Block Degeneracy for Graded Lie Superalgebras of Cartan Type

TL;DR

This paper demonstrates that certain graded Cartan type Lie superalgebras over algebraically closed fields of positive characteristic have a category of restricted supermodules consisting of a single block, simplifying their representation theory.

Contribution

It introduces a class of Lie superalgebras with a unified block structure for their restricted supermodules, extending to graded Cartan types W, S, and H under specific conditions.

Findings

Category of restricted supermodules is of one block for the introduced Lie superalgebras.

For p > 3, graded Cartan type Lie superalgebras of types W, S, H also have a single block category.

Simplifies the representation theory of these Lie superalgebras.

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic $ p>0. $ In this short note, we illustrate a class of Lie superalgebras over $ \mathbb{k} $ such that the category of restricted supermodules is of one block. As an application, if $ p>3 $ and $ \mathfrak{g} $ is a graded restricted Cartan type Lie superalgebra of type W, S and H, then the category of restricted $ \mathfrak{g} $ supermodules is of one block.

Equations

$$\mathfrak{g}={\mathfrak{g}}_{\bar{1}}^{-}\oplus {\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}\oplus {\mathfrak{g}}_{\bar{1}}^{+}$$

$${\text{dim}}_{\mathbb{K}}({\mathfrak{n}}^{-})<{\text{dim}}_{\mathbb{K}}({\mathfrak{n}}^{+})\text{ and }{\text{dim}}_{\mathbb{K}}({\mathfrak{g}}_{\bar{1}}^{-})<{\text{dim}}_{\mathbb{K}}({\mathfrak{g}}_{\bar{1}}^{+}).$$

$$V^{+}(\lambda ):=u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{+})}{\mathbb{K}}_{\lambda }$$

$$V^{-}(\lambda ):=u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{-})}{\mathbb{K}}_{\lambda },$$

$$\mathfrak{l}={\mathfrak{l}}_{\bar{1}}^{-}\oplus {\mathfrak{n}}_{\mathfrak{l}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}_{\mathfrak{l}}^{+}\oplus {\mathfrak{l}}_{\bar{1}}^{+}$$

$$[V^{-}(\lambda )]=\sum _{\mu \in \Lambda }{p}^s{2}^t[{\mathbb{K}}_{\mu }],$$

$$N={\text{span}}_{\mathbb{K}}\{X^{\underline{i}}{Y}^{\underline{j}}{Z}^{\underline{k}}\mid \underline{i}\in I^n\}$$

$$V^{-}(\lambda )={\text{span}}_{\mathbb{K}}\{X^{\underline{i}}{Y}^{\underline{j}}{Z}^{\underline{k}}\otimes 1_{\lambda }\mid \underline{i}\in I^n,\underline{j}\in I^s\text{ and }\underline{k}\in B(t)\},$$

$$\mathfrak{g}={\mathfrak{g}}_{\bar{1}}^{-}\oplus {\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}\oplus {\mathfrak{g}}_{\bar{1}}^{+},{\mathfrak{g}}_{\bar{0}}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}.$$

$$[V^{-}(\lambda )]=\sum _{\mu \in \Lambda }{p}^s{2}^t[V^{+}(\mu )],$$

$$[V^{-}(\lambda )]=[u(\mathfrak{g}){\otimes }_{u(\mathfrak{l})}[u(\mathfrak{l}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{l}}^{-})}{\mathbb{K}}_{\lambda }]]=\sum _{\mu \in \Lambda }{p}^{\alpha }{2}^{\beta }[u(\mathfrak{g}){\otimes }_{u(\mathfrak{l})}{\mathbb{K}}_{\mu }],$$

$$[u(\mathfrak{g}){\otimes }_{u(\mathfrak{h})}{\mathbb{K}}_{\lambda }]=[u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{\pm })}[u({\mathfrak{b}}_{\mathfrak{g}}^{\pm }){\otimes }_{u(\mathfrak{h})}{\mathbb{K}}_{\lambda }]]=\sum _{\mu \in \Lambda }{p}^{s_{\pm }}{2}^{t_{\pm }}[u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{\pm })}{\mathbb{K}}_{\mu }],$$

$$\sum _{\mu \in \Lambda }[u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{-})}{\mathbb{K}}_{\mu }]=\sum _{\mu \in \Lambda }{p}^{s_{+}-s_{-}}{2}^{t_{+}-t_{-}}[u(\mathfrak{g}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{g}}^{+})}{\mathbb{K}}_{\mu }].$$

$$p^{\dim (\mathfrak{h})}[V^{-}(\lambda )]=\sum _{\mu \in \Lambda }{p}^{s_{+}-s_{-}}{2}^{t_{+}-t_{-}}[V^{+}(\mu )],$$

$$[V^{-}(\lambda )]=\sum _{\mu \in \Lambda }{p}^{s_{+}-s_{-}-\dim (\mathfrak{h})}{2}^{t_{+}-t_{-}}[V^{+}(\mu )].$$

$$\mathfrak{g}={\mathfrak{g}}_{\bar{1}}^{-}\oplus {\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}\oplus {\mathfrak{g}}_{\bar{1}}^{+},{\mathfrak{g}}_{\bar{0}}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}.$$

