Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations
Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations
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发表时间：
2007-08
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通讯作者：
J. Du;Qiang Fu
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作者：
J. Du;Qiang Fu
在本文中，我们研究了量子群${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$的结构和表示。我们将根据Beilinson- Lusztig- MacPherson (BLM) \cite{BLM}给出$\mathbf{U}(\infty)$的实现，并证明从$\mathbf{U}(\infty)$到无限$q$ -Schur代数${\boldsymbol{\mathcal S}}(\infty,r)$的自然代数同态$\zeta_r$对于任何$r\geq 1$都不是满射。我们将给出图像$\mathbf{U}(\infty,r):=\zeta_r(\mathbf{U}(\infty))$的BLM类型实现，并从生成器和关系的角度讨论其表示。我们进一步构造了一个补全代数$\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$，使得$\zeta_r$可以推广为一个代数上的外胚$\tilde\zeta_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$。最后我们将研究${\bf U}(\infty)$的表示理论，特别是${\bf U}(\infty)$的多项式表示。
In this paper, we investigate the structure and representations of the quantum group ${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$. We will present a realization for $\mathbf{U}(\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism $\zeta_r$ from $\mathbf{U}(\infty)$ to the infinite $q$-Schur algebra ${\boldsymbol{\mathcal S}}(\infty,r)$ is not surjective for any $r\geq 1$. We will give a BLM type realization for the image $\mathbf{U}(\infty,r):=\zeta_r(\mathbf{U}(\infty))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra $\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$ so that $\zeta_r$ can be extended to an algebra epimorphism $\tilde\zeta_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will investigate the representation theory of ${\bf U}(\infty)$, especially the polynomial representations of ${\bf U}(\infty)$.