B2型量子群的PBW形变及其对称性

量子群的负部分是量子群理论中出现的一类重要的连通分次代数,其PBW形变简称为量子群的PBW形变。 除了A2情形,它们的定义关系式均是具有混合次数的齐次量子Serre关系式。 特别地, B2型量子群的负部分的定义关系式分别是次数为3和4的2个量子Serre关系式。 在连通分次代数的PBW形变理论的框架下, 本文明确刻画了B2型量子群的所有PBW形变并研究了它们的对称性, 即给出了B2型量子群的4类对合反自同构下的对称PBW形变。

PBW-deformations of type B2 quantum group and their symmetry property

The negative parts of quantum groups are very important connected graded algebras appeared in the quantum group theory. Their PBW-deformations are called the PBW-deformations of quantum groups. Except the A2 case, the defining relations of these algebras are homogeneous quantum Serre relations with mixed degrees. In particular, the defining relations of the negative part of type B₂ quantum group are respectively the homogeneous quantum Serre relations with degree 3 and 4. In this paper, in the framework of PBW-deformations of connected graded algebras, we explicitly characterize all the PBW-deformations of type B₂ quantum group and investigate their symmetry property, i.e., all the symmetric PBW-deformations of type B₂ quantum group under four kinds of involution anti-automorphisms are obtained.