$$[u(\mathfrak{l}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{l}}^{-})}{\mathbb{K}}_{\lambda }]=[u(\mathfrak{l}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{l}}^{+})}{\mathbb{K}}_{\psi (\lambda )}].$$

$$[V^{-}(\lambda )]=\sum _{\mu \in \Lambda }{p}^s{2}^t[V^{+}(\mu )],$$

$$[u({\mathfrak{b}}_{\mathfrak{g}}^{+}){\otimes }_{u({\mathfrak{b}}_{\mathfrak{l}}^{+})}{\mathbb{K}}_{\lambda }]=\sum _{\mu \in \Lambda }{p}^s{2}^t[{\mathbb{K}}_{\mu }],$$

$$\tau{(i)}=\left\{\begin{tabular}[]{ll}$0,$&$1\leq i\leq m;$\\ $1,$&$m+1\leq i\leq m+n.$\end{tabular}\right.$$

$$W(m,n,1)={\oplus }_{i\in \mathbb{Z}}W(m,n,1{)}_i$$

$$W(m,n,1{)}_i=\left\{\sum _{j=1}^{m+n}{f}_j{D}_j\mid |f_j|=i+1\right\}.$$

$$D_{ij}(f)=f_i{D}_i+f_j{D}_j$$

$$f_i=-(-1{)}^{\text{d}(f)(\tau (i)+\tau (j))}{D}_j(f),f_j=(-1{)}^{\tau (i)\tau (j)}{D}_i(f).$$

$$S(m,n,1)=⟨D_{ij}(f)\mid f\text{ is homogeneous},1\le i<j\le m+n⟩.$$

$$i^{\prime}=\left\{\begin{tabular}[]{ll}$i+l,$&$1\leq i\leq l,$\\ $i-l,$&$l+1\leq i\leq m,$\\ $i,$&$m<i\leq m+n;$\end{tabular}\right.\sigma(i)=\left\{\begin{tabular}[]{ll}$1,$&$1\leq i\leq l,$\\ $-1,$&$l+1\leq i\leq m,$\\ $1,$&$m<i\leq m+n.$\end{tabular}\right.$$

$$D_H:$$

$$\bar{H}(m,n,1)=⟨D_H(f)\mid f\text{ is homogeneous}⟩,$$

$$H(m,n,1)=[\bar{H}(m,n,1),\bar{H}(m,n,1)].$$

$$\tilde{\sigma }(\lambda )(h)=-\lambda (\sigma (h))$$

Full text

1. Introduction

A Lie superalgebra $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ over $\mathbb{K}$ is called restricted if $(\mathfrak{g}_{\bar{0}},[p])$ is a restricted Lie algebra with $p$-mapping $[p]:\mathfrak{g}_{\bar{0}}\rightarrow\mathfrak{g}_{\bar{0}}$ and $\mathfrak{g}_{\bar{1}}$ is a restricted $\mathfrak{g}_{\bar{0}}$ module via the adjoint action (cf. [5]). Let $(\mathfrak{g},[p])$ be a restricted Lie superalgebra and $U(\mathfrak{g})$ be the enveloping superalgebra of $\mathfrak{g}.$ One can define the so-called restricted enveloping superalgebra $u(\mathfrak{g})=U(\mathfrak{g})/I_{p}$ where $I_{p}$ is the $\mathbb{Z}_{2}$-graded two-sided ideal generated by $\{x^{p}-x^{[p]}\mid x\in\mathfrak{g}_{\bar{0}}\}.$ A $\mathfrak{g}$ supermodule $(V=V_{\bar{0}}\oplus V_{\bar{1}},\rho)$ is called restricted if $\rho$ satisfies $\rho(x^{[p]})=\rho(x)^{p}$ for all $x\in\mathfrak{g}_{\bar{0}}.$ All restricted $\mathfrak{g}$-supermodules constitute a full subcategory of the $\mathfrak{g}$-supermodule category which coincide with the $u(\mathfrak{g})$-supermodule category denoted by $u(\mathfrak{g})$-smod. We call $u(\mathfrak{g})$ is of one block if $u(\mathfrak{g})$-smod is of one block.

Over the past decades, the study of modular representations of restricted Lie (super)algebras in prime characteristic has made significant progress (see [3, 4, 7, 8, 9] for examples). When $\mathfrak{g}=W(0,n)$ over $\mathbb{C},$ Shomron proves in [6] that the category of finite-dimensional representations decomposes into blocks parametrized by $(\mathbb{C}/\mathbb{Z})\times\mathbb{Z}_{2}$. In contrast to complex case, if either $\mathfrak{g}=X(m,1)$ is a Cartan type Lie algebra where $X\in\{W,S,H,K\}$ ([3]) or $\mathfrak{g}=W(0,n,1)$ is a Cartan type Lie superalgebra ([7]) over $\mathbb{K}$, the category of restricted (super)modules has only one block. In this paper, we generalize this degeneracy phenomenon of restricted supermodules to the so-called restricted Cartan type Lie superalgebras $X(m,n,1)$ where $X\in\{W,S,H\}.$

