Lie Superalgebras Graded by the Root System $A(m,n)$

Structures of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras

This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.

Graded identities of some simple Lie superalgebras

We study $\mathbb{Z}_2$-graded identities of Lie superalgebras of the type $b(t), t\ge 2$, over a field of characteristic zero. Our main result is that the $n$-th codimension is strictly less than $(\dim b(t))^n$ asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra $b(t), t\ge 3$.

Maximal Subalgebras for Modular Graded Lie Superalgebras of Odd Cartan Type

The purpose of this paper is to determine all maximal graded subalgebras of the four infinite series of finite-dimensional graded Lie superalgebras of odd Cartan type over an algebraically closed field of characteristic $p>3$. All maximal graded subalgebras consist of three types (\MyRoman{1}), (\MyRoman{2}) and (\MyRoman{3}). Maximal graded subalgebras of type (\MyRoman{3}) fall into reducible maximal graded subalgebras and irreducible maximal graded subalgebras. In this paper we classify maximal graded subalgebras of types (\MyRoman{1}), (\MyRoman{2}) and reducible maximal g raded subalgebra…

Maximal graded subalgebras of Witt and Special superalgebras

This paper has been withdrawn by the author due to that the main results and approaches are closedly parallel to the ones in Lie algebra case.

Superderivations for Modular Graded Lie Superalgebras of Cartan-type

Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic $p>3$ are completely determined by a uniform approach: The infinite dimensional case is reduced to the finite dimensional case and the latter is further reduced to the restrictedness case, which proves to be far more manageable. In particular, the outer superderivation algebras of those Lie superalgebras are completely determined.

B(0,N)-graded Lie superalgebras coordinatized by quantum tori

We use a fermionic extension of the bosonic module to obtain a class of B(0,N)-graded Lie superalgebras with nontrivial central extensions.

Some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras

This paper is primarily concerned with generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of $\mathbb{Z}$-graded modular Lie superalgebras of Cartan type.

Pauli gradings on Lie superalgebras and graded codimension growth

We introduce grading on certain finite dimensional simple Lie superalgebras of type $P(t)$ by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of $(\mathbb Z_2\times \mathbb Z_2)$-grading on Lie algebra of $(2\times 2)$-traceless matrices.We use this grading for studying numerical invariants of polyomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.

Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras…

Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-gr…

关于自然分次的李超代数与莱布尼茨超代数

On naturally graded Lie and Leibniz superalgebras

本文将自然分次概念推广至 Lie 与非 Lie Leibniz 超代数，给出定义，并刻画最大超幂零指数情形下的自然分次。

带超对合或分次对合的超代数的钩子定理

Hook Theorem for Superalgebras with Superinvolution or Graded Involution

本文在特征零域上具有超对合或分次对合的PI-超代数中，证明了多重线性#-超恒等式的钩子定理，并由此导出Amitsur恒等式的类似物，以对称群表示刻画恒等式的组合特征。

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。

On the codimension growth of simple color Lie superalgebras

We study polynomial identities of finite dimensional simple color Lie superalgebras over an algebraically closed field of characteristic zero graded by the product of two cyclic groups of order $2$. We prove that the codimensions of identities grow exponentially and the rate of exponent equals the dimension of the algebra. A similar result is also obtained for graded identities and graded codimensions.

Maximal Subalgebras for Lie Superalgebras of Cartan Type

The maximal graded subalgebras for four families of Lie superalgebras of Cartan type over a field of prime characteristic are studied. All maximal reducible graded subalgebras are described completely and their isomorphism classes, dimension formulas are found. The classification of maximal irreducible graded subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie superalgebras $\frak{gl}(m,n)$, $\frak{sl}(m,n)$ and $\frak{osp}(m,n)$.

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…

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。

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.

Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras

The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of $Γ$-graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a $\mathbb{Z}_2$-graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction…

Hom-Alternative, Hom-Malcev and Hom-Jordan superalgebras

Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are $\mathbb{Z}_{2}$-graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcev-admissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the $\mathbb{Z}_{2}$-graded Bruck-Kleinfled function are obtained.

Superspecies and their representations

Superspecies are introduced to provide the nice constructions of all finite-dimensional superalgebras. All acyclic superspecies, or equivalently all finite-dimensional (gr-basic) gr-hereditary superalgebras, are classified according to their graded representation types. To this end, graded equivalence, graded representation type and graded species are introduced for finite group graded algebras.

