An orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is introduced. These representations are particular lowest weight representations V (p), with a lowest weight of the form [−p2 , . . . ,− p 2 | p 2 , . . . , p 2 ], p being a positive integer. Explicit expressions for the transformation of the basis unde...

We introduce the concept of the Grassmann real form of an arbitrary matrix supergroup and of its Lie superalgebra. This new notion includes as special cases the ordinary real forms (considered and classified by Serganova) but it is more general. Indeed, the Grassmann real forms are characterized by certain Lie superal-gebra automorphisms of order four, while the ordinary real forms by those of ...

The quantization of the simple one-dimensional Hamiltonian H = xp is of interest for its mathematical properties rather than for its physical relevance. In fact, the Berry-Keating conjecture speculates that a proper quantization of H = xp could yield a relation with the Riemann hypothesis. Motivated by this, we study the so-called Wigner quantization of H = xp, which relates the problem to repr...

Howe dualities lead to diagrammatic categories which describe the representations of Lie-type objects as a monoidal category (that is, via generators and relations). Applying this philosophy type Q duality Cheng–Wang Sergeev, we introduce web supercategories relations show they full subcategory supermodules for Lie superalgebra given by tensor products supersymmetric powers natural supermodule.

In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology H•K(V) for any quasiconstant coefficient l × l matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z-graded Lie sup...

In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology H•K(V) for any quasiconstant coefficient l × l matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z-graded Lie sup...

Here we describe a Gauss decomposition of the Yangian Y (glm|n) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y (A(m− 1, n− 1)) defined and studied in papers by Stukopin. We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian Y (glm|n). This is a conjecture of Nazaro...

We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a Z2-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in ...

We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra D(1)(2, 1;x). Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck’s analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid. Preprint numbers: MIT-CTP 38...

We consider a class of Lie superalgebra, called spinc supersym-metry algebras, constructed from spinor representation. They are motivated by supersymmetry algebras used by physicists. On a Riemannian manifold, a KK ahler manifold, and a hyperkk ahler manifold respectively, it is known that some natural operators on the space of diierential forms span spinc (2), spinc(3) and spinc (5) supersymme...