Our paper is organized as follows. In section 2, we illustrate a class of Lie superalgebras over $\mathbb{K}$ such that the category of restricted supermodules is of one block. Section 3 is concerned with the structure of the Cartan type Lie superalgebras. Applying the results in section 2, we obtain the following main theorem in section 4:

Theorem 1.1**.**

$\operatorname{(see\ Theorem\ \ref{main thm})}$* Let $\mathbb{K}$ be an algebraically closed field with characteristics $p>3,$ and $\mathfrak{g}=X(m,n,1),\ X\in\{W,S,H\},$ be a graded restricted Lie superalgebra of Cartan type over $\mathbb{K}$ except if $X=H$ with $n=4.$*

Then $u(\mathfrak{g})$ is of one block.

As I know, F.Duan, B.Shu and Y.Yao obtain similar results in [2] by a different method.

Entire the whole paper, denote $I=\{0,1,\cdots,p-1\}.$

2. Restricted Lie superalgebras with triangular Decomposition

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a restricted Lie superalgebra which admits a triangular decomposition relative to a maximal torus $\mathfrak{h}$ of $\mathfrak{g}_{\bar{0}}:$

[TABLE]

where $\mathfrak{g}_{\bar{0}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}.$ Recall that $\mathfrak{n}^{\pm}$ are $p$-nilpotent subalgebras. Set $\mathfrak{b}_{\mathfrak{g}}^{\pm}=\mathfrak{g}_{\bar{1}}^{\pm}\oplus\mathfrak{n}^{\pm}\oplus\mathfrak{h}$ and $\mathfrak{b}_{\mathfrak{g}_{\bar{0}}}^{\pm}=\mathfrak{n}^{\pm}\oplus\mathfrak{h}.$ Analogue to [3], this decomposition for $\mathfrak{g}$ is long if

[TABLE]

By [9], the iso-classes of simple restricted $\mathfrak{g}$ modules are parametrized by restricted weights $\Lambda=\{\lambda\in\mathfrak{h}^{*}\mid\lambda(h^{[p]})=\lambda(h)^{p},\ \forall h\in\mathfrak{h}\}.$ If $\text{dim}(\mathfrak{h})=n,$ then $\Lambda\simeq I^{n}=\{\lambda=(\lambda_{1},\cdots,\lambda_{n})\mid\lambda_{i}\in I,i=1,\cdots,n\}.$ More precisely, for a given $\lambda\in\Lambda,$ there is a one-dimensional restricted $\mathfrak{b}_{\mathfrak{g}}^{+}$ module $\mathbb{K}_{\lambda}=\mathbb{K}\cdot 1_{\lambda}$ on which $\mathfrak{h}$ acts as a scalar determined by $\lambda$ while $\mathfrak{g}_{\bar{1}}^{+}\oplus\mathfrak{n}^{+}$ acts trivially. Then one has the so-called baby Verma module

[TABLE]

with simple head $L(\lambda).$ Moreover, For any restricted simple module $\mathfrak{m},$ there is a $\lambda\in\Lambda,$ such that $V^{+}(\lambda)\twoheadrightarrow\mathfrak{m}$ (cf. [9]).

Note that $\mathfrak{b}_{\mathfrak{g}}^{-}$ also satisfies the conditions of [9, lemma 2.2], then for each $\lambda\in\Lambda,$ the one-dimensional $u(\mathfrak{b}_{\mathfrak{g}}^{-})$ module $\mathbb{K}_{\lambda}$ induces an $u(\mathfrak{g})$ module

[TABLE]

which is indecomposable with simple head.

For $M\in u(\mathfrak{g})$-smod, let $[M]$ denote the formal sum of composition factors in the Grothendick ring of $u(\mathfrak{g})$-smod.

Lemma 2.1**.**

Let $\mathfrak{l}=\mathfrak{l}_{\bar{0}}\oplus\mathfrak{l}_{\bar{1}}$ be a restricted Lie superalgebra which admits a triangular decomposition relative to a maximal torus $\mathfrak{h}$ of $\mathfrak{l}_{\bar{0}}:$

[TABLE]

where $\mathfrak{l}_{\bar{0}}=\mathfrak{n}^{-}_{\mathfrak{l}}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}_{\mathfrak{l}}.$

Assume the following:

1. (1)

$\mathfrak{l}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}_{\mathfrak{l}}\oplus\mathfrak{n}^{+}_{\mathfrak{l}}\oplus\mathfrak{l}_{\bar{1}}^{+}$* is a $p$-nilpotent $\mathbb{Z}_{2}$-graded ideal.* 2. (2)

$\mathfrak{n}^{+}_{\mathfrak{l}}$* contains $\textup{dim}(\mathfrak{h})$ linear independent weight vectors having linearly independent weights in $\Lambda.$*

Then for each $\lambda\in\Lambda,$ $[V^{-}(\lambda)]$ is independent of $\lambda$ and

[TABLE]

where $s=\textup{dim}(\mathfrak{n}^{+}_{\mathfrak{l}})-\textup{dim}(\mathfrak{h}),\ t=\textup{dim}(\mathfrak{l}_{\bar{1}}^{+})$ and $\mathbb{K}_{\mu}$ is the one dimensional simple $u(\mathfrak{l})$ module of weight $\mu.$

Proof.

