量子hom-spaces的实现，不变理论和量子行列式理想
Realizations of quantum hom-spaces, invariant theory and quantum determinantal ideals
摘要：
对于Hecke算子$R$，定义了矩阵双代数$\E_R$，它被认为是与$R$相关联的量子空间的自同态的量子空间上的函数代数。一个推广这个概念，定义函数代数$\M_{RS}$的量子空间的同态的两个量子空间的两个Hecke运营商$R$和$S$分别。$\M_{RS}$可以看作是函数代数在某种程度上的矩阵簇上的量子模拟（或变形）。给出了$\M_{RS}$作为商代数和作为张量代数的子代数的两种实现，由此得到了关于$\M_{RS}$的一些有趣的信息，例如Koszul性质，一个计算Poincaré级数的公式.在$\M_{RS}$上，双代数$\E_R$和$\E_S$相互作用.我们研究了$\M_{RS}$中的双边理想，特别是行列式理想，它们对这些作用是不变的。我们证明了这些量子群和量子hom-spaces不变理论的基本定理的类比。
英文摘要：
For a Hecke operator $R$, one defines the matrix bialgebra $\E_R$, which is considered as the function algebra on the quantum space of endomorphisms of the quantum space associated to $R$. One generalizes this notion, defining the function algebra $\M_{RS}$ on the quantum space of homomorphisms of two quantum spaces associated to two Hecke operators $R$ and $S$ respectively. $\M_{RS}$ can be considered as a quantum analogue (or a deformation) of the function algebra on the variety of matrices of a certain degree. We provide two realiztions of $\M_{RS}$ as a quotient algebra and as a subalgebra of a tensor algebra, whence derive interesting informations about $\M_{RS}$, for instance the Koszul property, a formula for computing the Poincaré series. On $\M_{RS}$ coact the bialgebras $\E_R$ and $\E_S$. We study the two-sided ideals in $\M_{RS}$, invariant with respect to these actions, in particular, the determinantal ideals. We prove analogies of the fundamental theorems on invariant theory for these quantum groups and quantum hom-spaces.
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