Irreducible highest weight representations of the quantum algebra $U_h(A_\infty)$

Details Irreducible highest weight representations of the quantum algebra $U_h(A_\infty)$ Irreducible highest weight representations of the quantum algebra $U_h(A_\infty)$

1998-07-28

arxiv.org/abs/math/9807157v1

Mathematics

Quantum Algebra

32 pages, TeX, an extended version of q-alg/9709004, containing all proofs. To appear in Journ. Math. Phys

Scientific paper

A class of highest weight irreducible representations of the algebra $U_h(A_\infty)$, the quantum analogue of the completion and central extension $A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably larger than the class of representations known so far. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly given. The verification of the quantum algebra relations is shown to reduce to a set of nontrivial $q$-number identities. All representations are restricted in the terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. {\bf 140}, 399-414 (1991)).

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