量子群的De Rham上同调与Hodge分解
De Rham Cohomology and Hodge decomposition for Quantum Groups
摘要：
设G=G（t，z）是量子群GL_q（N），SL_q（N），O_q（N）或Sp_q（N）的N ~ 2维双共变一阶微分演算之一，其中q是超越复数，z是正则参数.证明了Woronowicz外代数G^的de Rham上同调与它的左上不变子复形、右上不变子复形和（双侧）上不变子复形的de Rham上同调是一致的.在GL_q（N）和SL_q（N）情形下，上同调环同构于上不变外代数G^_{inv}和调和型向量空间.在这些情况下，我们证明了霍奇分解定理。主要的技术工具是量子Laplace-Beltrami算子的谱分解。关键词：量子群，双协变微分演算，德拉姆上同调，拉普拉斯-贝尔特拉米算子，霍奇理论
英文摘要：
Let G=G(t,z) be one of the N^2-dimensional bicovariant first order differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or Sp_q(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra G^ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) the cohomology ring is isomorphic to the coinvariant external algebra G^_{inv} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. Keywords: quantum groups, bicovariant differential calculi, de Rham cohomology, Laplace-Beltrami operator, Hodge theory
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