By (1), $\text{rad}(\mathfrak{l})=\mathfrak{l}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}.$ Since $\text{rad}(\mathfrak{l})$ is $p$-nilpotent and finite dimension, each restricted representations of $\mathfrak{l}$ is one dimension [9, lemma 2.2]. Let $\{\mathbb{K}_{\mu}\mid\mu\in\Lambda\}$ represent the set of non-isomorphic simple $u(\mathfrak{l})$ modules.

The composition factors of a module can be obtained by computing its weight spaces. From (2), suppose $n=\text{dim}(\mathfrak{h}),$ $\mathfrak{l}_{\bar{1}}^{+}$ has basis $\{z_{1},\cdots,z_{t}\}$ and $\mathfrak{n}^{+}$ has basis $\{x_{1},\cdots,x_{n},y_{1},\cdots,y_{s}\}$ where $x_{i}$ is of weight $\alpha_{i}\in\Lambda$ for each $i=1,\cdots,n$ such that $\alpha_{1},\cdots,\alpha_{n}$ are linear independent. Then $x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$ has weight $i_{1}\alpha_{1}+\cdots+i_{n}\alpha_{n}$ for each $(i_{1},\cdots,i_{n})\in I^{n}.$

For each choice of $\underline{j}\in I^{s}$ and $\underline{k}\in B(t),$ as $u(\mathfrak{h})$ module,

[TABLE]

must have all weights occurring with multiplicity 1 . Since

[TABLE]

then all possible weights occuring with the same multiplicity $p^{s}2^{t}$ in $V^{-}(\lambda).$

Namely, $[V^{-}(\lambda)]=\sum_{\mu\in\Lambda}p^{s}2^{t}[\mathbb{K}_{\mu}]$ which is independent of $\lambda.$ ∎

Proposition 2.2**.**

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a restricted Lie superalgebra which admits a long triangular decomposition relative to a maximal torus $\mathfrak{h}$ of $\mathfrak{g}_{\bar{0}}:$

[TABLE]

Assume the following:

1. (1)

$\mathfrak{g}$* has a restricted subalgebra $\mathfrak{l}$ satisfies the assumptions of lemma 2.1.* 2. (2)

$\mathfrak{b}^{-}_{\mathfrak{l}}=\mathfrak{b}_{\mathfrak{g}}^{-}.$** 3. (3)

$\mathfrak{n}^{-}_{\mathfrak{l}}=\mathfrak{n}^{-}$* has at least $\operatorname{dim}(\mathfrak{h})$ linearly independent vectors having linearly independent weights in $\Lambda.$*

Then for each $\lambda\in\Lambda,$

[TABLE]

where $s=\textup{dim}(\mathfrak{n}^{+})-\textup{dim}(\mathfrak{n}^{-})-\textup{dim}(\mathfrak{h}),$ $t=\textup{dim}(\mathfrak{g}_{\bar{1}}^{+})-\textup{dim}(\mathfrak{g}_{\bar{1}}^{-}).$

Proof.

By lemma 3.1, for each $\lambda\in\Lambda,$

[TABLE]

where $\alpha=\textup{dim}(\mathfrak{l})-\dim(\mathfrak{n}^{-}),\ \beta=\dim(\mathfrak{g}_{\bar{1}}^{+}).$

In particular, $[V^{-}(\lambda)]$ is independent of $\lambda.$

By assumption (3), $\mathfrak{b}_{\mathfrak{g}}^{\pm}$ satisfies the assumptions of lemma 3.1. Therefore,

[TABLE]

where $s_{\pm}=\dim(\mathfrak{n}^{\pm}),\ t_{\pm}=\dim(\mathfrak{g}_{\bar{1}}^{\pm}).$ Since the triangular decomposition is long, i.e. $s_{+}>s_{-}\textup{ and }t_{+}>t_{-},$ we have

[TABLE]

Note that $[V^{-}(\lambda)]$ is independent of $\lambda.$ Therefore, for all $\lambda\in\Lambda,$

[TABLE]

[TABLE]

∎

Proposition 2.3**.**

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a restricted Lie superalgebra which admits a triangular decomposition relative to a maximal torus $\mathfrak{h}$ of $\mathfrak{g}_{\bar{0}}:$

[TABLE]

Assume that there exists a subalgebra $\mathfrak{l}$ such that:

1. (1)

$\mathfrak{b}^{-}_{\mathfrak{l}}=\mathfrak{b}_{\mathfrak{g}}^{-}.$** 2. (2)

$\mathfrak{l}$* is a classical Lie superalgebra and there is a bijection $\psi:\Lambda\rightarrow\Lambda,$ such that*

[TABLE] 3. (3)

A vector space complementary to $\mathfrak{l}$, in $\mathfrak{g},$ has at least $\operatorname{dim}(\mathfrak{h})$ linearly independent vectors having linearly independent weights in $\Lambda.$

Then

[TABLE]

where $s=\textup{dim}(\mathfrak{n}^{+})-\textup{dim}(\mathfrak{n}^{-})-\textup{dim}(\mathfrak{h}),t=\textup{dim}(\mathfrak{g}_{\bar{1}}^{+})-\textup{dim}(\mathfrak{g}_{\bar{1}}^{-}).$

Proof.

