Cyclic homology and the Macdonald conjectures
Cyclic homology and the Macdonald conjectures
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循环同调和麦克唐纳猜想
DOI：
10.1007/bf01391498
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发表时间：
1986
影响因子：
3.1
通讯作者：
P. Hanlon
中科院分区：
文献类型：
--
作者：
P. Hanlon
设A+（k）表示环G [t]/tk+1，G是约化复李代数，其指数为sm 1，…，m n.本文研究了G <$A+（k）作为双分次代数的李代数上同调（其中一个分次是同调度，另一个分次是从G <$A+（k）的显分次继承而来的，我们称之为权）.我们猜想这个李代数上同调是一个有k +1个同调度为2 ms +1的生成元的外代数，其中s = 1，2，…，n.在这k +1个2 ms +1度的生成元中，一个具有权重0，而其他的具有权重（k+1）ms+t_（fort）= 1，2，...，k.证明了关于λ A+（k）的李代数上同调的猜想蕴含Macdonald根系图.其次，我们考虑G是一个典型的李代数，其根系为An，Bn，Cn或Dn。结果表明，当n趋于无穷大时，我们的猜想在n的极限下成立，这相当于计算A ~+（k）的循环上同调和二面体上同调.最后，我们讨论的相关性，这种情况下的有限的情况下，在这种情况下的限制。
SummaryLetA+(k) denote the ring ℂ[t]/tk+1 and letG be a reductive complex Lie algebra with exponentsm1, ...,m n. This paper concerns the Lie algebra cohomology ofG⊗A+(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofG⊗A+(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2ms+1 fors=1,2, ...,n. Of thesek+1 generators of degree 2ms+1, one has weight 0 and the others have weights (k+1)ms+t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of ⊗A+(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemAn,Bn,Cn, orDn. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.