Matrix 3-Lie superalgebras and BRST supersymmetry

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras if instead of the trace of a matrix we make use of the super trace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded comm…

Ovsienko-Roger 超代数上单 Harish-Chandra 模的分类

Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra

本文构造 Ovsienko-Roger 超代数的 $Ω$-算子，据此分类其 $\Z$ 与 $\ rac12\Z$ 分次单尖点模，并推广到 $N=1$ BMS$_3$、超 $W(2,2)$ 等相关 Lie 超代数的单 Harish-Chandra 模。

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.

关于简单李超代数的规范构造

On the Canonical Construction of Simple Lie Superalgebras

本文提出了一种基于抽象根系统构造简单李超代数的统一方法，通过选择最少各向同性根的基来实现

具有几乎多项式增长的超代数簇子簇的中心余特征标

Central cocharacters of the subvarieties of varieties of superalgebras with almost polynomial growth

本文聚焦PI-理论中具有附加结构的代数，研究5个仅有的具有几乎多项式增长分次余维数的超代数簇，确定其中心多项式空间生成元、中心分次余维数及中心分次余特征标的分解。

带Cartan矩阵的例外模李超代数的根

The roots of exceptional modular Lie superalgebras with Cartan matrix

本研究针对带不可分解Cartan矩阵的例外李超代数，给出对应最多两两正交迷向奇单根的不等价Cartan矩阵的根列表与Chevalley基，工具为SuperLie代码与Python。

Irreducible representations of q-Schur superalgebras at a root of unity

Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur superalgebra introduced H. Rui and the first Author.

Universal central extensions of $sl_{m|n}$ over $Z/2Z$-graded algebras

We study central extensions of the Lie superalgebra $sl_{m|n}(A)$, where $A$ is a $Z/2Z$-graded superalgebra over a commutative ring $K$. The Steinberg Lie superalgebra $st_{m|n}(A)$ plays a crucial role. We show that $st_{m|n}(A)$ is a central extension of $sl_{m|n}(A)$ for $m+n\geq 3$. We use a $Z/2Z$-graded version of cyclic homology to show that the center of the extension is isomorphic to $HC_1(A)$ as $K$-modules. For $m+n\geq 5$, we prove that $st_{m|n}(A)$ is the universal central extension of $sl_{m|n}(A)$. For $m+n=3,4$, we prove that $st_{2|1}(A)$ and $st_{3|1}(A)$ are both centrally…

The existence of superinvolutions

Superinvolutions on graded associative algebras constitute a source of Lie and Jordan superalgebras. Graded versions of the classical Albert and Albert-Riehm Theorems on the existence of superinvolutions are proven. Surprisingly, the existence of superinvolutions of the first kind is a rare phenomenon, as nontrivial central division superalgebras are never endowed with this kind of superinvolutions.

关于分裂莱布尼茨超代数

On split Leibniz superalgebras

本文研究了分裂莱布尼茨超代数的结构，证明其可分解为一个阿贝尔分次子代数的子空间与若干两两交换的分次理想之和，并在最大长度情形下用根连接刻画了其单性。

Construction of free Lie Rota-Baxter superalgebra via Gröbner-Shirshov bases theory

In this paper, we construct free Lie Rota-Baxter superalgebra by using Gröbner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish Gröbner-Shirshov bases theory for operated Lie superalgebras. Thirdly, we find a Gröbner-Shirshov basis of a free Lie Rota-Baxter superalgebra on a $\mathbb{Z}_2$-graded set. Consequently, we can obtain a linear basis of a free Lie Rota-Baxter superalgebra by the composition-diamond lemma for operated Lie superalgebras.

正则Hom-Lie超代数中的同源性与因子集

Isoclinism and factor set in regular Hom-Lie Superalgebras

本文在正则Hom-Lie超代数中引入同源性与因子集概念，证明有限维同维数正则Hom-Lie超代数同源当且仅当同构。

Second homology of generalized periplectic Lie superalgebras

Let $(R,{}^-)$ be an arbitrary unital associative superalgebra with superinvolution over a commutative ring $\Bbbk$ with $2$ invertible. The second homology of the generalized periplectic Lie superalgebra $\mathfrak{p}_m(R,{}^-)$ for $m\geqslant3$ has been completely determined via an explicit construction of its universal central extension. In particular, this second homology could be identified with the first $\mathbb{Z}/2\mathbb{Z}$-graded dihedral homology of $R$ with certain superinvolution whenever $m\geqslant 5$.

