ON THE COHOMOLOGY OF THE LIE ALGEBRA ARISING FROM THE LOWER CENTRAL SERIES OF A p-GROUP
نویسنده
چکیده
We study the cohomology H(A) = Ext∗A(k, k) of a locally finite, connected, cocommutative Hopf algebra A over k = Fp. Specifically, we are interested in those algebras A for which H∗(A) is generated as an algebra by H(A) and H(A). We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras F → A → B with F monogenic and B semiKoszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for A to be semi-Koszul. Special attention is given to the case in which A is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-p lower central series of a p-group. We show that the algebras arising in this way from extensions by Z/(p) of an abelian p-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 p-groups, and it is shown that these are all semiKoszul for p ≥ 5.
منابع مشابه
Some properties of nilpotent Lie algebras
In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.
متن کاملOn continuous cohomology of locally compact Abelian groups and bilinear maps
Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...
متن کاملCohomology of aff(1|1) acting on the space of bilinear differential operators on the superspace IR1|1
We consider the aff(1)-module structure on the spaces of bilinear differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra aff(1) with coefficients in space Dλ,ν;µ of bilinear differential operators acting on weighted densities. We study also the super analogue of this problem getting the same results.
متن کاملOn dimensions of derived algebra and central factor of a Lie algebra
Some Lie algebra analogues of Schur's theorem and its converses are presented. As a result, it is shown that for a capable Lie algebra L we always have dim L=Z(L) 2(dim(L2))2. We also give give some examples sup- porting our results.
متن کاملMODULE GENERALIZED DERIVATIONS ON TRIANGULAUR BANACH ALGEBRAS
Let $A_1$, $A_2$ be unital Banach algebras and $X$ be an $A_1$-$A_2$- module. Applying the concept of module maps, (inner) modulegeneralized derivations and generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $A_i$ into the dual space $A^*_i$ (for$i=1,2$) and such derivations from the triangular Banach algebraof t...
متن کامل