Lie Algebra Cohomology Research Articles

A Compendium of Lie Structures on Tensor Products

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.

The cohomology of filiform Lie algebras of maximal rank

We describe the structure of the cohomology of the filiform Lie algebras Ln and Qn as a module over their (2-dimensional) torus of derivations. Our approach relies on the fact that both filiform algebras have an ideal h of codimension 1 for which the structure of its cohomology under the action of the Levi factor of the algebra of derivations of h is known.

Inequalities for the second cohomology of finite dimensional Lie algebras

Inequalities for the second cohomology of finite dimensional Lie algebras

Dirac cohomology, K -characters and branching laws

Inspired by work of Enright and Willenbring, we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandzic, and Renard.

The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product

We consider the Belkale-Kumar cup product ⊙ t \odot _t on H ∗ ( G / P ) H^*(G/P) for a generalized flag variety G / P G/P with parameter t ∈ C m t \in \mathbb {C}^m , where m = dim ⁡ ( H 2 ( G / P ) ) m=\dim (H^2(G/P)) . For each t ∈ C m t\in \mathbb {C}^m , we define an associated parabolic subgroup P K ⊃ P P_K \supset P . We show that the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) contains a graded subalgebra A A isomorphic to H ∗ ( P K / P ) H^*(P_K/P) with the usual cup product, where P K P_K is a parabolic subgroup associated to the parameter t t . Further, we prove that ( H ∗ ( G / P K ) , ⊙ 0 ) (H^*(G/P_K), \odot _0) is the quotient of the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) with respect to the ideal generated by elements of positive degree of A A . 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 original approach of Hochschild and Serre.

About Leibniz cohomology and deformations of Lie algebras

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of the Leibniz (and Lie) universal infinitesimal deformation of the 4-dimensional diamond Lie algebra, used in the WZW model, and study the case of its 5-dimensional analogue by computing Massey products.

Split abelian chief factors and first degree cohomology for Lie algebras

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As an application we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.

Embeddings of semisimple complex Lie groups and cohomological components of modules

Let G↪G˜ be an embedding of semisimple complex Lie groups, B⊂B˜ a pair of nested Borel subgroups and G/B↪G˜/B˜ the associated embedding of flag manifolds. Let O˜(λ˜) be an equivariant invertible sheaf on G˜/B˜ and O(λ) be its restriction to G/B. Consider the G-equivariant pullbackπλ˜:H(G˜/B˜,O˜(λ˜))→H(G/B,O(λ)). The Borel–Weil–Bott theorem and Schurʼs lemma imply that πλ˜ is either surjective or zero. If πλ˜ is nonzero, the image of the dual map (πλ˜)⁎ is a G-irreducible component in a G˜-irreducible module, called a cohomological component.We establish a necessary and sufficient condition for nonvanishing of πλ˜. Also, we prove a theorem on the structure of the set of pairs of dominant weights (μ,μ˜) with V(μ)⊂V˜(μ˜) cohomological. Here V(μ) and V˜(μ˜) denote the respective highest weight modules. Simplified specializations are formulated for regular and diagonal embeddings. In particular, we give an alternative proof of a recent theorem of Dimitrov and Roth. Beyond the regular and diagonal cases, we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad-invariant polynomials on a semisimple Lie algebra can be obtained as cohomological components. Our methods rely on Kostantʼs theory of Lie algebra cohomology.

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.

SDiff(2) and uniqueness of the Plebański equation

The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive the second Plebański equation and its geometry. We do not use Kähler or other additional structures but obtain the equation solely from the geometry of area preserving transformations group. We conclude that the Plebański equation is Lie remarkable.

The minimal model for the Batalin–Vilkovisky operad

The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then, we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne–Mumford–Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.

Galileons as Wess-Zumino terms

We show that the galileons can be thought of as Wess-Zumino terms for the spontaneous breaking of space-time symmetries. Wess-Zumino terms are terms which are not captured by the coset construction for phenomenological Lagrangians with broken symmetries. Rather they are, in d space-time dimensions, d-form potentials for (d+1)-forms which are non-trivial co-cycles in Lie algebra cohomology of the full symmetry group relative to the unbroken symmetry group. We introduce the galileon algebras and construct the non-trivial (d+1)-form co-cycles, showing that the presence of galileons and multi-galileons in all dimensions is counted by the dimensions of particular Lie algebra cohomology groups. We also discuss the DBI and conformal galileons from this point of view, showing that they are not Wess-Zumino terms, with one exception in each case.

The Belkale–Kumar Cup Product and Relative Lie Algebra Cohomology

We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters.

A van Est Isomorphism for Bicrossed Product Hopf Algebras

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.

On Hochschild–Serre spectral sequence of Lie algebras

Using non-abelian exterior product and free presentation of a Lie algebra the Hochschild–Serre spectral sequence for cohomology of Lie algebras will be extended a step further. Also, some results about this sequence are obtained.

Rigid Schubert varieties in compact Hermitian symmetric spaces

Given a singular Schubert variety X w in a compact Hermitian symmetric space X, it is a long-standing question to determine when X w is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong. Key tools include (i) a new characterization of Schubert varieties that generalizes the well-known description of the smooth Schubert varieties by connected sub-diagrams of a Dynkin diagram and (ii) an algebraic Laplacian (à la Kostant), which is used to analyze the Lie algebra cohomology group associated with the problem.

Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$, and the Segre product $\mathit{Seg}(\mathbb{P}^1\times \mathbb{P}^n)$, both varieties with osculating sequences of length two, are flexible at order two. In the $SL_3\mathbb{C}$ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques.

Cyclic cohomology of Lie algebras

We define and completely determine the category of Yetter-Drinfeld modules over Lie algebras. We prove a one to one correspondence between Yetter-Drinfeld modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. We associate a mixed complex to a Lie algebra and a stable-Yetter-Drinfeld module over it. We show that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G2-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.

A Gröbner-Bases Algorithm for the Computation of the Cohomology of Lie (Super) Algebras

We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a field \({\mathbb{K}}\) of characteristic zero. In order to reach explicit representatives of some generators of the quotient space \({\fancyscript{Z}^ k /\fancyscript{B}^k}\) of cocycles \({\fancyscript{Z}^k}\) modulo coboundaries \({\fancyscript{B}^k}\) , we apply Gröbner bases techniques (in the appropriate linear setting) and take advantage of their strength. Moreover, when the considered Lie (super) algebras enjoy a grading — a case which often happens both in representation theory and in differential geometry—, all cohomology spaces \({\fancyscript{Z}^k /\fancyscript{B}^k}\) naturally split up as direct sums of smaller subspaces, and this enables us, for higher dimensional Lie (super) algebras, to improve the computer speed of calculations. Lastly, we implement our algorithm in the Maple software and evaluate its performances via some examples, most of which have several applications in the theory of Cartan-Tanaka connections.