras of $W(n)$ which are related to Borel subalgebras such that firstly, they could be trigonalizable; and secondly, they essentially belong to the ones investigated in [15]. In this paper, the conjugation classes of these subalgebras and representative for each class will be determined. Then some properties such as filtration and dimension will be investigated.

1. Introduction

If $A$ is a non-associative algebra of finite dimension over a (commutative) field, its maximal solvable subalgebras will be called the Borel subalgebras of A. In this paper, we study the Borel subalgebras of non-classical simple Lie algebras in positive characteristic with certain conditions.

Over last few decades, Borel subalgebras has been investigated and generalised by many authors. In [5], J.Green investigates so-called Borel subalgebras of the Schur algebra associated. In [4], J.Du and H.Rui investigate the existence of the Borel type subalgebras of a $q\operatorname{-Schur}^{m}$ algebra. In [7, Appendix], the notion of Borel subalgebras for a quasi-hereditary algebra has been introduced by Scott. Then in [7, 8, 9], S.Konig introduces and investigates the exact Borel subalgerbas and strong exact Borel subalgebras for an quasi-hereditary algebra. In [3], I. Dimitrov and I.Penkov study the Borel subalgebras of the Lie algebra $\mathfrak{gl}(\infty)$ of finitary infinite matrices.

As we known, Borel subalgebras of a Lie algebra play an important role in the structure and representation theory. However, there is less study on them for non-classical restricted simple Lie algebras. We neither know the number of conjugacy classes of Borel subalgebras nor what kind of role the Borel subalgebras play in the representation theory although their Cartan subalgebras are well-known (cf. [17]).

It seems very difficult to classify all Borel subalgebras of non-classical restricted simple Lie algebras. Therefore, [15] and this paper study two kinds of subalgebras which are easier to understand and highly related to Borel subalgebras. In [15], the last author investigates a class of special Borel subalgebras, the so-called homogeneous Borel subalgebras. This paper is a continuous of [15] to focus on completely solvable subalgebras of non-classical restricted simple Lie algebras.

Among non-classical simple Lie algebras, the Jacobson-Witt algebras $W(n)$ are primary, which are also the main objects in present paper. Since 1960’s, they have been extensively studied (cf. [2, 10, 11, 14, 16] etc.).

The theorem of Borel([1]) and Morozov([12]) asserts that the Borel subalgebras of a semisimple complex Lie algebra $\mathfrak{g}$ are conjugate with respect to automorphism group. The same result is also true for a classical simple Lie algebra over an algebraically closed field of prime characteristic with some mild restriction on the characteristic ([6]), as well as alternative and Jordan algebras([13]).

However, this conjugation phenomenon will fail for Borel subalgebras of non-classical Lie algebras over positive characteristic field. There are $(n+1)$ conjugacy classes of homogeneous Borel subalgebras of $W(n)$ (cf. [15]). As a corollary, there are at least $(n+1)$ conjugation classes of Borel subalgebras of $W(n).$

Based on [15], the present paper focuses on a special class of completely solvable subalgebras such that firstly, they could be trigonalizable; and secondly, they essentially belong to the ones investigated in [15]. In this paper, the conjugation classes of these subalgebras and representatives for each class will be determined. Then some properties such as filtration and dimension will be investigated.

We collect such subalgebras, endow the set of them with a variety structure and establish an analogy of classical Springer theory for Cartan type Lie algebras in other papers.

2. Preliminaries

Entire the whole paper, denote $I=\{0,1,\cdots,p-1\}$ and we always assume the ground filed $\mathbb{K}$ to be algebraically closed of odd characteristic $p>0.$ Unless mentioned otherwise, all vector spaces are assumed to be finite-dimensional . Given a restricted Lie algebra $(\mathfrak{g},[p]),$ we have an adjoint group $G:=\operatorname{Aut}_{p}(\mathfrak{g})^{\circ},$ the identity component of its restricted automorphism group. The term filtration stands a descending filtration.

