Lie algebra cohomology and the product in the free loop space homology

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

Symmetric (co)homologies of Lie algebras

Compatible L∞-algebras

A new algorithmic approach for computing cohomologies of Lie (super)algebras is described. This approach is based on splitting a complete cochain complex into minimal (in some noninvariant sense) subcomplexes; this makes the computations much more efficient due to the fact that the proposed splitting allows the cochain complex spaces (which, normally, have very large dimensions) to be decomposed into much smaller subspaces. The algorithms in a pseudocode are presented. The approach is illustrated by comprehensive examples.

This book provides an introduction to the cohomology theory of Lie groups and Lie algebras and to some of its applications in physics. The mathematical topics covered include the differential geometry of Lie groups, fibre bundles and connections, characteristic classes, index theorems, extensions of Lie groups and algebras, Chevalley - Eilenberg cohomology of Lie algebras, symplectic cohomology and an introduction to infinite-dimensional Lie groups and algebras. The physical applications include the U(1) (Dirac) monopole, SU(2) instantons and various aspects of anomalies (Wess - Zumino - Witten terms, Abelian and non-Abelian anomaly, path-integral derivation and descent equations). The material presented is essentially self-contained and at a basic graduate text level. The material is also well organized and the book reads very well. The book would be most useful for graduate students and researchers in theoretical and mathematical physics who are interested in applications of Lie group and Lie algebra cohomology in particle physics. Even though most of the proofs of the mathematical theorems are presented, the focus is more on explaining the ideas than striving for mathematical rigour, therefore the book would be less suitable for those students who are interested in the mathematical foundations per se. Since the book seems to aim for physics students, it is a pity that - apart from the topic of anomalies which is covered very thoroughly - the applications are only briefly touched upon. Other applications of current interest, such as non-Abelian monopoles and instantons for gauge groups other than SU(2) as well as their moduli spaces, are not discussed at all even though the necessary mathematical background is presented and thus they seem well within the scope of the book. It must be said, however, that the lack of different applications is compensated for by an excellent set of bibliographical notes and references at the end of each chapter. The book is warmly recommended.

On the cohomology of Lie algebras associated with graphs

COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS

Note on the cohomology of color Hopf and Lie algebras

Given a graded vector space V, the variety of complexes Com(V) consists of all differentials making V into a cochain complex. This variety was first introduced by Buchsbaum and Eisenbud and later studied by Kempf, De Concini, Strickland and many other people. It is highly singular and can be seen as a proto-typical singular moduli space in algebraic geometry. We introduce a natural derived analog of Com(V) which is a smooth derived scheme RCom(V). It can be seen as the derived scheme classifying twisted complexes. We study the cohomology of the dg-algebra of regular functions on RCom(V). It turns out that the natural action of the group GL(V) (automorphisms of V as a graded space) on the cohomology has simple spectrum. This generalizes the known properties of Com(V) and the classical theorem of Kostant on the Lie algebra cohomology of upper triangular matrices.

We study the derived representation scheme DRep _n(A) parametrizing the n -dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep _n(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra \mathfrak {gl}_n^*(\bar{C}) defined over a Koszul dual coalgebra of A . This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras \mathfrak {gl}_n(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra \mathfrak g , we define the derived affine scheme DRep _{\mathfrak g}(\mathfrak a) parametrizing the representations (in \mathfrak g ) of a Lie algebra \mathfrak{a} ; we show that the homology of DRep_ {\mathfrak g}(\mathfrak a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra \mathfrak g^*(\bar{C}) , where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra \mathfrak a . We construct a canonical DG algebra map \Phi_{\mathfrak g}(\mathfrak a): \mathrm {DRep}_{\mathfrak g}(\mathfrak a)^G \to \mathrm {DRep}_{\mathfrak h}(\mathfrak a)^W , relating the G -invariant part of representation homology of a Lie algebra \mathfrak a in \mathfrak g to the W -invariant part of representation homology of \mathfrak a in a Cartan subalgebra of \mathfrak g . We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map. We conjecture that, for a two-dimensional abelian Lie algebra \mathfrak{a} , the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for \mathfrak {gl}_2 and \mathfrak {sl}_2 as well as for \mathfrak {gl}_n, \mathfrak {sl}_n, \mathfrak{so}_n and \mathfrak{sp}_{2n} in the inductive limit as n \to \infty . For any complex reductive Lie algebra \mathfrak g , we compute the Euler characteristic of DRep _{\mathfrak g}(\mathfrak a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep _{\mathfrak h}(\mathfrak a)^W . This yields an interesting combinatorial identity, which we prove for \mathfrak {gl}_n and \mathfrak{sl}_n (for all n ). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [Ha1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in [32]. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.

Deformations, cohomologies and integrations of relative difference Lie algebras

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.

Post-Lie algebra structures on pairs of Lie algebras

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic p>h, where h is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras of semisimple and simply connected algebraic groups. To describe the cohomology of simple modules, we will use the properties of the connections between ordinary and restricted cohomology of restricted Lie algebras.