Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations
Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations
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DOI：
10.1215/21562261-1214375
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发表时间：
2010-02
影响因子：
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通讯作者：
B. Feigin;E. Feigin;M. Jimbo;T. Miwa;E. Mukhin
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文献类型：
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作者：
B. Feigin;E. Feigin;M. Jimbo;T. Miwa;E. Mukhin
我们开始研究量子连续的表示理论$\mathfrak{gl}_\infty$，我们用$\mathcal E$表示。该代数依赖于两个参数，是作用于一元洛朗多项式空间的差分算子李代数的包络代数的变形版本。$\mathcal E$的基本表示用一个连续参数$u\in {\mathbb C}$来标记。$\mathcal E$的表示理论有许多与$\mathfrak{gl}_\infty$的表示理论相似的性质：向量表示、Fock模块、模块的半无限构造。利用向量表示的张量积，构造了从$\mathcal E$到所有$N$的球面双仿射Hecke代数$S\ddot H_N$的满射同态。这个构造的关键步骤是确定向量表示与麦克唐纳多项式的张量积的自然基。我们还证明了其中一个Fock表示与先前利用Hilbert格式的$K$ -理论构造的模块是同构的。
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, semi-infinite 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.