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

A class of highest weight irreducible representations of the algebra $U_h(A_{\mathrm{\infty }})$, the quantum analogue of the completion and central extension $A_{\mathrm{\infty }}$ of the Lie algebra $g{l}_{\mathrm{\infty }}$, 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)).