2.1. Graded dimension of a graded algebra

Definition 2.1**.**

For a given graded algebra $(\mathfrak{l},(\mathfrak{l}_{n})),$ set $G:=\text{Aut}_{p}(\mathfrak{l})^{\circ}.$ We call $(\mathfrak{l},(\mathfrak{l}_{n}),G,G_{0},U)$ satisfies graded assumption if $G=G_{0}\ltimes U$ where $G_{0}$ consists of homogeneous automorphisms and $u\cdot x-x\in\mathfrak{l}_{\geq i+1}$ for all $u\in U,x\in\mathfrak{l}_{i}.$

Lemma 2.2**.**

Let $(\mathfrak{l},(\mathfrak{l}_{n})_{n\in\mathbb{Z}})$ be a graded Lie algebra such that $(\mathfrak{l},(\mathfrak{l}_{n}),G,G_{0},U)$ satisfying graded assumption, and $\mathfrak{b}$ be a graded subalgebra of $\mathfrak{l}.$ Then every subalgebra conjugates to $\mathfrak{b}$ is filtered, namely $C:=g\cdot\mathfrak{b}$ admits a filtration structure $(C_{(n)})_{(n\in\mathbb{Z})}$ for all $g\in G.$ Moreover, $(C_{(n)})$ is restricted if all $\mathfrak{l},$ $(\mathfrak{l}_{(n)})$ and $\mathfrak{b}$ are restricted and $g$ is a restricted automorphism.

Proof.

Denote $\mathfrak{b}_{i}:=\mathfrak{b}\cap\mathfrak{l}_{i},$ then $\mathfrak{b}=\oplus_{i\in\mathbb{Z}}\mathfrak{b}_{i}.$ Moreover, $\mathfrak{b}_{\geq n}:=\oplus_{i\geq n}\mathfrak{b}_{i}=\mathfrak{b}\cap\mathfrak{l}_{\geq n}$ for $n\in\mathbb{Z}$ defines a filtered structure for $\mathfrak{b}.$

Set $C_{(n)}:=g\cdot\mathfrak{b}_{\geq n}.$ One can check the lemma by definitions. ∎

Definition 2.3**.**

Let $(\mathfrak{l},(\mathfrak{l}_{i})_{i\in\mathbb{Z}})$ be a $\mathbb{Z}$-graded Lie algebras with finite dimensional homogeneous spaces. The graded dimension associated with $L$ is defined by:

[TABLE]

Moreover, if $(A,(A_{(n)})_{n\in\mathbb{Z}})$ is a filtered Lie algebra such that $\mathsf{gr}(A)$ has finite dimensional homogeneous spaces, we can also define its graded dimension as

[TABLE]

Lemma 2.4**.**

Let $(\mathfrak{l},(\mathfrak{l}_{i})_{(i\in\mathbb{Z})})$ be a $\mathbb{Z}$-graded Lie algebras with finite dimensional homogeneous spaces. Suppose $(\mathfrak{l},(\mathfrak{l}_{i})_{(i\in\mathbb{Z})},G,G_{0},U)$ satisfies graded assumption, and $\mathfrak{b}$ is a graded subspace of $\mathfrak{l}.$ Then $\mathsf{gdim}(g\cdot\mathfrak{b})=\mathsf{gdim}(\mathfrak{b})$ for all $g\in G.$

Namely, graded dimension is $G$-invariant.

Proof.

By the assumptions of $(G,G_{0},U),$ $\mathsf{gr}(u\cdot g_{0}\cdot\mathfrak{b})=g_{0}\cdot\mathfrak{b},\ \forall g_{0}\in G_{0},u\in U.$ ∎

2.2. Completely solvable subalgebras

If $L$ is arbitrary Lie algebra over $\mathbb{K},$ define $L^{[n+1]}:=[L,L^{[n]}],\ L^{(n+1)}:=[L^{(n)},L^{(n)}]$ and $L^{[0]}=L^{(0)}=L\text{ for all }n\in\mathbb{N}.$ Then $L^{[1]}=L^{(1)}.$ We say that $L$ is nilpotent (resp. solvable) if $L^{[n]}=0$ (resp. $L^{(n)}=0$) for some $n.$

Definition 2.5**.**

A Lie algebra $\mathfrak{b}$ is called completely solvable if $[\mathfrak{b},\mathfrak{b}]$ is nilpotent (cf. [18]).

The importance of completely solvable Lie algebra comes from the following feature.

Lemma 2.6**.**

[18, Lemma 8.6]** Let $(\mathfrak{l},[p])$ be a restricted, completely solvable Lie algebra of finite dimension and such that $C(\mathfrak{l})^{[p]}=0.$

1. (1)

If $\mathfrak{t}$ is a maximal torus of $\mathfrak{l}$, then $\mathfrak{l}=\mathfrak{t}\oplus\operatorname{rad}_{p}(\mathfrak{l}).$ 2. (2)

Every irreducible restricted representation of $\mathfrak{l}$ is one-dimensional.

