arXiv 1102.51652011-02math.RT

量子广义Kac-Moody代数与晶体基的范畴化

Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases

Seok-Jin Kang, Se-jin Oh, Euiyong Park

备注：: We corrected typos (and a few small errors) and changed the definition of KLR algebras to more general version

摘要：

构造了Khovanov-Lauda-Rouquier代数$R$及其分圆子$R^λ$，并研究了它们的结构.设$U_\A(\g)$是与Borcherds-Cartan矩阵$A=(a_{ij})_{i,j \in I}$相关联的量子广义Kac-Moody代数的积分形式，$K_0(R)$是广义生成的投射分次$R$ -模的Grothedieck群.本文证明了存在一个单射代数同态$Φ: U_\A^-(\g) \to K_0(R)$，并且如果对所有的$i\in I$都有$a_{ii}\ne 0$，则$Φ$是一个同构.令$B(\infty)$和$B(λ)$分别是$U_q^-(\g)$和$V(λ)$的晶体，其中$V(λ)$是不可约的最高权$U_q(\g)$ -模。我们分别用$\mathfrak{B}(\infty)$和$\mathfrak{B}(λ)$表示$R$和$R^λ$上不可约分次模的同构类.如果对所有的$i\in I$都有$a_{ii}\ne 0$，我们定义了$\mathfrak{B}(\infty)$和$\mathfrak{B}(λ)$上的$U_q(\g)$ -晶体结构，并证明了存在晶体同构$\mathfrak{B}(\infty) \simeq B(\infty)$和$\mathfrak{B}(λ) \simeq B(λ)$ .我们的方法的关键成分之一是广义Kac-Moody代数的完美基理论。

英文摘要：

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $Φ: U_\A^-(\g) \to K_0(R)$ and that $Φ$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(\infty)$ and $B(λ)$ be the crystals of $U_q^-(\g)$ and $V(λ)$, respectively, where $V(λ)$ is the irreducible highest weight $U_q(\g)$-module. We denote by $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$ the isomorphism classes of irreducible graded modules over $R$ and $R^λ$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we define the $U_q(\g)$-crystal structures on $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$, and show that there exist crystal isomorphisms $\mathfrak{B}(\infty) \simeq B(\infty)$ and $\mathfrak{B}(λ) \simeq B(λ)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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