Using compatible actions, we construct new resolutions to compute the cohomology of semidirect products of lie algebras and give an alternative way to construct the hochschildserre spectral. Let me try and push infres as far as it can reasonably go. The lerayserre spectral sequence associated to a bration there is the lerayserre spectral. We prove the above results by using basic facts about the hochschild serre spectral sequence for relative lie algebra cohomology, and most of the paper consists of proving these facts using the. Hochschildserre spectral sequence for not normal subalgebra. Browse other questions tagged liealgebras homologicalalgebra groupcohomology or ask your own question. The derivations, central extensions and automorphism group. Also, the construction of the relative hochschild serre spectral sequence in bw uses the identi. Kathryn hessbellwald supervisedby marc stephan academic year. Proof the spectral sequence is that of hochschildserre see 21, pp. Lie algebra lie algebras lie groups algebra manifolds. Using nonabelian exterior product and free presentation of a lie algebra the hochschildserre spectral sequence for cohomology of lie algebras will b.
Using nonabelian exterior product and free presentation of a lie algebra the hochschildserre spectral sequence for cohomology of lie. The spectral sequence is named after roger lyndon, gerhard hochschild, and jean. The hochschildserre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature. Moreover, if the related spectral sequences collapse over a eld of positive characteristic that is, the di erential. In the case where m is the 1dimensional gmodule with zero operators, this filtration and the corresponding spectral sequence have. M and studying the corresponding spectral sequence.
The hochschild serre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. Using nonabelian exterior product and free presentation of a lie algebra the hochschild serre spectral sequence for cohomology of lie algebras will b. Lyndonhochschildserre spectral sequence for not normal. Also, some results about this sequence are obtained. I have a hopefully simple question about the hochschildserre spectral sequence which may just be a simple question about general spectral sequences. Hochschildserre type spectral sequence attached to an extension of lie algebroids. First quadrant spectral sequences in algebraic ktheory via homotopy colimits. The associated mod 2 cohomology spectral sequence collapses to give the formula h. First we note that for every ideal i of a reductive lie algebra l, we. In this section we fix a graded lie algebra, l l i i, with universal enveloping algebra ul.
Let l be a lie algebra, k be an ideal of l and m be a trivial lmodule. Furthermore, a detailed analysis of the above spectral sequence shows. We use this to identify lmodules as ulmodules and conversely. Lie algebras and lie groups 1964 lectures given at harvard university. Hochschild cohomology versus lie algebra cohomology 8 5. We will see that there is a natural surjective homomorphism from the universal covering algebra. Lie algebra cohomology is an important tool in many branches of mathematics. Stefan st, we see that this spectral sequence degenerates at e2term. A spectral sequence for hochschild cohomology 11 6. A generalization of the cartanleray spectral sequence.
A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. We provide physical insight into useful tools to compute the cohomology, namely morse theory and the hochschildserre spectral sequence. Pdf the relative hochschildserre spectral sequence and. In mathematics, the serre spectral sequence sometimes lerayserre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology. The relative hochschildserre spectral sequence and. The exact sequence of low degree terms in the hochschild serre spectral sequence for cohomology of lie algebras states that 0 hom lk,m. We prove the above results by using basic facts about the hochschildserre spectral sequence for relative lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of hochschild and serre topics. In this subsection we describe one of the main computational tools in lie algebra cohomology the. Pdf let m be a complex manifold and f a ommodule with a gholomorphic action where g is a complex lie algebra cf. The hochschild serre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature. A hochshildserre spectral sequence 11 references 16 date. Here we consider the precisely analogous filtration for the case where g is a lie algebra and k is a subalgebra of g.
Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the lie algebra cohomology. G k\subset g a normal subgroup and a a a left g gmodule. Browse other questions tagged abstractalgebra grouptheory homologicalalgebra spectralsequences or ask your own question. The relative hochschildserre spectral sequence and the belkalekumar product. In the second chapter, we construct the serre spectral sequence of a. Cohomology of quaternion groups 15 acknowledgments 17 references 18 1. Furthermore, by the hochschildserre spectral sequence. The supergravity lie 3algebra is the extension of the super poincare lie algebra by a 4cocycle. The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. Introduction the cohomology of lie groups has a long history. Pdf hochschildserre statement for the total cohomology.
With the aid of an analogue of the hochschildserre type spectral sequence due to d. A generalization of the lyndon hochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. The hochschildserre spectral sequence for an ideal i. Revised 14 april 2017 2000 mathematics subject classi. Calculating the cohomology of a lie algebra using maple and the. Hochschild homology and the homology of free loop spaces winter term 201516. The hochschildserre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. Dedicated to the memory of our friend jeanlouis loday. Thanks to laurent berger and david harari for pointing out independent blunders in earlier versions. The hochschildserre spectral sequence springerlink.
Lyndonhochschildserrehochschild, g serre, jeanpierre 1953, cohomology of group extensions, transactions of the american mathematical society american mathematical society 74 1. On hochschildserre spectral sequence of lie algebras. In w 2 we describe h1sn in all cases, use theorem 2. Serre spectral sequence theorem for group extensions of discrete groups. Thanks for contributing an answer to mathematics stack exchange. We show that the second cohomology group with trivial coefficients for the lie algebra w is two dimensional.
It expresses, in the language of homological algebra, the singular cohomology of the total space x of a serre fibration in terms of the cohomology of the base space b and the fiber f. Let l be a reductive lie algebra and v a semisimple lrepresentation. In this section the ground ring is an arbitrary field k of characteristic zero. The purpose of this paper is to study the structure of the lie algebra w and its universal central extension. Using nonabelian exterior product and free presentation of a lie algebra the hochschildserre spectral sequence for cohomology of lie algebras will be extended a step further. We provide physical insight into useful tools to compute the cohomology, namely morse theory and the hochschild serre spectral sequence. The string lie 2algebra is the extension of a semisimple lie algebra induced by the canonical 3cocycle coming from the killing form. Hochschildserre spectral sequence kevin buzzard june 11, 20 last modi ed 110620. If his a normal subgroup of g, then ghacts freely on bh egh, and so we have a. Relative lie algebra cohomology revisited mark muzere communicated by maurice auslander. In this section we will use the hochschildserre spectral sequence to derive various isomorphism theorems in lie algebra cohomology. We list explicit generators for the lie algebra cohomology ring. Also, the construction of the relative hochschildserre spectral sequence in bw uses the identi.