Representation of q-analogue of rational Brauer algebras
Representation of q-analogue of rational Brauer algebras
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有理布劳尔代数的 q 模拟的表示
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发表时间：
1997
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通讯作者：
M. Kosuda
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作者：
M. Kosuda
令q 和a 为特征为0 的域K 上的不定式，并令K(a,q) 表示有理函数域。我们通过生成元和关系定义 K(a,q) 上的代数 Hm^n{a,q)。（参见定义 2.1。）如果我们在定义中将不定 a 替换为 q~，我们就得到了有理布劳尔代数 Hrmn{q) 的 q 类似物，我们在之前与 J. Murakami 的论文中介绍过它 [8]。（在论文 [8] 中，我们将代数 Hrmn{q）称为广义赫克代数。）正如我们在[8]中观察到的，在 r>m + n 的情况下，代数 Hrmn{q) 是半简单的。该观察结果扩展到代数 Hm>n(a,q)。也就是说，Hmjn(a,q)也是半简单的。在本文中，我们构造了代数 Hmjn(a,q) 和 Hrmn(q) 的新表示。这些表示是不可约的，它们分别从 Hmjn(a,q) 和 Hrmn{q) 的左正则表示中获得。我们之前的论文最初是为了研究量子代数 ^q{gln{C)) 的混合张量表示的中心化代数，这是 Benkart 等人工作的 ^-模拟版本。 [1]。（他们的论文[1]的初步版本的存在是由冈田教授告知作者的。）他们最初的情况如下。令 G 表示 r x r 可逆复矩阵的一般线性群 GL(r, C)，令 V 为 G 自然作用的向量空间。令 V* 为 V 的对偶空间。 m 个 V 副本和 n 个 V* 副本的混合张量 T 定义为 T = ((x)wF) (x)(0nV*)。在这种情况下，他们通过定位混合张量 T 中的最大向量，构造了居中代数 End^T) 的不可约表示。用 %q{gln{C)) 替换 G 并将基础域 C 扩展到 C(q)，我们
Let q and a be Indeterminates over a field K of characteristic 0, and let K(a,q) denote the field of rational functions. We define the algebra Hm^n{a,q) over K(a,q) by generators and relations.(See the Definition 2.1.)If we replace the indeterminate a with q~rin the definition,we have a q-analogue of rational Brauer algebra Hrmn{q), which we have introduced in the previous paper with J. Murakami [8].(In the paper [8], we called the algebra Hrmn{q) the generalized Hecke algebra.) The algebra Hrmn{q) is semisimple in case r>m + n, as we already observed in [8]. This observation is extended to the algebra Hm>n(a,q). That is to say, Hmjn(a,q) is also semisimple. In this paper, we construct new representations of the algebras Hmjn(a,q) and Hrmn(q). These representations are irreducible and they are obtained from the left regular representations of Hmjn(a,q) and Hrmn{q) respectively. Our previous paper was written originally to investigate the centralizer algebra of mixed tensor representations of quantum algebra ^q{gln{C)), which was ^-analogue version of the work of Benkart et al. [1].(The existence of their preliminary version of the paper [1] was informed to the author by Professor Okada.) Their original situation was as follows. Let G denote the general linear group GL(r, C) of r x r invertible complex matrices and let V be the vector space on which G acts naturally. Let V* be the dual space of V. The mixed tensor T of m copies of V and n copies of V* is defined by T = ((x)wF) (x)(0nV*). In this situation, they constructed the irreducible representations of the centralizer algebra End^T), by locating the maximal vectors in the mixed tensor T. Replacing G with %q{gln{C)) and extending the underlying field C to C(q), we