Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebras and root vectors and which make it possible to construct representations by operators acting according to Gel’fand–Tsetlin-type formulas. Unitary representations of the q-deformed algebras Uq(son,1) are found. AMS subject classifications (1980). 16...

A proof of Poincaré-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type An and Mp,q,ǫ(n, K), which is a nonstandard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the...

We show that for general linear groups GLn(q) as well as for q-Schur algebras the knowledge of the modular Alvis-Curtis duality over fields of characteristic l, l ∤ q, is equivalent to the knowledge of the decomposition numbers.

Let Q be a smooth manifold acted upon smoothly by a Lie group G, and let N be the space of G-orbits. The G-action lifts to an action on the total space T∗Q of the cotangent bundle of Q and hence on the ordinary symplectic Poisson algebra of smooth functions on T∗Q, and the Poisson algebra of G-invariant functions on T∗Q yields a Poisson structure on the space (T∗Q) / G of G-orbits. We develop a...

The. theory of reduced incidence algebras of binomial posets furnishes a unified treatment of several types of generating functions that arise in enumerative combinatorics. Using this theory as a tool, we study ‘reduced covering algebras’ of binomial lattices and show that they are isomorphic to various algebras of q-binomial generating functions for certain modular binomial lattices.

We determine explicit quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centers and block diagonal forms of these algebras. In the case where q is an arbitrary root of unity, this further determines the degrees.

We obtain results on algebras which have many of the properties of cellular algebras except for the existence of a certain anti-involution. We show that they are applicable to q-walled Brauer-Clifford superalgebras.

Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their immediate generalizations that we call pseudo-bosons and pseudo-fermions, and also matrix algebras, in the finitely generated case. We then recover q-bosons (a...

Two families of q-Schur algebras associated to Hecke algebras of type D are introduced, and related to a family used by Geck, Gruber and Hiss [10], [11]. We prove that the algebras in one family, called the q-Schur algebras, are integrally free, stable under base change, and are standardly stratified if the base field has odd characteristic. In the so-called linear prime case of [10],[11], all ...