Highest weight irreducible representations of the quantum algebra $U_h(A_{\infty })$

Abstract

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 $g{l}_{\infty }$, is constructed. It is considerably larger than the known so far representations. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written. The verification of the quantum algebra relations to be satisfied is shown to reduce to a set of nontrivial $q$-number identities. All our representations are restricted in the terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. 140, 399-414 (1991)).

17 pages, TeX

References in corpus (5)

• Many-Body Wigner Quantum Systems
• Finite-dimensional representations of the quantum superalgebra $U_q[gl(n/m)]$ and related q-identities
• Many-body States and Operator Algebra for Exclusion Statistics
• Finite-Dimensional Representations of the Quantum Superalgebra U${}_q$[gl(2/2)]: II. Nontypical representations at generic $q$
• Highest weight representations of the quantum algebra U_h(gl_\infty)

Cited by in corpus (3)

• Gelfand-Tsetlin bases for classical Lie algebras
• Quantum ${\mathfrak{gl}}_{\infty }$, infinite $q$-Schur algebras and their representations
• Two-Parameter Quantum Groups and Ringel-Hall algebras of $A_{\infty }-$type