Remark 2.7*.*

If a solvable subalgebra can be embedded into a Lie algebra of classical type, it must be complete. This will fail for Cartan type Lie algebras. There are examples of subalgebras which are solvable other than completely solvable.

2.3. Basics on Jacobson-Witt algebra

Let $W(n)$ be Jacobson-Witt algebra over $\mathbb{K}$, which is the derivation algebra over the truncated polynomial ring $A(n)=\mathbb{K}[x_{1},\cdots,x_{n}]/(x_{1}^{p},\cdots,x_{n}^{p}),$ namely,

[TABLE]

where $\partial_{i}(x_{j})=\delta_{i,j},$ for all $i,j=1,\cdots,n.$

Set $G:=\operatorname{Aut}_{p}(W(n))^{\circ}.$ There is an isomorphism from $\operatorname{Aut}(A(n))$ to $G$ by sending $\psi$ to $\Psi_{\psi}$ via $\Psi_{\psi}:D\mapsto\psi\circ D\circ\psi^{-1}$ (cf. [19]).

For each automorphism $\psi\in\operatorname{Aut}(A(n)),$ set $\tilde{\psi}_{i}:=\psi(x_{i})\in A(n)_{\geq 1}(i=1,\cdots,n)$ and J(\psi):=\big{(}\partial_{i}(\tilde{\psi}_{j})\big{)}_{n\times n}\in\operatorname{Mat}_{n\times n}(A(n)). Then $\psi$ is determined by $\tilde{\psi}_{1},\cdots,\tilde{\psi}_{n}$ and $\psi\in G$ if and only if $J(\mu)$ is invertible.

We list some basic material on $W(n)$ in the following proposition.

Proposition 2.8**.**

Keep notations as above, then we have

1. (1)

$\{x^{\underline{a}}\partial_{i}\mid 1\leq i\leq n;\underline{a}\in I^{n}\}$* is a basis of $W(n)$ over $\mathbb{K}.$ In particular, $\mathsf{dim}_{\mathbb{K}}W(n)=np^{n}.$* 2. (2)

$[D,fE]=D(f)E+f[D,E],$* for all $D,E\in W(n),f\in A(n).$* 3. (3)

There is a so-called standard grading structure of $W(n):$

[TABLE]

[TABLE] 4. (4)

Suppose $n\geq 2.$ Then $W(n)$ is generated by $W(n)_{-1}\oplus W(n)_{1}.$ 5. (5)

$A(n)$* is a $W(n)$-module by $f\partial_{i}\cdot g:=f\partial_{i}(g)$ for $f\partial_{i}\in W(n),$ $g\in A(n).$ We call the corresponding representation $\rho:W(n)\rightarrow\mathfrak{gl}(A(n))$ the natural representation of $W(n).$* 6. (6)

The representation $\bar{\rho}:W(n)_{0}\rightarrow\mathfrak{gl}(A(n)_{1})$ induces from $\rho$ is an isomorphism. Remind that $W(n)_{0}=\langle x_{i}\partial_{j}\mid i,j=1,\cdots,n\rangle,$ $A(n)_{1}=\langle x_{1},\cdots,x_{n}\rangle,$ and $\bar{\rho}(x_{i}\partial_{j})=E_{ij},$ which sends $x_{i}$ to $x_{j}$.

Remark 2.9*.*

If $p=3,\ W(1)\simeq sl(2)$ as restricted Lie algebras. We will omit this very special case throughout this paper.

For convenience, we fix the following notations.

[TABLE]

For more details of $W(n),$ reader refers to [18, chapter 4].

2.4. Automorphisms of $W(n)$.

Lemma 2.10**.**

([19])* Let $\mathfrak{g}=W(n)$ over $\mathbb{K}$ with $p\geq 3$ (unless $n=1$ with assumption $p>3$). The following statements hold.*

1. (1)

$G=\operatorname{Aut}(\mathfrak{g})$* coincides with the adjoint group $\operatorname{Aut}_{p}(\mathfrak{g})^{\circ}.$ Hence it is a connected algebraic group.* 2. (2)

$G$* is a semi-direct product $G=G_{0}\ltimes U,$ where $G_{0}\simeq\operatorname{GL}(n,\mathbb{K})$ consists of those automorphisms preserving the $\mathbb{Z}$-grading of $\mathfrak{g}$, and*

[TABLE]

Remark 2.11*.*

$(\mathfrak{g},(\mathfrak{g}_{i})_{i\in\mathbb{Z}},G,G_{0},U)$ satisfies graded assumption as definition 2.1.

The following lemma is an algorithm to compute automorphisms.

