Lie Algebra Cohomology Research Articles

On the cohomology of Lie algebras associated with graphs

We describe a canonical decomposition of the cohomology of the Dani-Mainkar 2-step nilpotent Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the cohomology in all degrees of Lie algebras associated with arbitrary star graphs. We also describe a procedure to reduce the calculation of the cohomology of solvable Lie algebras associated with graphs through the Grantcharov-Grantcharov-Iliev construction to the cohomology of Dani-Mainkar Lie algebras.

Classification of 2-term L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document}-algebras

We classify all 2-term L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document}-algebras up to isomorphism. We show that such L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document}-algebras are classified by a Lie algebra, a vector space, a representation (all up to isomorphism) and a cohomology class of the corresponding Lie algebra cohomology.

Canonical torsor bundles of prescribed rational functions on complex curves

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve [Formula: see text], we construct explicitly a bundle [Formula: see text] with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

On the cohomology of restricted Heisenberg Lie algebras

We show that the Heisenberg Lie algebras over a field F of characteristic p>0 admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and ⋅[p]-operators.

2-Step nilpotent L∞-algebras and hypergraphs

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.

Cohomologies of relative Rota-Baxter Lie algebras with derivations and applications

The purpose of the present paper is to investigate cohomologies of relative Rota-Baxter Lie algebras with derivations and applications. First, we introduce a notion of relative Rota-Baxter LieDer pairs. Moreover, we construct cohomologies of relative Rota-Baxter LieDer pairs. Finally, we discuss abelian extensions, crossed modules and skeletal relative Rota-Baxter Lie 2-Der pairs in terms of cohomology groups.

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.

Morphismes de périodes et cohomologie syntomique

We give the geometric version of a construction of Colmez-Niziol which establishes a comparison theorem between arithmetic p-adic nearby cycles and syntomic sheaves. The local construction of the period isomorphism uses $(\phi,\Gamma)$-modules theory and is obtained by reducing the period isomorphism to a comparison theorem between cohomologies of Lie algebras. By applying the method of "more general coordinates" used by Bhatt-Morrow-Scholze, we construct a global isomorphism. In particular, we deduce the semi-stable conjecture of Fontaine-Jannsen. This result was also proved by (among others) Tsuji, via the Fontaine-Messing map, and by Cesnavicius and Koshikawa, which generalized the proof of the crystalline conjecture by Bhatt, Morrow and Scholze. We use the previous map to show that the period morphism of Tsuji and the one of Cesnavicius-Koshikawa are the same.

Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras. Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras. In particular, we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted L∞-algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group globally in a functorial way.

Solid locally analytic representations of 𝑝-adic Lie groups

We develop the theory of locally analytic representations of compact p p -adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard’s isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.

Compatible L∞-algebras

A compatible L∞-algebra is a graded vector space together with two compatible L∞-algebra structures on it. Given a graded vector space, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely compatible L∞-algebra structures on it. We provide examples of compatible L∞-algebras arising from Nijenhuis operators, compatible V-datas and compatible Courant algebroids. We define the cohomology of a compatible L∞-algebra and as an application, we study formal deformations. Next, we classify ‘strict’ and ‘skeletal’ compatible L∞-algebras in terms of crossed modules and cohomology of compatible Lie algebras. Finally, we introduce compatible Lie 2-algebras and find their relationship with compatible L∞-algebras.

A survey on deformations, cohomologies and homotopies of relative Rota–Baxter Lie algebras

In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\infty$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\infty$-algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative Rota-Baxter Lie algebras with close connection to pre-Lie$_\infty$-algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras of nonzero weights.

On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p>h, where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G1 and g, respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G1, we describe them as the cohomology of G1 in terms of the cohomology for G1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms Hn(G1,V)≅Hn(G,V) and Hn(g,V)≅Hn(G,V), and a necessary condition for the isomorphism Hn(g,V)≅Hn(G1,V), where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G, G1, and g with coefficients in the considered simple modules.

Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras

Let X X be a smooth affine variety over a field k \mathbf k of characteristic 0 0 and T ( X ) T(X) be the Lie algebra of regular vector fields on X X . We compute the Lie algebra cohomology of T ( X ) T(X) with coefficients in k \mathbf k . The answer is given in topological terms relative to any embedding k ⊂ C \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C ∞ C^\infty -manifold. Unlike the C ∞ C^\infty -case, our setup is purely algebraic: no topology on T ( X ) T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classifying abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.

On Cohomology of Simple Modules for Modular Classical 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.

Associative algebra twisted bundles over compact topological spaces

For the associative algebra A(g) of an infinite-dimensional Lie algebra g, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic completion of the space of formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. Homotopical invariance as well as covariance in terms of trivial bundles of twisted A(g)-bundles is proven. Further applications of the paper's results useful for studies of the cohomology of infinite-dimensional Lie algebras on smooth manifolds, K-theory, as well as for purposes of conformal field theory, deformation theory [10,14,15], and the theory of foliations are mentioned.

On the Cohomology of Lie Algebras with an Invariant Inner Product

In this work we consider the moduli space of all noncommutative metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \(\mathbb C\) and \(\mathbb R\) up to dimension 5 and all metric Lie algebras over \(\mathbb C\) in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.