Restricted Lie Superalgebras Research Articles

Central Extensions of Restricted Lie Superalgebras and Classification of p-Nilpotent Lie Superalgebras in Dimension 4

Central Extensions of Restricted Lie Superalgebras and Classification of <i>p</i>-Nilpotent Lie Superalgebras in Dimension 4

Representations of the Restricted Lie Superalgebra p(n)

Representations of the Restricted Lie Superalgebra p(n)

Irreducible modules of modular Lie superalgebras and super version of the first Kac–Weisfeiler conjecture

Abstract Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p&gt;2$ . In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$ , as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

The Modular Representations of Basic Classical Restricted Lie Superalgebras

The Modular Representations of Basic Classical Restricted Lie Superalgebras

Symplectic Double Extensions for Restricted Quasi-Frobenius Lie (Super)Algebras

In this paper, we present a method of symplectic double extensions for restricted quasi-Frobenius Lie superalgebras. Certain cocycles in the restricted cohomology represent obstructions to symplectic double extension, which we fully describe. We found a necessary condition for which a restricted quasi-Frobenius Lie superalgebras is a symplectic double extension of a smaller restricted Lie superalgebra. The constructions are illustrated with a few examples.

Formula omitted]-graded modules for [formula omitted]-graded modular vertex superalgebras

In this paper, we study Z-graded vertex superalgebras over a general field of an odd prime characteristic. In particular, we study affine vertex superalgebras Vgˆ(ℓ,0) and a family of quotient vertex superalgebras Vgˆχ(ℓ,0) with g assumed to be a restricted Lie superalgebra. Among the main results, we obtain a necessary and sufficient condition for a gˆ-module of level ℓ to be a Vgˆχ(ℓ,0)-module. Furthermore, we study the special case with g=g1¯ and introduce a quotient vertex superalgebra Lgˆ(ℓ,0) of Vgˆ(ℓ,0). As the main results, we prove that for ℓ≠0, vertex superalgebra Lgˆ(ℓ,0) is simple, the adjoint module is the only irreducible N-graded module up to equivalence, and every Lgˆ(ℓ,0)-module is isomorphic to the direct sum of some copies of Lgˆ(ℓ,0).

Restricted Cohomology of Restricted Lie Superalgebras

Suppose the ground field \(\mathbb{F}\) is an algebraically closed field characteristic of p > 2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra \(L_{p,p}^\lambda\) structures parameterized by elements \(\lambda \in {\mathbb{F}^p}\). We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.

Constructions and Properties for a Finite-Dimensional Modular Lie Superalgebra K n , m

In this paper, a finite-dimensional Lie superalgebra K n , m over a field of prime characteristic is constructed. Then, we study some properties of K n , m . Moreover, we prove that K n , m is an extension of a simple Lie superalgebra, and if m = n − 1 , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.

On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type

Let g be a restricted Lie superalgebra of Cartan type W(n), S(n)or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac’s definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.

Double Extensions of Restricted Lie (Super)Algebras

A double extension ( $${\mathscr {D}}$$ -extension) of a Lie (super)algebra $${\mathfrak {a}}$$ with a non-degenerate invariant symmetric bilinear form $${\mathscr {B}}$$ , briefly, a NIS-(super)algebra, is an enlargement of $${\mathfrak {a}}$$ by means of a central extension and a derivation; the affine Kac–Moody algebras are the best known examples of double extensions of loops algebras. Let $${\mathfrak {a}}$$ be a restricted Lie (super)algebra with a NIS $${\mathscr {B}}$$ . Suppose $${\mathfrak {a}}$$ has a restricted derivation $${\mathscr {D}}$$ such that $${\mathscr {B}}$$ is $${\mathscr {D}}$$ -invariant. We show that the double extension of $${\mathfrak {a}}$$ constructed by means of $${\mathscr {B}}$$ and $${\mathscr {D}}$$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $${\mathscr {D}}$$ -extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $${\mathscr {D}}$$ -extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;1)\otimes\wedge(n)$$, where $$\mathfrak{A}(m;1)$$ is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type I.

Computer-Aided Study of Double Extensions of Restricted Lie Superalgebras Preserving the Nondegenerate Closed 2-Forms in Characteristic 2

A Lie (super)algebra with a nondegenerate invariant symmetric bilinear form B is called a nis-(super)algebra. The double extension of a nis-(super)algebra is the result of simultaneous adding to a central element and a derivation so that is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2, the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman’s Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the nondegenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7—sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.

On support varieties for Lie superalgebras and finite supergroup schemes

We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by Duflo and Serganova in characteristic zero but which have no known cohomological interpretation. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.

Restricted Lie (Super)Algebras in Characteristic 3

We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under certain conditions) their simple subquotients over an algebraically closed field of characteristic 3, as well as one type of the deformed divergence-free Lie superalgebras with any number of indeterminates in any characteristic.

Restricted Lie (Super)Algebras in Characteristic 3

Мы дадим явные формулы, доказывающие ограниченность следующих (супер)алгебр Ли: известных исключительных (супер)алгебр Ли векторных полей в характеристике 3, продеформированных (супер)алгебр Ли с неразложимой матрицей Картана и их простых подфакторов над алгебраически замкнутым полем характеристики 3, а также одного типа продеформированных бездивергентных супералгебр Ли от любого числа переменных в характеристиках большей 2.

On modular representations of finite-dimensional Lie superalgebras

In this paper, we studied representations of finite-dimensional Lie superalgebras over an algebraically closed field \begin{document} $\mathbb{F}$ \end{document} of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over \begin{document} $\mathbb{F}$ \end{document} are finite-dimensional, and there exists an upper bound on the dimensions of simple modules. Moreover, a finite-dimensional Lie superalgebra can be embedded into a finite-dimensional restricted Lie superalgebra. We gave a criterion on simplicity of modules over a finite-dimensional restricted Lie superalgebra \begin{document} ${\mathfrak{g}}$ \end{document} , and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of \begin{document} ${\mathfrak{g}}$ \end{document} and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.

On restricted representations of restricted contact Lie superalgebras of odd type

Simple restricted modules are considered for the restricted contact Lie superalgebras of odd type over an algebraically closed field with characteristic [Formula: see text]. In particular, a sufficient and necessary condition in terms of typical or atypical weights is given for the restricted Kac modules to be simple. Furthermore, the number of the simple restricted Kac modules is obtained.

Engel condition on enveloping algebras of Lie superalgebras

Let L be a Lie superalgebra over a field of characteristic p≠2 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field of characteristic p>2 with restricted enveloping algebra u(L). In this note, we establish when u(L) or U(L) is bounded Lie Engel.

On representations of restricted Lie superalgebras

Simple modules for restricted Lie superalgebras are studied. The indecom-posability of baby Kac modules and baby Verma modules is proved in some situation. In particular, for the classical Lie superalgebra of type A(n|0), the baby Verma modules Z χ (λ) are proved to be simple for any regular nilpotent p-character χ and typical weight λ. Moreover, we obtain the dimension formulas for projective covers of simple modules with p-characters of standard Levi form.

Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes

We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic p &gt; 2 p &gt; 2 is a finitely-generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup G L ( m | n ) GL(m|n) that are similar to the universal extension classes for G L n GL_n exhibited by Friedlander and Suslin.