Lemma 2.12**.**

Keep assumptions and notations as above, for all $\mu\in G,$

[TABLE]

Proof.

It is a direct computation that \big{(}\partial_{i}(\tilde{\mu}_{j})\big{)}(\Psi_{\mu}(\partial_{i})(x_{j}))=I_{n}. Hence lemma holds. ∎

2.5. Maximal Torus of $W(n)$

A torus $\mathfrak{t}$ is an abelian restricted subalgebra consisting of semisimple elements, i.e. $X\in(X^{[p]})_{p}$ for all $X\in\mathfrak{t}$, where $(X^{[p]})_{p}$ denotes for the restricted subalgebra generated by $X^{[p]}$ (see [18]). According to Demuskin’s result [2], we have the following conjugacy property for maximal torus of $W(n).$

Theorem 2.13**.**

Let $\mathfrak{g}=W(n).$ Then the following statements hold.

1. (1)

Two maximal torus $\mathfrak{t},\mathfrak{t}^{\prime}$ belong to the same $G$-orbit if and only if

[TABLE] 2. (2)

There are $(n+1)$ conjugacy classes for the maximal torus of $\mathfrak{g}$. Each maximal torus of $\mathfrak{g}$ is conjugate to one of

[TABLE]

where $z_{i}=x_{i}$ for $i=1,\cdots,n-r,$ and $z_{i}=1+x_{i}$ for $i=n-r+1,\cdots,n.$

Theorem 2.2. We call these $\mathfrak{t}_{r}$ the standard maximal torus of $W(n)$.

2.6. Gradings associated with $\mathfrak{t}_{r}$

Note that $A(n)$ can be presented as the quotient algebra $\mathbb{K}[T_{1};\cdots;T_{n}]/(T_{1}^{p}-1,\cdots,T_{n}^{p}-1).$ Denote the image of $T_{i}$ by $y_{i}$ in the quotient algebra. Then we can write $A(n)$ as $\mathbb{K}[y_{1},\cdots,y_{n}].$ Comparing with the notations in section 1.3, we have $y_{i}=1+x_{i},\ i=1,\cdots,n.$

Now, fix $r=0,1,\cdots,n,$ $A(n)$ can be presented as a truncated polynomial

[TABLE]

with generator $z_{i}=x_{i},\ z_{j}=y_{j},\text{ where }i=1,\cdots,n-r;j=n-r+1,\cdots,n,$ and defining relations:

[TABLE]

$\text{Moreover, }W(n)=\sum_{i=1}^{n}\sum_{\underline{a}(i)}\mathbb{K}Z^{\underline{a}(i)}\partial_{i}$ where $Z^{\underline{a}(i)}=z_{1}^{a_{i1}}z_{2}^{a_{i2}}\cdots z_{n}^{a_{in}}$ with $\underline{a}(i)=(a_{i1},\cdots,a_{in})\in I^{n}.$

There is a space decomposition as below, called $\mathbb{Z}(\mathfrak{t}_{r})$-grading:

[TABLE]

In fact, every homogenous space $W(n)_{s}^{(\mathfrak{t}_{r})}$ is a $\mathfrak{t}_{r}$-module. For the case $r=0,$ the $\mathbb{Z}(\mathfrak{t}_{0})$-graded structure coincides with the standard graded structure of $W(n),$ namely $W(n)_{s}=W(n)_{s}^{(\mathfrak{t}_{0})}$ for all $s.$

Let $\mathfrak{h}$ be a subalgebra of $W(n).$ Call $\mathfrak{h}$ a $\mathbb{Z}(\mathfrak{t}_{r})$-graded subalgebra if $\mathfrak{h}=\sum_{i}\mathfrak{h}_{i}^{(\mathfrak{t}_{r})},$ where $\mathfrak{h}_{i}^{(\mathfrak{t}_{r})}=\mathfrak{h}\cap W(n)_{i}^{(\mathfrak{t}_{r})}.$

We refine $\mathbb{Z}(\mathfrak{t}_{r})$-grading. If a subalgebra $\mathfrak{h}$ is $\mathbb{Z}(\mathfrak{t}_{r})$-graded, we set for every $\alpha\in I^{n}$ with $\lvert\alpha\rvert=s+1,\mathfrak{h}_{\alpha}^{(\mathfrak{t}_{r})}=\{v\in\mathfrak{h}_{s}^{(\mathfrak{t}_{r})}\mid\text{ad}(z_{i}\partial_{i})(v)=\alpha_{i}v,i=0,\cdots,n\}.$ Then we call $\mathfrak{h}$ is $\mathfrak{t}_{r}$-graded if $\mathfrak{h}$ is $\mathbb{Z}(\mathfrak{t}_{r})$-graded and

