The least degree of identities in the subspace M 1 (m,k) (F) of the matrix superalgebra M (m,k)(F)

We determine the least degree of identities in the subspace M ₁ ^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over a field F for arbitrary m and k. For the subspace M ₁ ^(m,k) (F) (k > 1) we obtain concrete minimal identities and generalize some results by Chang and Domokos.     

On the graded identities ofM 1,1(E)

We determine theS _n ×S _m -cocharacterX _n,m of the algebraM _1,1(E) and prove that theT ₂-ideal of its graded identities is generated by the polynomialsy ₁ y ₂−y ₂ y ₁ andz ₁ z ₂ z ₃+z ₃ z ₂ z ₁.     

Not all graded identities for Mn(E) come from Mn,n

We introduce here two classes of infinite dimensional Lie Algebras called potentially nilpotent and potentially solvable and we define the notion of Fine Lie Algebra. The study of the Fine Lie Algebras allows to obtain their complete classification in the natural or standard graded cases.     

Basis for identities of the algebra of simplified insertion

We prove that all identities of the algebra of simplified insertion on countably many generators over a field of characteristic zero follow from the right-symmetric identity. We prove that the bases of the free special Jordan algebra and the special algebra of simplified insertion coincide. We construct an infinite series of relations in the algebra of simplified insertion which hold for the words of length k, k ε ℕ.     

H-Module Algebra Structures On M2(F)

Let F be an algebraically closed field of characteristic 0,H be an eight-dimensional non-semisimple Hopf algebra which is neither pointed nor unimodular and M2(F)be the full matrix algebra of 2×2 over F.In this paper,we discuss and classify all H-module algebra structures on M2(F).     

The Yangian double of the Lie superalgebra A(m, n)

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. Normally ordered bases in the Yangian and its dual in the quantum double are introduced. We calculate the pairing between the elements of these bases and obtain a formula for the universal R-matrix of the Yangian double as well as a formula for the universal R-matrix (introduced by Drinfeld) of the Yangian.     

Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.     

A note on the homogenized lie superalgebra osp(l, 2)

The homogenization of a superenveloping algebra is Auslander regular iff this enveloping algebra is a domain. This note determines points and lines in the quantum space of the homogenized envelope of the classical simple Lie superalgebra osp(l,2). It is also concerned with homogenized sl(2), which is graded embedded in homogenized osp(l,2), and describes the conies in the associated quantum space.     

The superalgebra of W(2,2) and its modules of the intermediate series

In this paper, we study a new Lie superalgebra constructed by a 2|2-dimensional Balinsky–Novikov superalgebra, which is called the superalgebra of W(2,2). It can be realized from semi product of the W-algebra W(2,2) and its module of the intermediate series. Finally, we determine all modules of the intermediate series over this superalgebra. Since it is di?cult to do so directly, we make it by using modules of the intermediate series over the trivial super extension of the Witt algebra.     

Quantum group structure of Lie superalgebra osp(1/2)

In this paper we quantise the enveloping Lie superalgebra U(osp(1/2)) in the non-standard simple root system with two odd simple roots. The Quantum supergroup structure associated with the non-standard simple root system is also given.     

Classification of finite dimensional modules of the lie superalgebra sl(2/1)

In this paper, we not only give a complete classification ofinite dimensional indecomposablemodules of the Lie superalgebra 5/(2/1) with diagonal Cartan subalgebra, but also compely classify those modules with nondiagonal Cartan subalgebra. Thus, by taking all possibe direct sums of such modules, we have effectively classified all the finite dimensional sl(2/\)-modules.     

A Complete classification of indecomposable harish-chandra modules of the lie superalgebra B(0,1)

In this paper, we give a classification of indecomposable B(0,1)—modules which admit a weight space decomposition with all weight spaces finite dimensional, i.e., the so—called Harish—Chandra modules.     

The homology of graded infinite-dimensional Lie algebras in connection with Macdonald identities

D. Fucks' monograph devoted to the cohomology of infinite-dimensional Lie algebras contains an error in calculating the homology of a graded affine Kac-Moody algebra of type A _n ⁽¹⁾ , so that the proof of the corresponding Macdonald identity, which is based on that calculation, is incorrect. In the present paper, a revised proof is suggested.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1996, pp. 5–17.The author is grateful to A. M. Vershik, who suggested the topic of this paper, for his constant attention and useful comments. The author also thanks M. A. Vsemirnov for his information on non-coincidence of the coefficients in two sides of the equality in [4, §3.2.3] for n=7, which he observed while proving Macdonald identities in a new (analytical) way, and on the sign of a summand on the right (see his paper in this issue).     

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on ²ⁿ are investigated under suitable continuity restrictions on the cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket or arbitrary n > 0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n > 1.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 163–194, May, 2005.