Crystal bases, dilogarithm identities and torsion in algebraic $K$-theory
Crystal bases, dilogarithm identities and torsion in algebraic $K$-theory
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代数 $K$ 理论中的晶体基、双对数恒等式和挠率
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
A. Szenes
中科院分区：
文献类型：
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作者：
E. Frenkel;A. Szenes
和~j = sin2ill / sin21c ~:i)·它们先前由Kirillov和Reshetikhin[18]证明。在Lewin b[23]、Richmond和Szekeres b[30]的早期作品中也出现了相关的身份。2恒等式（1.1）的右侧等于~乘以仿射Kac-Moody李代数Si的可积层k表示的中心电荷；这个数出现在这种表示的渐近特征中。按照我们之前的工作[9]中采用的一般方法，我们将通过对该渐近性的另一种计算得到左边。我们将使用可积Sl的所谓晶体基[17]（cf. also[25]）的组合描述；-由Jimbo， Misra， Miwa和Okado在b[12]中发现的k级模块V k（参见[15,16]）。他们在晶体基向量上定义了一个权函数w，这样，如果我们将qW加起来，我们就得到了V k的特征。我们将引入一个由所有晶体基向量集合的有限子集组成的递增系统，并证明相应的部分特征与q递归关系相关。这就给出了一个公式
and ~j = sin2 ill/ sin2 1C~:i)· They were previously proved by Kirillov and Reshetikhin [18]. Related identities also appeared earlier in the works of Lewin [23], and Richmond and Szekeres [30]. 2 The right-hand side of the identity (1.1) is equal to ~ times the central charge of an integrable level k representation of the affine Kac-Moody Lie algebra Si;. This number appears in the asymptotics of the character of such a representation. Following the general approach taken in our previous work [9], we will obtain the left-hand side as a result of an alternative calculation of this asymptotics. We will use a combinatorial description of the so-called crystal basis [17] (cf. also [25]) of an integrable Sl;-module V k of level k, found by Jimbo, Misra, Miwa, and Okado in [12] (cf. also [15, 16]). They defined a weight function w on the crystal basis vectors, such that if we sum up qW over them, we obtain the character of V k • We will introduce an increasing system of finite subsets of the set of all crystal basis vectors and show that the corresponding partial characters are related by a q-recurrence relation. This will give us a formula for