[TABLE]

Moreover, $\mathfrak{h}$ is called torus graded if $\mathfrak{h}$ contains a maximal torus of $W(n)$ and for every maximal torus $\mathfrak{t}\subseteq\mathfrak{h},$ $\varphi\cdot\mathfrak{h}$ is $\mathfrak{t}_{r}$-graded where $\mathfrak{t}_{r}=\varphi\cdot\mathfrak{t}$ for some $\varphi\in G$ and $r=0,\cdots,n.$

3. Completely solvable subalgebras of Jacobson-Witt algebras

3.1. Homogeneous Borel subalgebras of restricted Lie algebras

B.Shu introduces homogeneous Borel subalgebras of $W(n)$ in [15] which is by definition a torus graded and maximal solvable subalgebras of $W(n).$ He proves that there are $n+1$ conjugation classes of homogeneous Borel subalgebras of $W(n)$ with representatives $B_{0},\cdots,B_{n}.$ The definition of $B_{i}(i=0,\cdots,n)$ refers to [15].

We mention here that $B_{r}$ is not completely solvable except $r=0.$ For example, $B_{1}\subseteq W(2)$ is a subspace spanned by

[TABLE]

One can check the following statements.

1. (1)

$B_{1}$ is a maximal solvable subalgebra ([15]). 2. (2)

$\partial_{2}\in(B_{1}^{[1]})^{[n]}$ for all $n\geq 0.$ Therefore, $\mathfrak{l}^{[1]}$ is not nilpotent and $\mathfrak{l}$ is not completely solvable.

In fact, since $\{x_{2}\partial_{2},\partial_{2}\}\subseteq B_{1}^{[1]},$ and $[x_{2}\partial_{2},\partial_{2}]=-\partial_{2},$ one can prove it by induction.

3.2.

We introduce the following subspaces which will be useful.

$\bullet$ $\mathfrak{b}_{0}=\mathfrak{b}\oplus W(n)_{\geq 1}$, where $\mathfrak{b}$ consists of all upper triangular matrices of $\mathfrak{gl}(n)$.

$\bullet$ $\mathfrak{b}_{n}=\mathfrak{t}_{0}\oplus C_{n}$, where $C_{n}=\langle X^{\underline{a}(i)}\partial_{i}\mid\underline{a}(i)=(a_{1},\cdots,a_{i-1},0,\cdots,0),a_{j}\in I\rangle,$ and $\mathfrak{t}_{0}=\langle x_{i}\partial_{i}\mid i=1,\cdots,n\rangle$ is the standard torus of $W(n).$

$\bullet$ For arbitrary $q=1,2,\cdots,n-1$, let

[TABLE]

[TABLE]

[TABLE]

where $(\underline{a}(i),\underline{b}(i),i):=(a_{1},\cdots,a_{n-q},b_{1},\cdots,b_{q},i)\in I^{n-q}\times I^{q}\times\{1,\cdots,n\}$,

[TABLE]

Remark 3.1*.*

Note that $\mathfrak{t}_{0}\subset\mathfrak{b}_{q},\ q=0,\cdots,n$. After proposition 3.6, we will see that all $\mathfrak{b}_{r}(r=0,\cdots,n)$ are torus graded.

Lemma 3.2**.**

$\mathfrak{b}_{0}^{[1]}=\mathfrak{n}\oplus W(n)_{\geq 1}$* is nilpotent, where $\mathfrak{n}$ is nilradical of $\mathfrak{b}.$ Moreover, $\mathfrak{b}_{0}$ is a maximal completely solvable subalgebra.*

Proof.

Since $W(n)_{\geq 1}$ is a $W(n)_{0}$-module and $[\mathfrak{b},\mathfrak{b}]=\mathfrak{n},$ $\mathfrak{b}_{0}^{[1]}=\mathfrak{n}\oplus W(n)_{\geq 1}$ which is nilpotent. Moreover, [15] has proven that $\mathfrak{b}_{0}$ is a Borel. Lemma holds. ∎

Lemma 3.3**.**

$\mathfrak{b}_{n}$* is a maximal completely solvable subalgebra.*

Proof.

Thanks to the definition, $\mathfrak{b}_{n}$ contains a maximal torus $\mathfrak{t}_{0}$.

