对称型折叠扩张代数中PBW基与典范基的构造
Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types
浏览论文内容
中文总结 AI 辅助
研究源于Lusztig构造的扩张代数，扩展Varagnolo和Vasserot方法构造斜群代数，证明相关过渡矩阵性质，为对称型情形下PBW基与典范基及其关系提供几何范畴化。
中文摘要 AI 辅助
我们研究了源于Lusztig在配备可允许自同构\(a\)的箭图表示空间上构造的反常层的扩张代数，从而得到折叠的Khovanov - Lauda - Rouquier（KLR）代数。将M. Varagnolo和E. Vasserot的方法扩展到Lusztig的对称型情形，我们构造了斜群代数\(R(\nu)\langle\mathbf{a}\rangle\)，其中\(\mathbf{a}\)由可允许自同构\(a\)诱导，并表明\(\Ext^{\bullet}(\,L)\)保持带符号基和晶体结构。在这个框架下，对于有限型，我们定义了标准和对偶标准模，分析它们在对偶下的行为，并研究所得的Grothendieck群。我们证明从给出典范基的不可分解投射模类到给出PBW基的对偶标准模类的过渡矩阵是对角元素为1的上三角矩阵。这为这些基及其在对称型情形下的关系提供了几何范畴化。
英文摘要
We study extension algebras arising from Lusztig's construction of perverse sheaves on quiver representation spaces equipped with an admissible automorphism $a$, leading to folding Khovanov--Lauda--Rouquier (KLR) algebras. Extending the method of M. Varagnolo and E. Vasserot to Lusztig's symmetrizable setting, we construct the skew group algebra $R(ν)\langle\mathbf{a}\rangle$, where $\mathbf{a}$ is induced by the admissible automorphism $a$, and show that $\Ext^{\bullet}(\ ,L)$ preserves the signed basis and the crystal structure. In this framework, for finite types, we define standard and dual standard modules, analyze their behavior under dualities, and study the resulting Grothendieck groups. We prove that the transition matrix from the classes of indecomposable projective modules, which give the canonical basis, to the dual standard modules, which give the PBW basis, is upper triangular with diagonal entries equal to one. This provides a geometric categorification of these bases and of their relation in the symmetrizable setting.