Similar to the proof of lemma 2.1, for all $\lambda\in\Lambda,$ we have

[TABLE]

where $s$ and $t$ are defined in proposition.

By assumption (1) and (2), we have

[TABLE]

Proposition holds. ∎

Remark 2.4*.*

The Lie algebra version of lemma 2.1 and proposition 2.2 (resp. proposition 2.3) are investigated in [3] (resp. [4]).

Corollary 2.5**.**

If $\mathfrak{g}$ is a restricted Lie superalgebra satisfies all assumptions in proposition 2.2 or 2.3, then $u(\mathfrak{g})$ is of one block.

Proof.

Note that $V^{-}(\lambda)$ is indecomposable with simple head for all $\lambda\in\Lambda.$ The proof of corollary 2.4 in [3] still works. Hence, the corollary holds. ∎

3. Restricted Cartan Type Lie Superalgebras

For given positive integers $m$ and $n,$ put

[TABLE]

For $0\leq k\leq n,$ set $\mathbb{B}_{k}=\{(i_{1},\cdots,i_{k})\mid m+1\leq i_{1}<\cdots<i_{k}\leq m+n\}$ and $\mathbb{B}(n)=\cup_{i=0}^{n}\mathbb{B}_{k}$ where $\mathbb{B}_{0}=\emptyset.$

Let $A(m,1)$ denote the truncated divided power algebra over $\mathbb{K}$ with a basis $\{x^{(\alpha)}\mid\alpha\in I^{m}\}.$ For $\epsilon_{i}=(\delta_{i1},\cdots,\delta_{im}),$ we abbreviate $x^{(\epsilon_{i})}$ to $x_{i},\ i=1,\cdots,m.$ Let $\Lambda(n)$ be the Grassmann superalgebra over $\mathbb{K}$ in $n$ variables $x_{m+1},\cdots,x_{m+n}$ with basis $\{x^{(\beta)}\mid\beta\in\mathbb{B}(n)\}$ where $x^{(\beta)}=x_{i_{1}}\cdots x_{i_{k}}$ if $\beta=(i_{1},\cdots,i_{k}).$ Denote the tensor product by $A(m,n,1)=A(m,1)\otimes\Lambda(n).$ Then $A(m,n,1)$ is an associative superalgebra with a $\mathbb{Z}_{2}$-gradation induced by the trivial $\mathbb{Z}_{2}$-gradation of $A(m,1)$ and the natural $\mathbb{Z}_{2}$-gradation of $\Lambda(n).$ Denote $\text{d}(f)$ the parity of $f\in A(m,n,1).$

Let $D_{1},\cdots,D_{m+n}$ be the superderivations of the superalgebra $A(m,n,1)$ such that $D_{i}(x_{j})=\delta_{ij}$ for $1\leq i,j\leq m+n.$ Define $W(m,n,1)=\left\{\sum_{i=1}^{m+n}f_{i}D_{i}\mid f_{i}\in A(m,n,1),1\leq i\leq m+n\right\}.$

Then $W(m,n,1)$ is a restricted Lie superalgebra of Witt type. The $\mathbb{Z}$-grading of

[TABLE]

is induced by $\lvert x_{i}\rvert=1$ and $\lvert D_{i}\rvert=-1$ for all $1\leq i\leq m+n.$ Namely,

[TABLE]

For each pair $1\leq i,j\leq m+n$ defines $D_{ij}:A(m,n,1)\rightarrow W(m,n,1)$ by

[TABLE]

where $f$ is homogeneous and

[TABLE]

The special superalgebra $S(m,n,1)$ is defined by

[TABLE]

$S(m,n,1)$ is a $\mathbb{Z}$-graded restricted subalgebra of $W(m,n,1).$ The $\mathbb{Z}$-grading structure is given by $S(m,n,1)_{i}:=S(m,n,1)\cap W(m,n,1)_{i}.$

Next we define the Hamiltonian type Lie superalgebra $H(m,n,1),$ where $m=2l$ is even and $n>3.$ Let

[TABLE]

The Hamiltonian operator $D_{H}$ is defined as follows:

[TABLE]

where $f$ is homogeneous and $f_{i}=\sigma(i^{\prime})(-1)^{\tau(i^{\prime})\text{d}(f)}D_{i^{\prime}}(f).$

The Hamiltonian superalgebra $H(m,n,1)$ is defined by

[TABLE]

[TABLE]

$H(m,n,1)$ is a $\mathbb{Z}$-graded restricted subalgebra of $W(m,n,1).$ The $\mathbb{Z}$-grading structure is given by $H(m,n,1)_{i}:=H(m,n,1)\cap W(m,n,1)_{i}.$

4. Blocks of Cartan Type Lie Superalgebra

Entire this section, assume $p>3.$

4.1. Type W

For $W(m,n,1),$ there is no subalgebra $\mathfrak{l}$ satisfying the hypothesis of proposition 3.2 (in fact, assumption (3) fails). Hence, we need proposition 3.3.

