代数学系列报告I

代数学系列报告I

2022年9月16日

会议报告摘要

Hall algebras and quantum symmetric pairs

卢明

四川大学

A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). A quantum group can be viewed as an example of i-quantum groups associated to symmetric pairs of diagonal type.

In this talk, we present a new Hall algebra construction of i-quantum groups. Our approach leads to monomial bases, PBW bases, and braid group actions for i-quantum groups. In case of symmetric pairs of diagonal type, our work reduces to a reformulation of Bridgeland's Hall algebra realization of a quantum group, which in turn was a generalization of earlier constructions of Ringel and Lusztig for half a quantum group.

This is joint work with Weiqiang Wang.

（一个量子对称对包含一个量子群和它的一个余理想子代数，称为i-量子群。量子群可以看作为特殊的i-量子群。本报告中，我们会展示i-量子群的一个新的Hall代数构造。我们的构造能够导出i-Hall代数的单项式基，PBW基以及辫子群作用等。）

$\imath$Hall algebra of the projective line and $q$-Onsager algebra

阮诗佺

厦门大学

The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations. This is joint work with Ming Lu and Weiqiang Wang.  

（投射线的i-Hall代数可以通过相应凝聚层的1周期复形范畴的Hall代数的扭得到。这类i-Hall代数可以实现Onsager代数。接下来我们考虑Kronecker箭图的i-Hall代数，之后我们会证明这两个i-Hall代数的导出等价的。