QUANTUM gl∞, INFINITE q-SCHUR ALGEBRAS AND THEIR REPRESENTATIONS
نویسندگان
چکیده
In this paper, we investigate the structure and representations of the quantum group U(∞) = Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [1], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞, r) is not surjective for any r > 1. We will give a BLM type realization for the image U(∞, r) := ζr(U(∞)) and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra K̂ † (∞) so that ζr can be extended to an algebra epimorphism ζ̃r : K̂ † (∞) → S(∞, r). Finally we will investigate the representation theory of U(∞), especially the polynomial representations of U(∞).
منابع مشابه
QUANTUM gln, q-SCHUR ALGEBRAS AND THEIR INFINITE/INFINITESIMAL COUNTERPARTS
We present a survey of recent developments of the Beilinson–Lusztig–MacPherson approach in the study of quantum gl n , infinitesimal quantum gl n , quantum gl ∞ and their associated q-Schur algebras, little q-Schur algebras and infinite q-Schur algebras. We also use the relationship between quantum gl ∞ and infinite q-Schur algebras to discuss their representations.
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