Let $\mathfrak{l}$ be a restricted Lie superalgebra of classical type with triangular decomposition $\mathfrak{l}=\mathfrak{l}^{-}\oplus\mathfrak{h}\oplus\mathfrak{l}^{+}$ with respect to $\mathfrak{h}.$ Suppose $\sigma$ is an even restricted automorphism of $\mathfrak{l}$ such that $\sigma(\mathfrak{h})\subseteq\mathfrak{h}.$ Then it induces $\tilde{\sigma}:\mathfrak{h}^{*}\rightarrow\mathfrak{h}^{*}$ by

[TABLE]

where $\lambda\in\mathfrak{h}^{*},\ h\in\mathfrak{h}.$ Moreover, $\tilde{\sigma}(\Lambda)\subseteq\Lambda.$

Denote $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{l}^{+}$ a solvable subalgebra of $\mathfrak{l}$ and $V(\lambda)=u(\mathfrak{l})\otimes_{u(\mathfrak{b})}\mathbb{K}_{\lambda}$ the baby Verma module where $\mathbb{K}_{\lambda}$ is a one-dimensional $u(\mathfrak{b})$ module with weight $\lambda.$ Let $V^{\sigma}(\lambda)$ be the twisted baby Verma module. Namely, $V^{\sigma}(\lambda)\simeq V(\lambda)$ as vector spaces while $x\cdot m:=\sigma(x)(m)$ for all $x\in u(\mathfrak{l}),\ m\in V^{\sigma}(\lambda).$

The following lemma is a straightforward calculation.

Lemma 4.1**.**

Keep assumptions as above, $V^{\sigma}(\lambda)\simeq u(\mathfrak{l})\otimes_{u(\sigma^{-1}(\mathfrak{b}))}1_{-\tilde{\sigma}(\lambda)}$ by sending $x\otimes 1_{\lambda}$ to $\sigma^{-1}(x)\otimes 1_{-\tilde{\sigma}(\lambda)}.$ In particular, $[V(\lambda)]=[V^{\sigma}(\lambda)]=[u(\mathfrak{l})\otimes_{u(\sigma^{-1}(\mathfrak{b}))}1_{-\tilde{\sigma}(\lambda)}].$

Let $\mathfrak{g}=W(m,n,1)=\mathfrak{g}^{-}\oplus\mathfrak{h}\oplus\mathfrak{g}^{+}$ be the triangular decomposition related to maximal torus $\mathfrak{h}=\langle h_{1},\cdots,h_{m+n}\rangle$ where $h_{i}:=x_{i}D_{i}$ for $i=1,\cdots,m+n.$

Now, set

[TABLE]

where $p_{i}=x_{i}\sum_{j=1}^{m+n}x_{j}D_{j}\in\mathfrak{g}_{1}.$

Thanks to [1, lemma 3.1], $\mathfrak{l}\simeq\mathfrak{pgl}(m+1|n).$ Let $e_{i}:=E_{i,i+1},\ f_{i}:=E_{i+1,i}.$ Then $\{e_{i},f_{i}\mid i=1,\cdots,m+n-1\}$ generates $\mathfrak{pgl}(m+1|n).$ There is an even restricted automorphism $\alpha$ of $\mathfrak{l}$ induced by $\alpha(e_{i})=f_{i}$ and $\alpha(f_{i})=e_{i}.$ Note that $\alpha(\mathfrak{b}_{\mathfrak{l}}^{\pm})=\mathfrak{b}_{\mathfrak{l}}^{\mp}$ and $\tilde{\alpha}$ keeps $\Lambda.$ By lemma 4.1, we have

[TABLE]

Therefore, $\mathfrak{l}$ is a subalgebra satisfying (1) and (2) of proposition 3.3.

For each $i=1,\cdots,m+n,\ x_{i}^{3}D_{i}$ has weight $2\gamma_{i}$ where $\gamma_{i}\in\Lambda$ such that $\gamma_{i}(h_{j})=\delta_{ij}.$ Therefore, assumption (3) of proposition 3.3 satisfies.

To sum above up, all assumptions of proposition 3.3 hold for $W(m,n,1)$ and hence we have the following proposition by corollary 3.4.

Proposition 4.2**.**

Keep assumptions as above, and let $\mathfrak{g}=W(m,n,1),$ then $u(\mathfrak{g})$ is of one block.

