Lie superalgebras based on ᵓ0ᵒ9(2,ᵓd)

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $\sigma$ of order $T$ with $M$ as a $\sigma$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a $\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.

In this paper, we determine the first cohomology groups of centerless N = 1 super Virasoro algebras with coefficients in their tensor density modules. As applications, we classify all central extensions and derivations of the complete spectrum-generating superalgebra.

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superal- gebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

On δ-derivations of Lie algebras and superalgebras

For every finite-dimensional nilpotent complex Lie algebra or superalgebran, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensionalg whose maximal nilpotent subalgebra isn, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots ofg. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super) algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.

Computation of cohomology of Lie (super)algebra: algorithm, implementation and new results

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (C3+A) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (C3+A), the Lie superbialgebra structures on the Lie superalgebra (C3+A) are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (C3+A) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ(Cp=12,𝜖⊕A1,1)(C3+A).

Let L be a Lie superalgebra over a field of characteristic different from 2, 3 and write for the Lie superalgebra consisting of superderivations mapping L to and the central elements to zero. In this paper we first give an upper bound for the superdimension of by means of linear vector space decompositions. Then we characterize the -superderivation superalgebras for the nilpotent Lie superalgebras of class 2 and the model filiform Lie superalgebras by methods of block matrices.

For a commutative algebra A over a commutative ring k satisfying certain conditions, we construct the universal central extension of \( {\frak g}_k \otimes_k A \), regarded as a Lie superalgebra over k, where \( {\frak g}_k \) denotes a basic classical Lie superalgebra over k. To consider basic classical Lie superalgebras over an ring k, we also show the existence of their Chevalley basis. Our results contain not only the descriptions of the untwisted affine Lie superalgebras but also those of the toroidal Lie superalgebras.

This paper gives an enumeration of the outer automorphisms of simple finite-dimensional complex Lie superalgebras, the nonisomorphic infinite-dimensional Lie superalgebras associated with these automorphisms, and the systems of simple roots in contragredient Lie superalgebras (finite-dimensional and associated with a finite-dimensional simple Lie superalgebra and automorphisms of this Lie superalgebra). Bibliography: 6 titles.

Cohomolgies of Rota-Baxter operators on Lie superalgebras and some classifications on Witt superalgebras

We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex lattice polytopes, sharing useful properties such as the integer decomposition property. This paper is the first step towards extending the framework of PBW degenerations to the Lie superalgebra setting.

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.