One can check that $\mathfrak{b}_{n}^{[1]}=C_{n}$ by definition. Moreover,

[TABLE]

[TABLE]

for arbitrary $s.$

Therefore, $C_{n}^{[p^{n-1}]}=0$ and $\mathfrak{b}_{n}$ is completely solvable.

Maximality: $\forall D\not\in\mathfrak{b}_{n},$ denote $A_{n}$ be the algebra generated by $\mathfrak{b}_{n}$ and $D.$

Suppose $D=\sum D_{i},$ where $D_{i}\in W(n)_{i}.$ There must be $D_{k}=\sum_{l=1}^{n}f_{l}\partial_{l}\in A_{n}\backslash\mathfrak{b}_{n}$ by $\mathfrak{t}_{0}$-action. After proper $\partial_{j}$’s actions, we can assume $D_{k}=X^{\underline{a}(l)}\partial_{l}\in A_{n}\backslash\mathfrak{b}_{n},$ where $\underline{a}(l)=(\underline{a}(l)_{1},\cdots,\underline{a}(l)_{n}).$ Namely, $\underline{a}(l)_{s}\neq 0$ for some $s\geq l.$

1. (1)

If $\underline{a}(l)_{s}\neq 0$ for some $s>l,$ we have $x_{s}\partial_{l}\in A_{n},$ and hence the semisimple subalgebra $\langle x_{s}\partial_{l},x_{l}\partial_{s},x_{l}\partial_{l}-x_{s}\partial_{s}\rangle$ lies in $A_{n}.$ 2. (2)

If $\underline{a}(l)_{s}=0$ whence $s>l,$ and $\underline{a}(l)_{l}\geq 2,$ then $x_{l}^{2}\partial_{l}\in A_{n}.$ The semisimple subalgebra $\langle\partial_{l},x_{l}\partial_{l},x_{l}^{2}\partial_{l}\rangle$ lies in $A_{n}.$ 3. (3)

If $\underline{a}(l)_{s}=0$ whence $s>l,$ then $\underline{a}(l)_{l}=1,$ and $x_{i}x_{l}\partial_{l}\in A_{n}\backslash\mathfrak{b}_{n},\ i<l.$ Moreover, $x_{l}\partial_{l}=[\partial_{i},x_{i}x_{l}\partial_{l}]\in A_{n}^{[1]},\ \partial_{l}\in(A_{n}^{[1]})^{[t]}$ for all $t\in\mathbb{N}.$

Hence, $A_{n}^{[1]}$ is not nilpotent. Namely, $\mathfrak{b}_{n}$ is maximal. ∎

Lemma 3.4**.**

$\mathfrak{b}_{q}$* is a maximal completely solvable subalgebra for $q=1,\cdots,n-1$.*

Proof.

Thanks to the definition, $\mathfrak{b}_{q}$ contains a maximal torus $\mathfrak{t}_{0}$ and $\mathfrak{b}_{q}^{[1]}=C_{q}.$

Recall that $\mathfrak{b}_{q}=\mathfrak{t}_{0}\oplus C_{q1}\oplus C_{q2}=\mathfrak{b}_{0}(x_{1},\cdots,x_{n-q})\oplus\mathfrak{t}_{0}\oplus\mathfrak{b}_{q}(x_{n-q+1},\cdots,x_{n}),$ where $C_{qi}$ corresponding to $\Lambda_{i}$ as section 3.2 ($i=1,2$). Then the followings holds.

[TABLE]

Suppose $(\mathfrak{b}_{0}^{[1]}(x_{1},\cdots,x_{n-q}))^{[s]}=0$ and $(\mathfrak{b}_{q}^{[1]}(x_{n-q+1},\cdots,x_{n}))^{[t]}=0.$ Then

[TABLE]

Hence, $C_{q}$ is nilpotent and $\mathfrak{b}_{q}$ is completely solvable.

Maximality: Similar to the maximality of $\mathfrak{b}_{n},$ suppose $A_{q}$ contains $\mathfrak{b}_{q}$ as a proper subset, then either there is an element $\partial_{l}$ lies in $(A_{q}^{[1]})^{[m]}$ for all $m\in\mathbb{N}$ which violates the nilpotence of $A_{q}^{[1]}$ or there exists a semisimple subalgebra in $A_{q},$ say either $\langle x_{s}\partial_{l},x_{l}\partial_{s},x_{l}\partial_{l}-x_{s}\partial_{s}\rangle(l\neq s)$ or $\langle\partial_{l},x_{l}\partial_{l},x_{l}^{2}\partial_{l}\rangle.$ ∎

3.3. Conjugation classes of maximal conpletely solvable subalgebras

By using similar idea and methods of [15], we will classify conjugation classes of all maximal complete solvable subalgebras which are torus graded.