4.2. Type S

Let $\mathfrak{g}=S(m,n,1)=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}\oplus\mathfrak{g}_{\bar{1}}^{+}$ be the triangular decomposition related to maximal torus $\mathfrak{h}=\langle h_{i}\mid 1\leq i\leq m+n-1\rangle$ where $h_{i}:=x_{i}D_{i}-x_{i+1}D_{i+1},$ and $\mathfrak{g}_{\bar{0}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}.$

Set $\mathfrak{l}=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}$ where $\mathfrak{n}_{1}^{+}=\mathfrak{s}\oplus\mathfrak{t},$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

One can check the followings:

• •

$[\mathfrak{h},\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}]\subseteq\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+};$

• •

$\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}$ is $p$-nilpotent;

• •

$[\mathfrak{n}^{-},\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}]\subseteq\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+};$ $[\mathfrak{g}_{\bar{1}}^{-},\mathfrak{n}_{1}^{+}]\subseteq\mathfrak{l}_{\bar{1}}^{+},\ [\mathfrak{g}_{\bar{1}}^{-},\mathfrak{l}_{\bar{1}}^{+}]=0;$

• •

$[\mathfrak{s},\mathfrak{s}]=[\mathfrak{t},\mathfrak{t}]=[\mathfrak{s},\mathfrak{l}_{\bar{1}}^{+}]=[\mathfrak{l}_{\bar{1}}^{+},\mathfrak{l}_{\bar{1}}^{+}]=0,\ [\mathfrak{s},\mathfrak{t}]\subseteq\mathfrak{s},$ and $[\mathfrak{t},\mathfrak{l}_{\bar{1}}^{+}]\subseteq\mathfrak{l}_{\bar{1}}^{+}.$

Therefore, $\text{rad}(\mathfrak{l})=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}$ is a $p$-nilpotent ideal and $\mathfrak{l}$ is a subalgebra satisfying (1) in lemma 3.1.

Now define $\gamma_{i}\in\Lambda$ by $\gamma_{i}(h_{j})=\delta_{ij},$ $1\leq i,j\leq m+n-1.$

For $1\leq i\leq m$ and $1\leq j\leq n-1,$ $D_{i}$ has weight $-\gamma_{i}$ while $x_{m+j}D_{m+j+1}$ has weight $-\gamma_{m+j-1}+2\gamma_{m+j}-(1-\delta_{j,n-1})\gamma_{m+j+1}$ with respect to $\mathfrak{h}.$ One can check that $\mathfrak{n}^{-}$ contains $m+n-1$ linear independent vectors

[TABLE]

with linear independent weights. Therefore, assumption (2) and (3) of proposition 3.2 satisfy.

For $1\leq i\leq m-1,\ x_{i}^{2}D_{m}$ has weight $2\gamma_{i}-2(1-\delta_{1,i})\gamma_{i-1}+\gamma_{m-1}-\gamma_{m}$ while $x_{1}^{3}D_{m}$ has weight $3\gamma_{1}+\gamma_{m-1}-\gamma_{m}.$

For $1\leq j\leq n-1,\ x_{1}x_{m+j}D_{m+n}$ has weight $\gamma_{1}+\gamma_{m+j-1}-\gamma_{m+j}-\gamma_{m+n-1}.$

Hence, $\mathfrak{n}^{+}_{1}$ contains $m+n-1$ linear independent vectors

[TABLE]

with linear independent weights. Assumption (2) of lemma 3.1 satisfies.

To sum above up, all assumptions of proposition 3.2 hold for $S(m,n,1)$ and hence we have the following proposition by corollary 3.4.

Proposition 4.3**.**

Keep assumptions as above, and let $\mathfrak{g}=S(m,n,1),$ then $u(\mathfrak{g})$ is of one block.

4.3. Type H

Let $\mathfrak{g}:=H(m,n,1),$ where $m=2l,\ n>3.$ Denote $k=[n/2].$ For every $(a)=(a_{1},\cdots,a_{m})\in I^{m}$ and $(b)=(b_{1},\cdots,b_{u})\in\mathbb{B}_{u}\subseteq\mathbb{B}(n),$ denote

[TABLE]

By definition, $\mathfrak{g}=\langle D_{H}(X^{(a)}Y^{(b)})\mid X^{(a)}Y^{(b)}\neq x_{1}^{p-1}\cdots x_{m}^{p-1}x_{m+1}\cdots x_{m+n}\rangle.$

Fix a maximal torus $\mathfrak{h}$ with basis

[TABLE]

where $h_{i}=D_{H}(x_{i}x_{l+i}),\ h_{m+j}=D_{H}(\sqrt{-1}x_{m+j}x_{m+k+j}).$

For $1\leq i\leq k,$ set $e_{i}:=x_{m+i}+\sqrt{-1}x_{m+k+i}\text{ and }f_{i}:=x_{m+i}-\sqrt{-1}x_{m+k+i}.$ Then both $D_{H}(e_{i})$ and $D_{H}(f_{i})$ are homogeneous odd elements of degree $-1$.

Define the following subspaces of $\mathfrak{g}:$

[TABLE]

Suppose $\mathfrak{g}=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}\oplus\mathfrak{g}_{\bar{1}}^{+}$ be the triangular decomposition related to maximal torus $\mathfrak{h}$ and $\mathfrak{g}_{\bar{0}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}.$

One can check that $\mathfrak{n}^{-}=\mathfrak{n}_{1}^{-}\oplus\mathfrak{n}_{2}^{-}$ where

[TABLE]

Remark 4.4*.*

Above description for $\mathfrak{n}_{2}^{-}$ comes from [8, section 2.2].