Cohomology of Hom-Leibniz and $n$-ary Hom-Nambu-Lie superalgebras

The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.

Twisted Frobenius extensions of graded superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that $A$ is a twisted Frobenius extension of $B$ if and only if induction of $B$-modules to $A$-modules is twisted shifted right adjoint to restriction of $A$-modules to $B$-modules. A large (non-exhaustive) class of examples is given by the fact t…

One proof of the original Kemer's theorems (concerning the text of C. Procesi "What happened to PI-theory", arxiv.org/abs/1403.5673)

We consider associative algebras over a field of characteristic zero. We give a version of the proof of the Kemer's theorems concerning the Specht problem solution. It is proved that the ideal of graded identities of a finitely generated PI-superalgebra coincides with the ideal of graded identities of some finite dimensional superalgebra. This implies that the ideal of polynomial identities of any (not necessary finitely generated) PI-algebra coincides with the ideal of identities of the Grassmann envelope of a finite dimensional superalgebra, and is finitely generated as a T-ideal.

Generators of simple modular Lie superalgebras

Let $X$ be one of the finite-dimensional simple graded Lie superalgebras of Cartan type $W, S, H, K, HO, KO, SHO$ or $SKO$ over an algebraically closed field of characteristic $p>3$. In this paper we prove that $X$ can be generated by one element except the ones of type $W,$ $HO$, $KO$ or $SKO$ in certain exceptional cases, in which $X$ can be generated by two elements. As a subsidiary result, we also prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

Real forms of complex Lie superalgebras and supergroups

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected supergroups.

Fractal just infinite nil Lie superalgebra of finite width

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. Their natural analogues are self-similar nil Lie $p$-algebras. In characteristic zero, similar examples of Lie algebras do not exist (Martinez and Zelmanov). The second author recently constructed a 3-generated self-similar nil finely graded Lie superalgebra, which showed that an extension of Martinez-Zelmanov's result for Lie superalgebras of characteristic zero is not valid. Now, we suggest a more handy example. We construct a 2-generated self-similar Lie superalgebra $\mathbf{R}$ over arbitrary field. It has…

On some representations of nilpotent Lie algebras and superalgebras

Let $G$ be a simply connected, nilpotent Lie group with Lie algebra $\gee$. The group $G$ acts on the dual space $\gee^*$ by the coadjoint action. %% which partitions $\gee^*$ into coadjoint orbits. By the orbit method of Kirillov, the simple unitary representations of $G$ are in bijective correspondence with the coadjoint orbits in $\gee^*$, which in turn are in bijective correspondence with the primitive ideals of the universal enveloping algebra of $\gee$. The number of simple $\gee$-modules which have a common eigenvector for a particular subalgebra of $\gee$ and are annihilated by a parti…

The centroid of extended affine and root graded Lie algebras

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.

Central extension of graded Lie algebras

In this thesis we describe the universal central extension of two important classes of so-called root-graded Lie algebras defined over a commutative associative unital ring $k.$ Root-graded Lie algebras are Lie algebras which are graded by the root lattice of a locally finite root system and contain enough $\mathfrak{sl}_2$-triples to separate the homogeneous spaces of the grading. Examples include the infinite rank analogs of the simple finite-dimensional complex Lie algebras. \\ In the thesis we show that in general the universal central extension of a root-graded Lie algebra $L$ is not root…

Representations of Lie superalgebras with fusion flags

We study the category of finite--dimensional representations for a basic classical Lie superalgebra $\Lg=\Lg_0\oplus \Lg_1$. For the ortho--symplectic Lie superalgebra $\Lg=\mathfrak{osp}(1,2n)$ we show that certain objects in that category admit a fusion flag, i.e. a sequence of graded $\Lg_0[t]$--modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite--dimensional irreducible $\Lg$--modules, truncated Weyl modules and Demazure type modules. Moreover, we establish a presentation for these types of fusion products in t…

Nonlinear realisations of Lie superalgebras

For any decomposition of a Lie superalgebra $\mathcal G$ into a direct sum $\mathcal G=\mathcal H\oplus\mathcal E$ of a subalgebra $\mathcal H$ and a subspace $\mathcal E$, without any further resctrictions on $\mathcal H$ and $\mathcal E$, we construct a nonlinear realisation of $\mathcal G$ on $\mathcal E$. The result generalises a theorem by Kantor from Lie algebras to Lie superalgebras. When $\mathcal G$ is a differential graded Lie algebra, we show that it gives a construction of an associated $L_\infty$-algebra.