FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 2, PAGES 183-199

Categories of bounded $(\mathfrak{sp}(\mathrm S^2V \oplus \mathrm S^2V^*), \mathfrak{gl}(V))$- and $(\mathfrak{sp}(\Lambda^2V \oplus \Lambda^2V^*), \mathfrak{gl}(V))$-modules

A. V. Petukhov

Abstract

View as HTML     View as gif image

Let $ \mathfrak g $ be a reductive Lie algebra over C and $ \mathfrak k \subset \mathfrak g $ be a reductive in $ \mathfrak g $ subalgebra. We call a $ \mathfrak g $-module M a $ (\mathfrak g, \mathfrak k) $-module whenever M is a direct sum of finite-dimensional $ \mathfrak k $-modules. We call a $ (\mathfrak g, \mathfrak k) $-module M bounded if there exists CM Î Z³0 such that for any simple finite-dimensional $ \mathfrak k $-module E the dimension of the E-isotypic component is not more than CM dim E. Bounded $ (\mathfrak g, \mathfrak k) $-modules form a subcategory of the category of $ \mathfrak g $-modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ (\mathfrak{sp}(\mathrm S^2V \oplus \mathrm S^2V^*), \mathfrak{gl}(V)) $-modules and $ (\mathfrak{sp}(\Lambda^2V \oplus \Lambda^2V^*), \mathfrak{gl}(V)) $-modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V.

Main page Contents of the journal News Search

Location: http://mech.math.msu.su/~fpm/eng/k1112/k112/k11208h.htm
Last modified: March 6, 2012