量子李代数的BRST算子：与棒复形的关系
BRST Operator for Quantum Lie Algebras: Relation to Bar Complex
摘要：
量子李代数（一类重要的二次代数，产生于量子群上的Woronowicz演算）是李（超）代数的推广。李（超）代数理论中的许多概念都承认“量子”的推广。特别地，有一个BRST算子Q（Q^2=0），它产生了Woronowicz理论中的微分，并给出了量子李代数的（余）同调的信息。在我们以前的文章中，给出并解决了量子李代数中算子Q的一个递推关系。本文研究q-李代数的杆复形及其q-反对称链的子复形。建立了q-李代数的标准复形到反对称链的子复形的链映射（同构）。该构造需要辫群的群代数中的一组非平凡恒等式。我们还讨论了当q-李代数具有分次算子时标准复形的推广。
英文摘要：
Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super) algebras. Many notions from the theory of Lie (super)algebras admit ``quantum'' generalizations. In particular, there is a BRST operator Q (Q^2=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given and solved. Here we consider the bar complex for q-Lie algebras and its subcomplex of q-antisymmetric chains. We establish a chain map (which is an isomorphism) of the standard complex for a q-Lie algebra to the subcomplex of the antisymmetric chains. The construction requires a set of nontrivial identities in the group algebra of the braid group. We discuss also a generalization of the standard complex to the case when a q-Lie algebra is equipped with a grading operator.
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