Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank

We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation quantizations equivariant under the action ofG and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

Some Poisson structures associated to Drinfeld-Jimbo R-matrices and their quantization

We describe various ways of constructing Poisson structures associated to Drinfeld-Jimbo R-matrices on homogeneous spaces, research some relations between these structures, and quantize some of them.     

Decomposition of symmetric and exterior powers of the coadjoint representation of gl(n) in the unstable case

Central Institute of Economical Mathematics, RAN. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 4, pp. 70–73, October–December, 1992.     

Diode-pumped green Nd:YAG laser with Q-switch and mode locking

We propose a new method for achieving simultaneous operation of laser mode locking and the Q-switch technique using only a single acousto-optic modulator (AOM) with a traveling wave; this AOM was placed inside the cavity of a green-emission Nd:YAG laser. The further shortening of the lasing pulse duration from 40 ps to less than 3.25 ps was obtained by the formation of a Kerr lens in a doubling-frequency crystal. At average output power of 1.5 W and pulse repetition rate of the Q-switch equal to 2 kHz, the peak power in a steady-operating laser exceeded 50 MW.     

Method of Quantum Characters in Equivariant Quantization

 Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose 𝔸 _h (M) is a 𝕌 _h (g)-equivariant quantization of the function algebra 𝔸(M) on M. We develop a method of building 𝕌 _h (g)-equivariant quantization on G-orbits in M as quotients of 𝔸 _h (M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in 𝕌^* _h (g) and quotients of 𝔸 _h (M). It turns out that they are in one-to-one correspondence with characters of the algebra 𝔸 _h (M). We specialize our approach to the situation g=gl(n,ℂ), M=End(ℂ ⁿ ), and 𝔸 _h (M) the so-called reflection equation algebra associated with the representation of 𝕌 _h (g) on ℂ ⁿ . For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the 𝕌(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits. Received: 28 April 2002 / Accepted: 3 October 2002 Published online: 24 January 2003 RID="*" ID="*" This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01. Communicated by L. Takhtajan     

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group U_hg and is commutative with respect to an involutive operator . Such a multiplication is unique. Let M be a kählerian symmetric space with the canonical Poisson structure. Then we construct a U_hg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.