Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α ^k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.     

The Yangian of the strange Lie superalgebra and its quantum double

We construct the Yangian of the strange Lie superalgebra as a particular case of the general construction of a twisted Yangian. We describe a Poincaré-Birkhoff-Witt basis of the Yangian of the type-Q_n Lie superalgebra and construct the quantum double of the Yangian of the type-Q₂ strange Lie superalgebra.     

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 the q-deformed Lie superalgebra

In this paper a general procedure is given to get the quantum derformation of the Lie superalgebra spl(2,1) and its corresponding algebraic structure of a quantum group i.e. non-commutative, non-co-commutative Hopf superalgebra. This procedure would be suitable for another Lie superalgebra.     

Drinfeld Realization of Twisted Quantum Affine Algebras

The quantum affine algebra has two realizations, the usual Drinfeld–Jimbo definition and a new Drinfeld realization given by Drinfeld. In this article, we use the adjoint action to prove that these two realizations are isomorphic for the twisted quantum affine algebra.     

The extension of a quantized Borcherds superalgebra by a Hopf algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b_(ik), c_(ik), g_(ik), h_(ik)(i ∈I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra U_q(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A =(aij)i,j∈I. Therefore, we define an extended Hopf superalgebra HU_q(G). We also discuss the basis and the grouplike elements of HU_q(G).     

Clebsch-Gordan coefficients for the rank one quantum superalgebra. I.

We give general definitions of quantum Lie superalgebras using defining relations and the Cartan matrix, as well as the R-matrix. We present a realization of quantum superalgebra generators in terms of the q-oscillators and calculate the Clebsch—Gordan coefficients for the quantum superalgebra osp_q(1|2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 76–88, 1990.     

q-Wedge Modules for Quantized Enveloping Algebras of Classical Type

We use the fusion construction in twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a by-product we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the q-wedge products.     

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.     

Quantum superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.     

The t-analog of the level one string function for twisted affine Kac–Moody algebras

We study Lusztig?s t-analog of weight multiplicities, or affine Kostka–Foulkes polynomials, associated to level one representations of twisted affine Kac–Moody algebras. We obtain an explicit closed form expression for the unique t-string function, using constant term identities of Macdonald and Cherednik. This extends previous work on t-string functions for the untwisted simply-laced affine Kac–Moody algebras.     

A Kronecker factorization theorem for the exceptional Jordan superalgebra

We prove that a Jordan superalgebra J containing the 10-dimensional exceptional Kac superalgebra K₁₀ is isomorphic to (K₁₀⊗_FS)⊕J′, where S is an associative commutative algebra.     

Representations of quantum affine superalgebras

Mathematische Zeitschrift - We study the quantum affine superalgebra $$U_q(\mathcal {L}\mathfrak {sl}(M,N))$$ and its finite-dimensional representations. We prove a triangular decomposition and...     

Dual affine quantum groups

Let be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group , dual of . Studying its specializations at roots of 1 (in particular, its semi-classical limits), we prove that it yields quantizations of and (the formal Poisson group attached to ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type. Received January 27, 1999     

Quantum supergroups IV: the modified form

We construct a canonical basis for a class of tensor product modules of a quantum covering group associated to a Kac–Moody Lie superalgebra of anisotropic type, and use these bases to construct a canonical basis for the modified form of a quantum covering group.     

Noncommutative Batalin–Vilkovisky geometry and matrix integrals

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.     

Finite-dimensional simple modular Lie superalgebra M

A new family of finite-dimensional simple modular Lie superalgebra M is constructed based on results of Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601–3619]. The simplicity and generators of M are discussed and the derivation superalgebra of M is characterized. Furthermore, the invariance of the nonnatural filtration of M is determined by the method of minimal dimension of image spaces.     

From Macdonald polynomials to a charge statistic beyond type A

The charge is an intricate statistic on words, due to Lascoux and Schützenberger, which gives positive combinatorial formulas for Lusztig?s t-analogue of weight multiplicities and the energy function on affine crystals, both of type A. As these concepts are defined for all Lie types, it has been a long-standing problem to express them based on a generalization of charge. I present a method for addressing this problem in classical Lie types, based on the recent Ram-Yip formula for Macdonald polynomials and the quantum Bruhat order on the corresponding Weyl group. The details of the method are carried out in type A (where we recover the classical charge) and type C (where we define a new statistic).