Gradings on the Kac superalgebra

Group gradings on superalgebras

In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.

Group gradings on the Lie and Jordan superalgebras $Q(n)$

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism.

Derivations of the Cheng-Kac Jordan superalgebras

The derivations of the Cheng-Kac Jordan superalgebras are studied. It is shown that, assuming -1 is a square in the ground field, the Lie superalgebra of derivations of a Cheng-Kac Jordan superalgebra is isomorphic to the Lie superalgebra obtained from a simpler Jordan superalgebra (a Kantor double superalgebra of vector type) by means of the Tits-Kantor-Koecher construction. This is done by exploiting the S4-symmetry of the Cheng-Kac Jordan superalgebra.

经典李超代数的群分次

Group gradings on classical Lie superalgebras

本文在特征零代数闭域上对非例外经典单李超代数的群分次进行了同构分类，通过研究满足降链条件的分次单结合超代数，实现了对有限维单结合超代数阿贝尔群分次的分类。

The Kac Jordan superalgebra: automorphisms and maximal subalgebras

The group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras.

Leibniz superalgebras with a set grading

Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of ${\mathfrak L}$ being of maximal length, the simplicity of ${\mathfrak L}$ is also characterized in terms of connections.

例外单李超代数上的群分次

Group gradings on exceptional simple Lie superalgebras

本文在特征0代数闭域上同构分类了例外单李超代数$G(3)$、$F(4)$、$D(2,1;α)$及$A(1,1)$上的任意群分次，核心方法是将Elduque等的新方法应用于已知的精细分次等价分类。

Irreducible representations of the exceptional cheng-kac superalgebra

We classify all conformal irreducible modules of finite type over the Cheng Kac superalgebra CK(6).

On Kac's Jordan superalgebra

The group-scheme of automorphisms of the ten-dimensional exceptional Kac's Jordan superalgebra is shown to be isomorphic to the semidirect product of the direct product of two copies of SL2 by the constant group scheme C2. This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.

特征5下通过张量范畴从Albert代数到Kac十维Jordan超代数

From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5

本研究在特征5域下通过张量范畴半单化，由Albert代数与5阶自同构得到Kac十维Jordan超代数，解释了相关奇特结果，还得到特征5特有李超代数el(5;5)并给出超代数构造的一般方法。

On Kac-Weisfeiler modules for general and special linear Lie superalgebras

Let $\ggg:=\gl_{m|n}$ be a general linear Lie superalgebra over an algebraically closed field $\mathds{k}=\overline{\mathbb{F}}_p$ of characteristic $p>2$. A module of $\ggg$ is said to be of Kac-Weisfeiler if its dimension coincides with the dimensional lower bound in the super Kac-Weisfeiler property presented by Wang-Zhao in \cite{WZ}. In this paper, we verify the existence of the Kac-Weisfeiler modules for $\gl_{m|n}$. We also establish the corresponding consequence for the special linear Lie superalgebras $\mathfrak{sl}_{m|n}$ with restrictions $p>2$ and $p\nmid(m-n)$.

Gradings and Symmetries on Heisenberg type algebras

We describe the fine (group) gradings on the Heisenberg algebras, on the Heisenberg superalgebras and on the twisted Heisenberg algebras. We compute the Weyl groups of these gradings. Also the results obtained respect to Heisenberg superalgebras are applied to the study of Heisenberg Lie color algebras.

分级嵌入、根生成子代数与 $π$-系统在拟简单Kac-Moody超代数中的应用

Graded embeddings, root generated subalgebras and $π$-systems for quasisimple Kac-Moody superalgebras

本文研究了拟简单Kac-Moody超代数中根生成子代数与$π$-系统的关系，通过反射操作获得实根并建立Dynkin双射的类比，同时探讨了超设置中根字符串的性质。

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of…

Automorphisms and superalgebra structures on the Grassmann algebra

Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$ admits. By using the duality between superalgebras and automorphisms of order $2$ we prove that in many cases the $\mathbb{Z}_{2}$-graded polynomial identities for such structures coincide with the $\mathbb{Z}_{2}$-graded polynomial identities of the "typical" cases $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ where the vector space $L$ is homogeneous. Recall that…

Lie superalgebras of Krichever-Novikov type and their central extensions

Classically important examples of Lie superalgebras have been constructed starting from the Witt and Virasoro algebra. In this article we consider Lie superalgebras of Krichever-Novikov type. These algebras are multi-point and higher genus equivalents. The grading in the classical case is replaced by an almost-grading. The almost-grading is determined by a splitting of the set of points were poles are allowed into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to an almost-grading, it is shown that there is up to rescaling and equivalence a unique non-tri…

Wedderburn principal theorem for Jordan superalgebras I

We consider finite dimensional Jordan superalgebras $\jor$ over an algebraically closed field of characteristic 0, with solvable radical $\rad$ such that $\radd=0$ and $\jor/\rad$ is a simple Jordan superalgebra of one of the following types: Kac $\kac$, Kaplansky $\mathcal{K}_3$ superform or $\algdt$. We prove that an analogue of the Wedderburn Principal Theorem (WPT) holds if certain restrictions on the types of irreducible subsuperbimodules of $N$ are imposed, where $N$ is considered as a $J$-superbimodule, and $J$ is a simple Jordan superalgebra. Using counterexamples, it is shown that the…

Homogeneous irreducible supermanifolds and graded Lie superalgebras

A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation $\mathrm{ad}_{\mathfrak{g}^0}|_{\mathfrak{g}^{-1}}$ is irreducible and $\mathfrak{g}^1 \neq (0)$. An example is the full prolongation of an irreducible linear Lie superalgebra $\mathfrak{g}^0 \subset \mathfrak{gl}(\mathfrak{g}^{-1})$ of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra $\mathfrak{g}$ w…

格拉斯曼代数中 $\mathbb{Z}_2$-分次与自同构的相互作用：综述

Interplay between $\mathbb{Z}_2$-gradings and automorphisms in the Grassmann algebra: a survey

本文综述了特征不为二的域上格拉斯曼代数的二阶自同构及其诱导的超代数结构，给出非齐次自同构实例，并提出超代数结构分类猜想。

The minimal number of generators for simple Lie superalgebras

Using the classification theorem due to Kac we prove that any finite dimensional simple Lie superalgebra over an algebraically closed field of characteristic 0 is generated by one element.

几乎单的代数超群

Almost-simple algebraic supergroups

本文研究了几乎单的代数超群，构造了新的超群，其李超代数不在Kac定理列表中。