For each $(c)=(c_{1},\cdots,c_{u})\in\mathbb{B}(k),\ 0\leq u\leq k,$ denote $f^{(c)}=f_{c_{1}}\cdots f_{c_{u}}.$ In this case, the parity of $D_{H}(f^{(c)})$ equals to the parity of $u=\lvert(c)\rvert.$

Now, if $n=2k$ is even, define $\mathfrak{l}^{+}=\langle D_{H}(x^{(a)}f^{(c)})\mid a_{j}=0\text{ if }1\leq j\leq l;\lvert(a)\rvert+\lvert(c)\rvert\geq 3\rangle.$

If $n=2k+1$ is odd, define $\mathfrak{l}^{+}=\langle D_{H}(x^{(a)}f^{(c)}x_{m+n}^{\delta})\mid a_{j}=0\text{ if }1\leq j\leq l;\delta\in\{0,1\};\lvert(a)\rvert+\lvert(c)\rvert+\delta\geq 3\rangle.$

Let $\mathfrak{n}_{1}^{+}$ (resp. $\mathfrak{l}_{\bar{1}}^{+}$) be the even (resp. odd) part of $\mathfrak{l}^{+}.$

One can check that $\mathfrak{l}:=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}$ is a restricted subalgebra of $\mathfrak{g}.$ Note that for all $f,g\in A(m,n,1)$ homogeneous,

[TABLE]

[TABLE]

We have the followings:

• •

$[D_{H}(e_{i}),D_{H}(e_{j})]=[D_{H}(f_{i}),D_{H}(f_{j})]=0;$

• •

$[D_{H}(e_{i}),D_{H}(f_{j})]=[D_{H}(f_{j}),D_{H}(e_{i})]=-2\delta_{ij};$

• •

$[\mathfrak{h},\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}]\subseteq\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+};$

• •

$\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}$ is $p$-nilpotent;

• •

$[\mathfrak{n}^{-},\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}]\subseteq\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+};$ $[\mathfrak{g}_{\bar{1}}^{-},\mathfrak{n}_{1}^{+}]\subseteq\mathfrak{l}_{\bar{1}}^{+},\ [\mathfrak{g}_{\bar{1}}^{-},\mathfrak{l}_{\bar{1}}^{+}]=0;$

• •

$[\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+},\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}]=0.$

Therefore, $\text{rad}(\mathfrak{l})=\mathfrak{g}_{\bar{1}}^{-}\oplus\mathfrak{n}^{-}\oplus\mathfrak{n}_{1}^{+}\oplus\mathfrak{l}_{\bar{1}}^{+}$ is a $p$-nilpotent ideal and $\mathfrak{l}$ is a subalgebra satisfying (1) in lemma 3.1.

For each $1\leq i,j\leq l,\ 1\leq u,v\leq k,$ defines $\gamma_{i},\delta_{j}\in\Lambda$ by $\gamma_{i}(h_{j})=\delta_{ij},$ and $\delta_{u}(h_{m+v})=\delta_{uv}.$

For $1\leq i\leq l$ and $1\leq j\leq k,$ $D_{i}$ has weight $-\gamma_{i}$ while $D_{H}(e_{j})$ (resp. $D_{H}(f_{j})$) has weight $\delta_{j}$ (resp. $-\delta_{j}$) with respect to $\mathfrak{h}.$ Then $b_{uv}$ has weight $-\delta_{u}-\delta_{v}$ for each $1\leq u,v\leq k$ and $a_{12}$ has weight $-\delta_{1}+\delta_{2}.$

Therefore, $\mathfrak{n}^{-}$ contains $l+k$ linear independent vectors

[TABLE]

with linear independent weights. Assumption (2) and (3) of proposition 3.2 satisfies.

For $1\leq i\leq l,\ D_{H}(x_{l+i}^{3})$ has weight $3\gamma_{i}.$ For $1\leq u<v\leq k-1,\ D_{H}(x_{l+1}^{2}b_{u,v})$ has weight $2\gamma_{1}-\delta_{u}-\delta_{v},$ and $D_{H}(x_{l+1}^{2}f_{k}x_{m+n})$ has weight $2\gamma_{1}-\delta_{k}$ if $n=2k+1.$ Moreover, $\mathfrak{n}^{+}$ contains $l+k$ linear independent vectors $S$ with linear independent weights as following:

If $k\geq 3$ is odd,

[TABLE]

If $k\geq 3$ is even,

[TABLE]

If $n=5,$

[TABLE]

Therefore, assumption (2) of lemma 3.1 satisfies if $n>4.$

Proposition 4.5**.**

Keep assumptions as above, and let $\mathfrak{g}=H(m,n,1)$ with $n>4,$ then $u(\mathfrak{g})$ is of one block.

By Proposition 4.2, 4.3 and 4.4, we have our main theorem as follows.

Theorem 4.6**.**

Let $\mathbb{K}$ be an algebraically closed field with characteristics $p>3,$ and $\mathfrak{g}=X(m,n,1),\ X\in\{W,S,H\},$ be a graded restricted Lie superalgebra of Cartan type over $\mathbb{K}$ except if $X=H$ with $n=4.$

Then $u(\mathfrak{g})$ is of one block.