Finite-dimensional Lie Superalgebras Research Articles

Generalized Derivations of Lie Superalgebras

Let 𝔽 be a field of characteristic ≠ 2 and ℒ a finite-dimensional Lie superalgebra over 𝔽. In this article, we study the derivation superalgebra Der(ℒ), the quasiderivation superalgebra QDer(ℒ), and the generalized derivation superalgebra GDer(ℒ) of ℒ, which form a tower Der(ℒ) ⊆ QDer(ℒ) ⊆ GDer(ℒ) ⊆ pl(ℒ), where pl(ℒ) denotes the general linear Lie superalgebra. More precisely, we characterize completely those Lie superalgebras ℒ for which QDer(ℒ) = pl(ℒ). We prove that the quasiderivations of ℒ can be embedded as derivations in a larger Lie superalgebra and, furthermore, when the annihilator of ℒ is equal to zero, we obtain a semidirect sum decomposition of .

The Finite-dimensional Modular Lie Superalgebra Γ

A new family of finite-dimensional modular Lie superalgebras Γ is defined. The simplicity and generators of Γ are studied and an explicit description of the derivation superalgebra of Γ is given. Moreover, it is proved that Γ is not isomorphic to any known Lie superalgebra of Cartan type.

On the modular Lie superalgebra [formula omitted

A class of finite-dimensional simple modular Lie superalgebra Ω was constructed and its derivation superalgebra was determined in Zhang and Zhang (2009) [24]. The purpose of the present paper is to continue the investigation of finite-dimensional modular Lie superalgebra Ω . A nonnatural filtration of finite-dimensional modular Lie superalgebra Ω over a field of positive characteristic is determined. It is proved that the nonnatural filtration of Ω is invariant under the automorphism group of Ω . Moreover, we obtain the sufficient and necessary conditions of Ω ( r , m , q , s ¯ , H ) ≅ Ω ( r ′ , m ′ , q ′ , s ¯ ′ , H ′ ) , i.e., we classified the Lie superalgebra of Ω type.

ACTIONS OF LIE SUPERALGEBRAS ON SEMIPRIME ALGEBRAS WITH CENTRAL INVARIANTS

AbstractLet R be a semiprime algebra over a field of characteristic zero acted finitely on by a finite-dimensional Lie superalgebra L = L0 ⊕ L1. It is shown that if L is nilpotent, [L0, L1] = 0 and the subalgebra of invariants RL is central, then the action of L0 on R is trivial and R satisfies the standard polynomial identity of degree 2 ⋅ [$\sqrt{2^{\dim_{\mathbb{K}}L_1}}$]. Examples of actions of nilpotent Lie superalgebras, with central invariants and with [L0, L1] ≠ 0, are constructed.

δ-Superderivations of simple finite-dimensional Jordan and Lie superalgebras

We introduce the concept of a $\delta$-superderivation of a superalgebra. $\delta$-Derivations of Cartan-type Lie superalgebras are treated, as well as $\delta$-superderivations of simple finite-dimensional Lie superalgebras and Jordan superalgebras over an algebraically closed field of characteristic 0. We give a complete description of $\{1}{2}$-derivations for Cartan-type Lie superalgebras. It is proved that nontrivial $\delta$-(super)derivations are missing on the given classes of superalgebras, and as a consequence, $\delta$-superderivations are shown to be trivial on simple finite-dimensional noncommutative Jordan superalgebras of degree at least 2 over an algebraically closed field of characteristic 0. Also we consider $\delta$-derivations of unital flexible and semisimple finite-dimensional Jordan algebras over a field of characteristic not 2.

Chevalley's restriction theorem for reductive symmetric superpairs

Let ( g , k ) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a ⊂ p . The restriction map S ( p ∗ ) k → S ( a ∗ ) W where W = W ( g 0 : a ) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. ( k ⊕ k , k ) with the flip involution where k is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley's restriction theorem due to Sergeev and Kac, Gorelik. For general symmetric superpairs, the invariants exhibit a new and surprising behaviour. We illustrate this phenomenon by a detailed discussion in the example g = C ( q + 1 ) = osp ( 2 | 2 q , C ) , endowed with a special involution. Here, the invariant algebra defines a singular algebraic curve.

