Quantum continuous $\mathfrak{gl}_\infty$: Semiinfinite construction of representations
Abstract
We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal E$ to spherical double affine Hecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$theory of Hilbert schemes.
 Publication:

arXiv eprints
 Pub Date:
 February 2010
 DOI:
 10.48550/arXiv.1002.3100
 arXiv:
 arXiv:1002.3100
 Bibcode:
 2010arXiv1002.3100F
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory
 EPrint:
 23 pages