arXiv subjects

Dmitry Gurevich

Publications and source records attributed to Dmitry Gurevich.

3 recordsLinked to original sources

arXiv2023

On Symmetrizers in Quantum Matrix Algebras

In this note we are dealing with a particular class of quadratic algebras -- the so-called quantum matrix algebras. The well-known examples are the algebras of quantized functions on classical Lie groups (the RTT algebras). We consider the problem of constructing some projectors on homogenous components of such algebras, which are analogs of the usual symmetrizers. The main objective of this note is to present a method, which hopefully enables one to construct symmetrizers on all homogenous components of the RTT algebras. We illustrate the construction by two low-dimensional examples. A way of extending this method onto other quantum matrix algebras is also discussed.

math.QA↗

arXiv1998

Invariant quantization in one and two parameters on semisimple coadjoint orbits of simple Lie groups

We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.

math.QA↗

arXiv1996

Quantization of function algebras on semisimple orbits in $\g^*$

In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson brackets on any symmetric coadjoint orbit. This paper gives a more explicit algebraic construction which includes more general orbit types and which we prove to be flat in all parameters.

q-alg↗