A classification of Mal'tsev superalgebras of small dimensions

The classification of Lie superalgebras of small dimensions is well known. Here we set out a similar task regarding Mal'tsev superalgebras. It is established that there exist a number of diverse types of non-Lie Mal'tsev superalgebras of dimension 4. One of the types of 3-dimensional non-Lie Mal'tsev superalgebras allows us to construct a number of interesting examples in the theory of ordinary (finite-dimensional) Mal'tsev algebras. Supported by CMUC-JNICT. Supported by DGA (PCB 6/91). Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 629–654, November–December, 1996.     

Operations for modules on Lie-Rinehart superalgebras

Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. LetL be ak−A-Lie-Rinehart superalgebra. From these data, one can construct a superalgebra of differential operatorsV(A,L) (generalizing the enveloping superalgebra of a Lie superalgebra). We will give a difinition of Lie-Rinehart superalgebra morphisms allowing to generalize the notions of inverse image and direct image. We will prove that a Lie-Rinehart superalgebra morphism decomposes into a closed imbedding and a projection. Furthermore, we will see that, under some technical conditions, a closed imbedding decomposes into two closed imbeddings of different nature. The first one looks like a Lie superalgebra morphism. The second one looks like a supermanifold closed imbedding and satisfies a generalization of the Kashiwara’s theorem. Then, as in theD-module theory, we introduce a duality functor. Finally, we will prove that, in the closed imbedding case, the direct image and the duality functor commute.     

Prime graded modules

In this paper, properties of prime and strongly prime graded modules are studied. The class of strongly prime graded modules that determines a graded strongly prime radical is described.     

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u-homology computation are worked out.     

Simplicity of vacuum modules over affine Lie superalgebras

We prove an explicit condition on the level k for the irreducibility of a vacuum module $V^k$ over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level k is a non-negative integer.     

Jordan superalgebras of vector type and projective modules

We study connection between the Jordan superalgebras of vector type and the finitely generated projective modules of rank 1 over an integral domain.     

Composition factors of Kac modules for the general linear Lie superalgebras

The composition factors of Kac modules for the general linear Lie superalgebras are explicitly determined. In particular, a conjecture of Hughes, King and van der Jeugt in [J. Math. Phys. 33 (1992), 470–491] is proved.     

Prime modules of a gamma ring

We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only ifQ is the annihilator of somes-moduleG ofM. Relationships between prime ΓM-modules and primeR-modules are established, whereR is the right operator ring ofM. Similar results are obtained fors-modules.     

Constructing tensor products of modules for C 2-cofinite vertex operator superalgebras

For any C2-cofinite vertex product and the P（z）-tensor product finite length are proved to exist, which operator superalgebra V, the tensor of any two admissible V-modules of are shown to be isomorphic, and their constructions are given explicitly in this paper.     

Prime and primary submodules of certain modules

In this paper we characterize all prime and primary submodules of the free R-module R ⁿ for a principal ideal domain R and find the minimal primary decomposition of any submodule of R ⁿ . In the case n = 2, we also determine the height of prime submodules.     

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g₀ we associate a Clifford algebra over the field of rational functions on O. We find the rank, k(O) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U(g)-module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k(O) is in many cases, equal to the odd dimension of the orbit G⋅O, where G is a Lie supergroup with Lie superalgebra g.     

n-Ary Mal'tsev Algebras

By analogy with n-Lie algebras, which are a natural generalization of Lie algebras to the case of n-ary multiplication, we define the concept of an n-ary Mal'tsev algerba. It is shown that exceptional algebras of a vector cross product are ternary central simple Mal'tsev algebras, which are not 3-Lie algebras if the characteristic of a ground field is distinct from 2 and 3. The basic result is that every n-ary algebra of the vector cross product is an n-ary central simple Mal'tsev algebra.     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.