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