Let $\mathfrak{h}$ be any subalgebra of $W(n)$. Similar to [15], define

[TABLE]

and $\operatorname{r}(\mathfrak{h})=-1$ if $\mathfrak{h}$ does not contain any maximal torus.

The following lemma is easy to verify.

Lemma 3.5**.**

Keep notations as above. For every $r=0,\cdots,n,$

1. (1)

$\mathfrak{t}_{i}\subseteq\mathfrak{b}_{r}$* if and only if $0\leq i\leq r.$* 2. (2)

$\operatorname{r}(\mathfrak{b}_{r})=r.$**

The arguments of [15, lemma 4.2-4.4] still work in our case. However, one need to check that all of the algebras involved are completely solvable not only solvable. Thus we have

Proposition 3.6**.**

Assume $p>3.$ Let $\mathfrak{b}$ be a torus graded and maximal completely solvable subalgebra of $W(n)$ with $\operatorname{r}(\mathfrak{b})=r$. Then $\mathfrak{b}$ conjugates to $\mathfrak{b}_{r}$ with respect to $G.$

In particular, each $\mathfrak{b}_{r}(r=0,\cdots,n)$ is torus graded and maximal complete solvable.

Corollary 3.7**.**

Let $\mathfrak{b}$ be a torus graded and maximal completely solvable subalgebra of $W(n),$ then

[TABLE]

where $\mathsf{pr}_{0}:W(n)\twoheadrightarrow W(n)/W(n)_{\geq 0}$ is a linear map.

Proof.

$\mathsf{pr}_{0}(\mathfrak{b}_{r})=<\partial_{i}\mid i=n-r+1,\cdots,n>,$ hence $\operatorname{r}(\mathfrak{b}_{r})=r=\mathsf{dim}(\mathsf{pr}_{0}(\mathfrak{b}_{r})).$

Note that $\mathsf{pr}_{0}(\sigma(\mathfrak{b}_{r}))=\mathsf{pr}_{0}(\mathfrak{b}_{r})$ and $\mathsf{dim}(\mathsf{pr}_{0}(g(\mathfrak{b})))=\mathsf{dim}(\mathsf{pr}_{0}(\mathfrak{b}))$ for all $\sigma\in U,$ $g\in G_{0}.$ Corollary holds. ∎

Remark 3.8*.*

Thanks to proposition 3.6, the conjugation class of $\mathfrak{b}$ could be determined by $\mathsf{dim}(\mathsf{pr}_{0}(\mathfrak{b})).$

Since $\mathsf{dim}(\mathsf{pr}_{0}(\mathfrak{b}_{r}))=r,$ the following holds.

Corollary 3.9**.**

$\mathfrak{b}_{0},\cdots,\mathfrak{b}_{n}$* are not conjugate with each other.*

Moreover, we have the following classification theorem.

Theorem 3.10**.**

Assume $p>3.$ There are $(n+1)$ conjugacy classes of torus graded and maximal completely solvable subalgebras of $W(n)$ with representatives $\{\mathfrak{b}_{i}\mid i=0,\cdots,n,\}.$

3.4. Completely solvable subalgebras with standard grading

Recall that a subalgebra $\mathfrak{h}$ is of standard grading if $\mathfrak{h}=\sum_{i}\mathfrak{h}\cap W(n)_{i}.$

By definition, all $\mathfrak{b}_{r}$’s are torus graded and maximal completely solvable subalgebras of $W(n)$ with standard grading.

Theorem 3.11**.**

Assume $p>3.$ Let $\mathfrak{b}$ be a torus graded and maximal completely solvable subalgebra of $W(n)$ with standard grading and $\operatorname{r}(\mathfrak{b})=r$. Then $\mathfrak{b}$ conjugates to $\mathfrak{b}_{r}$ with respect to $G.$

Namely, there are $(n+1)$ conjugacy classes of torus graded and maximal completely solvable subalgebras of $W(n)$ with standard grading. The representatives are $\{\mathfrak{b}_{0},\cdots,\mathfrak{b}_{n}\}.$

Proof.

