Volume 1 (2001), Number 2. Abstracts

A. Belavin, S. Gubanov, and B. Feigin. Truncation of functional relations in the XXZ model [PDF]

The integrable XXZ model with a special open boundary condition is considered. We study the Sklyanin transfer matrices after the quantum group reduction at roots of unity. In this case, the Sklyanin transfer matrices satisfy a closed system of truncated functional equations. The algebraic reason for the truncation is found. An important role in the proof of the result is played by the Zamolodchikov algebra introduced in the paper.

Keywords. Quantum groups, quantum algebras, representation theory, integrable XXZ model

2000 Mathematics Subject Classification. 81R12

Yu. Neretin. Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants [PDF]

Consider the pseudounitary group $G=\U(p,q)$ and its compact subgroup $K=\U(p)\times\U(q)$. We survey the analysis of the Berezin kernels on the symmetric space $G/K$. We also explicitly construct unitary intertwining operators from the Berezin representations of $G$ to the representation of $G$ in the space $L^2(G/K)$. This implies the existence of a canonical action of the group $G\times G$ in $L^2(G/K)$.

Keywords. Symmetric space, Cartan domain, positive definite kernel, spherical function, hypergeometric function, Plancherel formula, Hahn polynomials, special functions

2000 Mathematics Subject Classification. 43A85, 22E46, 53C35, 32A25, 43A90, 33C05, 33E20, 15A15

D. Panyushev. Inductive formulas for the index of seaweed Lie algebras [PDF]

A seaweed subalgebra of a semisimple Lie algebra $\mathfrak{g}$ is a generalization of the notion of parabolic subalgebra. In the case $\mathfrak{g}=\mathfrak{sl}(V)$, seaweed subalgebras were recently introduced by Dergachev and Kirillov. We give an inductive procedure for computing the index of seaweed subalgebras of classical Lie algebras. This allows us to prove that the index of any seaweed in $\mathfrak{sl}(V)$ or $\mathfrak{sp}(V)$ is at most the rank of $\mathfrak{g}$. For $\mathfrak{so}(V)$, the problem is reduced to the case of parabolic subalgebras.

Keywords. Index of a Lie algebra, Frobenius Lie algebra, parabolic subalgebra, seaweed subalgebra

2000 Mathematics Subject Classification. 17B20, 17B99

V. Tarasov and A. Varchenko. Small Elliptic Quantum Group $e_{\tau,\gamma}(\mathfrak{sl}_N)$ [PDF]

The small elliptic quantum group $e_{\tau,\gamma}(\sl_N)$, introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U(\sl_N)$. We define highest weight modules, Verma modules, and contragradient modules over $e_{\tau,\gamma}(\sl_N)$, the dynamical Shapovalov form for $e_{\tau,\gamma}(\sl_N)$, and the contravariant form for highest weight $e_{\tau,\gamma}(\sl_N)$-modules. We show that any finite-dimensional $\sl_N$-module and any Verma module over $\sl_N$ can be lifted to the corresponding $e_{\tau,\gamma}(\sl_N)$-module on the same vector space. For the elliptic quantum group $E_{\tau,\gamma}(\sl_N)$ we construct the evaluation morphism $E_{\tau,\gamma}(\sl_N)\to e_{\tau,\gamma}(\sl_N)$, thus making any $e_{\tau,\gamma}(\sl_N)$-module into an evaluation module $E_{\tau,\gamma}(\sl_N)$-module.

Keywords. Dynamical Yang--Baxter equation, elliptic quantum group

2000 Mathematics Subject Classification. 17B37, 81R10

E. Vinberg. Equivariant symplectic geometry of cotangent bundles [PDF]

It is proved that, for any action of a reductive algebraic group $G$ on a quasiaffine algebraic variety $X$, there is a canonical $G$-equivariant symplectic rational Galois covering $f\colon T^*\Hor X\to\TX$, where $\Hor X$ is the variety of horospheres (orbits of maximal unipotent subgroups of $G$) in $X$.

Keywords. Cotangent bundle, symplectic geometry, algebraic group, algebraic variety, horosphere

2000 Mathematics Subject Classification. 14M17, 22E46, 53C30

Moscow Mathematical Journal
is distributed by the
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Independent University of Moscow
Online ISSN 1609-4514
© 2001, Independent University of Moscow
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