论文标题
紧凑型半圣量子基团的晶体极限作为高级图代数
Crystal limits of compact semisimple quantum groups as higher-rank graph algebras
论文作者
论文摘要
令$ o_q [k] $表示有理函数的字段上的量化坐标环$ \ mathbb {c}（q）$，对应于配备其 * - 结构的紧凑型半完整的lie组$ k $。令$ a_0 $ in $ \ mathbb {c}（q）$表示定期函数的子来$ q = 0 $。我们引入了$ a_0 $ -subalgebra $ o_q^{a_0} [k] $ o_q [k] $的$，相对于 * - 结构是稳定的，并且相对于Crystal Limit限制$ Q \ f具有以下属性。 $ o_q [k] $在每个$ q $ in $（0，\ infty）\ setMinus \ {1 \} $上的专业化允许在固定的Hilbert Space上忠实 * - 代表$π_Q$，这是Soibelman引起的结果。我们表明，对于每个元素，$ a $ a $ in $ o_q^{a_0} k $，运营商的家族$π_q（a）$将标准限制允许为$ q \ to 0 $。这些限制定义了a *-presendation $π_0$ $ o_q^{a_0} k $。我们表明，结果 *-algebra $ o [k_0] =π_0（o_q^{a_0} [k]）$是kumjian-pask代数，从Aranda Pino，Clark，huef和Raeburn的意义上讲。我们在晶体基理论方面对基础高级图进行了明确描述。结果，我们获得了$ c^*$ - 代数$（c（k_q））_ {q \ in [0，\ infty]} $的连续字段，其中$ q = 0 $＆$ \ infty $的光纤被明确定义为高级级别的高级图形。
Let $O_q[K]$ denote the quantized coordinate ring over the field $\mathbb{C}(q)$ of rational functions corresponding to a compact semisimple Lie group $K$, equipped with its *-structure. Let $A_0$ in $\mathbb{C}(q)$ denote the subring of regular functions at $q=0$. We introduce an $A_0$-subalgebra $O_q^{A_0}[K]$ of $O_q[K]$ which is stable with respect to the *-structure, and which has the following properties with respect to the crystal limit $q \to 0$. The specialization of $O_q[K]$ at each $q$ in $(0,\infty)\setminus\{1\}$ admits a faithful *-representation $π_q$ on a fixed Hilbert space, a result due to Soibelman. We show that for every element $a$ in $O_q^{A_0}K$, the family of operators $π_q(a)$ admits a norm-limit as $q \to 0$. These limits define a *-representation $π_0$ of $O_q^{A_0}K$. We show that the resulting *-algebra $O[K_0]=π_0(O_q^{A_0}[K])$ is a Kumjian-Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of $C^*$-algebras $(C(K_q))_{q\in[0,\infty]}$, where the fibres at $q = 0$ and $\infty$ are explicitly defined higher-rank graph algebras.
