Some Results on the Structure of Quantum Family Algebras
نویسندگان
چکیده
We continue the study of family algebras introduced by the author. In this paper we describe completely the structure of quantum family algebras for two cases of representations with a simple spectrum. 1. Generalities about family algebras 1.1. Basic definitions. A new class of associative algebras related to simple complex Lie algebras (or root systems) was introduced and studied in [K]. They were named classical and quantum family algebras. The aim of the this paper is to expose some results about the structure of quantum family algebras. In particular, we give a partial answer to the last of several open questions formulated in [K]: In general, it would be very interesting to find out which quantum family algebras are commutative and which classical algebras are spanned over I(g) by powers of M or analogous elements related to other generators of I(g). We assume that the reader is acquainted with the general background of the theory of semi-simple Lie algebras (see e.g. [OV]). Let g be a simple complex Lie algebra with the canonical decomposition g = n− ⊕ h⊕ n+. We denote by P (respectively by Q) the weight (resp. root) lattice in h∗ and by P+ (resp. Q+) the semigroup generated by fundamental weights ω1, ω2, . . . , ωl (resp. by simple roots α1, α2, . . . , αl). 1991 Mathematics Subject Classification. 17 B3.
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