FINITE-DIMENSIONAL SPECIAL ODD HAMILTONIAN SUPERALGEBRAS IN PRIME CHARACTERISTIC

In this paper, we study a new family of finite-dimensional simple Lie superalgebras of Cartan type over a field of characteristic p > 3, the so-called special odd Hamiltonian superalgebras. The spanning sets are first given and then the grading structures are described explicitly. Finally, the simplicity and the dimension formulas are determined. As application, using the dimension formulas, we make a comparison between the special odd Hamiltonian superalgebras and the other known families of finite-dimensional simple modular Lie superalgebras of Cartan type.

Differential operator realizations of superalgebras and free field representations of corresponding current algebras

Based on the particular orderings introduced for the positive roots of finite-dimensional basic Lie superalgebras, we construct the explicit differential operator representations of the osp ( 2 r | 2 n ) and osp ( 2 r + 1 | 2 n ) superalgebras and the explicit free field realizations of the corresponding current superalgebras osp ( 2 r | 2 n ) k and osp ( 2 r + 1 | 2 n ) k at an arbitrary level k. The free field representations of the corresponding energy–momentum tensors and screening currents of the first kind are also presented.

Integrable modules for affine Lie superalgebras

Irreducible nonzero level modules with finite-dimensional weight spaces are discussed for nontwisted affine Lie superalgebras. A complete classification of such modules is obtained for superalgebras of type A(m, n) and C(n)^ using Mathieu's classification of cuspidal modules over simple Lie algebras. In other cases the classification problem is reduced to the classification of cuspidal modules over finite-dimensional cuspidal Lie superalgebras described by Dimitrov, Mathieu and Penkov. Based on these results a complete classification of irreducible integrable (in the sense of Kac and Wakimoto) modules is obtained by showing that any such module is of highest weight, in which case the problem was solved by Kac and Wakimoto.

The finite-dimensional modular Lie superalgebra Ω

The finite-dimensional modular Lie superalgebra Ω is constructed. The simplicity of Ω is proved. Its derivation superalgebra is determined. Then it is obtained that Ω is not isomorphic to any known Z-graded modular Lie superalgebra of Cartan type.

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra.

Automorphisms of toroidal Lie superalgebras

We give a description of the algebraic group Aut ( g ) of automorphisms of a simple finite-dimensional Lie superalgebra g over an algebraically closed field k of characteristic 0, which is obtained by viewing g as a module over a Levi subalgebra of its even part. As an application, we give a detailed description of the group of automorphism of the k-Lie superalgebra g ⊗ k R for a large class of commutative rings R.

Simple Lie superalgebras and nonintegrable distributions in characteristic p

Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p ≤ 5) and several other little-known simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4), and Sp(10). The description was performed in terms of Cartan-Tanaka-Shchepochkina prolongs using Shchepochkina’s algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite-dimensional Lie superalgebras overℂ; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one. Bibliography: 30 titles.

The Cartan-Type Modular Lie Superalgebra KO

ABSTRACT The finite-dimensional odd contact Lie superalgebras KO(n, n + 1, t ) over a field of prime characteristic are studied, where n is a positive integer and t is an n-tuple of non-negative integers. In particular, it is proven that KO(n, n + 1, t ) is simple and has no non-singular associative bilinear forms. Moreover, an explicit description of the derivation superalgebra of KO(n, n + 1, t ) is given, and as a consequence it is shown that the outer derivation superalgebra of KO(n, n + 1, t ) is Abelian of dimension . Communicated by K. Misra.

Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture

We study the representation theory of the superconformal algebra $W_k(g,f_{\theta})$ associated with a minimal gradation of $g$. Here, $g$ is a simple finite-dimensional Lie superalgebra with a non-degenerate, even supersymmetric invariant bilinear form. Thus, $W_k(g,f_{\theta})$ can be one of the well-known superconformal algebras including the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnik algebras, the N=2 superconformal algebra, the N=4 superconformal algebra, the N=3 superconformal algebra and the big N=4 superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan and M. Wakimoto for $W_k(g,f_{\theta})$. In fact, we show that any irreducible highest weight character of $W_k(g,f_{\theta})$ at any level $k\in C$ is determined by the corresponding irreducible highest weight character of the Kac-Moody affinization of $g$.

Integration on Lie supergroups: A Hopf superalgebra approach

Let g = g 0 ¯ ⊕ g 1 ¯ be a finite-dimensional complex Lie superalgebra such that g 0 ¯ is reductive and the adjoint representation of g 0 ¯ in g 1 ¯ is completely reducible. A Lie supergroup associated to g is defined by fixing the Hopf superalgebra of regular functions on the supergroup. Then it is shown that on this Hopf superalgebra there exists a non-zero left integral. According to an earlier work by the authors, this integral is unique up to scalar multiples.

New simple Lie superalgebras in characteristic 3

Symplectic (respectively orthogonal) triple systems provide constructions of Lie algebras (respectively superalgebras). However, in characteristic 3, it is shown that this role can be interchanged and that Lie superalgebras (respectively algebras) can be built out of symplectic triple systems (respectively orthogonal triple systems) with a different construction. As a consequence, new simple finite dimensional Lie superalgebras, as well as new models of some nonclassical simple Lie algebras, over fields of characteristic 3, will be obtained.

Lie Superalgebra Structures in H? (g;g)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let “-” be the change of parity functor, C i and H i the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L * and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l * for some simple finite dimensional Lie superalgebras g and their “relatives” - the nontrivial central extensions or derivation algebras of the considered simple ones.

Classically semisimple locally finite Lie superalgebras

We introduce the class of classically semisimple locally finite Lie superalgebras over an algebraically closed field K of characteristic 0 and classify all Lie superalgebras in this class. By definition, a countably dimensional locally finite Lie superalgebra g= g0 ⊕ g1 over K is classically semisimple if it is semisimple (i.e. its locally solvable radical equals zero) and in addition it admits a generalized root decomposition, such that the root spaces generate g, and g0 is a root-reductive Lie algebra in the sense of [DP] and [PS]. We prove that any classically semisimple Lie superalgebra is isomorphic to a direct sum of copies of the infinite dimensional Lie superalgebras sl (∞|n), sl (∞|∞), osp(m|∞), osp(∞|2k), osp(∞|∞), sp(∞), sq(∞) and of copies of simple finite dimensional Lie superalgebras with reductive even part. In particular, the above listed infinite dimensional Lie superalgebras are all (up to isomorphism) countably dimensional simple locally finite Lie superalgebras which admit a generalized root decomposition with root-reductive even part. Finally, we describe all generalized root decompositions of any classically semisimple locally finite Lie superalgebra.

The Kac Construction of the Centre of for Lie Superalgebras

In 1984, Victor Kac [8] suggested an approach to a description of central elements of a completion of for any Kac-Moody Lie algebra . The method is based on a recursive procedure. Each step is reduced to a system of linear equations over a certain subalgebra of meromorphic functions on the Cartan subalgebra. The determinant of the system coincides with the Shapovalov determinant for . We prove that the Kac approach can also be applied to finite dimensional Lie superalgebras with Cartan matrix A (as claimed in [8]) and reproduce for them Sergeev’s description of the centers of [14]. In order to prove this, one needs to show that the recursive procedure stops after a finite number of steps. The original paper [8] does not indicate how to check this fact. Here we give a detailed presentation of the Kac approach and apply it to finite dimensional Lie superalgebras . In particular, we deduce the Kac formulas for the Shapovalov determinants and verify the finiteness of the recursive procedure.