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