Thanks to proposition 3.6, there is $\sigma\in U,g\in G_{0}$ such that $\mathfrak{b}=\sigma\cdot g\cdot\mathfrak{b}_{r}.$ Now, apply the gradation functor on both side, $\mathfrak{b}=\mathsf{gr}(\mathfrak{b})=\mathsf{gr}(\sigma\cdot g\cdot\mathfrak{b}_{r})=g\cdot\mathfrak{b}_{r}.$ ∎

4. (Graded) dimensions

Recall that $\mathsf{gdim}(L)$ is the graded dimension for a filtered algebra $L$ (definition 2.3). We have proved that $\mathsf{gdim}(L)$ is $G$-invariant if $L$ and $G$ satisfies the graded assumption (definition 2.1).

Lemma 4.1**.**

All torus graded and maximal completely solvable subalgebras of $W(n)$ are filtered subalgebras.

Proof.

Note that $(W(n),(W(n)_{i}),G,G_{0},U)$ satisfies the graded assumptions in Lemma 2.2, and all $\mathfrak{b}_{r},\ r=0,\cdots,n,$ are graded. Claim holds. ∎

Proposition 4.2**.**

Suppose $\mathfrak{b}$ is a torus graded and maximal completely solvable subalgebra of $W(n)$ with $\operatorname{r}(\mathfrak{b})=r,$ then

[TABLE]

[TABLE]

where $Q=\sum_{i=0}^{p-1}t^{i}.$

Proof.

One can check that $\mathsf{gdim}(A(n))=Q^{n}\ \text{and }\mathsf{gdim}(A(n)\partial_{i})=Q^{n}t^{-1},$ where $Q:=\mathsf{gdim}(A(1))=\sum_{i=0}^{p-1}t^{i}.$ Then $\mathsf{gdim}(\mathfrak{b}_{0})=nQ^{n}t^{-1}-nt^{-1}-\frac{n(n-1)}{2}.$

For $\mathfrak{b}_{n}$ case, recall that $\mathfrak{b}_{n}=\mathfrak{t}_{0}+\langle X^{\underline{a}}\partial_{i}\mid\underline{a}=(a_{1},\cdots,a_{i-1},0,\cdots,0)\rangle.$ Therefore,

[TABLE]

Now, for arbitrary $q\neq 0,n,$ $\mathfrak{b}_{q}=\mathfrak{b}_{0}(x_{1},\cdots,x_{n-q})\oplus Q_{q}\oplus\mathfrak{b}_{q}(x_{n-q+1},\cdots,x_{n}),$ where $Q_{q}=\langle u^{\underline{a}(i)}w^{\underline{b}(i)}\partial_{i}\mid(\underline{a}(i),\underline{b}(i),i)\in\Gamma_{1}\cup\Gamma_{2}\rangle$ as 3.2.

Set $R_{i}:=Q_{q}\cap A(n)\partial_{i},\ i=1,\cdots,n.$ We have the followings.

If $1\leq i\leq n-q,$ $\mathsf{gdim}(R_{i})=(Q^{n-q}-1)(Q^{q}-1)t^{-1}-(n-i+1)(Q^{q}-1);$

If $n-q+1\leq i\leq n,$ $\mathsf{gdim}(R_{i})=(Q^{n-q}-1)Q^{q}t^{-1}.$

Moreover,

[TABLE]

Now, suppose $\mathfrak{b}$ is an arbitrary torus graded and maximal completely solvable subalgebra of $W(n)$ with $\operatorname{r}(\mathfrak{b})=r.$ Then there exits $\Phi\in G$ such that $\Phi\cdot\mathfrak{b}=\mathfrak{b}_{r}.$ Moreover, $\mathsf{gdim}(\mathfrak{b})=\mathsf{gdim}(\Phi\cdot\mathfrak{b})=\mathsf{gdim}(\mathfrak{b}_{r}).$ ∎

Remark 4.3*.*

By using the formula $Q(0)=1$ and $\frac{dQ}{dt}|_{t=0}=1,$ one can get

[TABLE]

[TABLE]

which match the fact that

[TABLE]

Note that $Q(1)=p.$ We have the following corollary by taking $t=1$.

Corollary 4.4**.**

Suppose $\mathfrak{b}$ is a torus graded and maximal completely solvable subalgebra with $\operatorname{r}(\mathfrak{b})=r,$ then

[TABLE]

It is easy to verify that all homogeneous Borel subalgebras defined of $W(n)$ in [15] are filtered and hence one can compute their (graded) dimensions as well.

Proposition 4.5**.**

Suppose $B$ is a homogeneous Borel subalgebra with $\operatorname{r}(B)=q,$ then

[TABLE]

where $Q=\sum_{i=0}^{p-1}t^{i}.$ And

[TABLE]