Chevalley Cohomology Research Articles

Some Features of Rank One Real Solvable Cohomologically Rigid Lie Algebras with a Nilradical Contracting onto the Model Filiform Lie Algebra Qn

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , ⋯ , n + k − 3 , n + 2 k − 3 for k ≥ 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k ≤ 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined.

New examples of rank one solvable real rigid Lie algebras possessing a nonvanishing Chevalley cohomology

The class of rank one solvable Lie algebras possessing a maximal torus t with eigenvalue spectrum spec(t)=(1,4,5,…,n+2) is studied in the context of rigidity. It is shown that from the value n ≥ 18, three isomorphism classes of rigid Lie algebras exist, two of them being algebraically rigid, and the third being geometrically rigid with a two-dimensional cohomology space H2(g,g).

Classification of solvable real rigid Lie algebras with a nilradical of dimension [formula omitted

The complete classification of real solvable rigid Lie algebras possessing a nilradical of dimension at most six is given. Eleven new isomorphism classes of indecomposable algebras are obtained. It is further shown that the resulting solvable Lie algebras have a vanishing second Chevalley cohomology group, thus correspond to algebraically rigid Lie algebras.

Chevalley cohomology for aerial Kontsevich graphs

Chevalley cohomology for aerial Kontsevich graphs

Chevalley Cohomology for linear graphs

The space of linear polyvector fields on \(\mathbb{R}^d\) is a Lie subalgebra of the (graded) Lie algebra \(T_{\rm poly}(\mathbb{R}^d)\), equipped with the Schouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in \(T_{\rm poly}(\mathbb{R}^d)\), restricting ourselves to the case of cochains defined with purely aerial Kontsevich’s graphs, as in Pac. J. Math. 218(2):201–239, 2005. We find a result which is very similar to the cohomology for the vector case Pac. J. Math. 229(2):257–292, 2007.

Pure spinors, free differential algebras, and the supermembrane

The Lagrangian formalism for the supermembrane in any 11d supergravity background is constructed in the pure spinor framework. Our gauge-fixed action is manifestly BRST, supersymmetric, and 3d Lorentz invariant. The relation between the Free Differential Algebras (FDA) underlying 11d supergravity and the BRST symmetry of the membrane action is exploited. The “gauge-fixing” has a natural interpretation as the variation of the Chevalley cohomology class needed for the extension of 11d super-Poincaré superalgebra to M-theory FDA. We study the solution of the pure spinor constraints in full detail.

M-theory FDA, twisted tori and Chevalley cohomology

The FDA algebras emerging from twisted tori compactifications of M-theory with fluxes are discussed within the general classification scheme provided by Sullivan's theorems and by Chevalley cohomology. It is shown that the generalized Maurer–Cartan equations which have appeared in the literature, in spite of their complicated appearance, once suitably decoded within cohomology, lead to trivial FDAs, all new p-form generators being contractible when the 4-form flux is cohomologically trivial. Non-trivial D = 4 FDAs can emerge from non-trivial fluxes but only if the cohomology class of the flux satisfies an additional algebraic condition which appears not to be satisfied in general and has to be studied for each algebra separately. As an illustration an exhaustive study of Chevalley cohomology for the simplest class of SS algebras is presented but a general formalism is developed, based on the structure of a double elliptic complex, which, besides providing the presented results, makes possible the quick analysis of compactification on any other twisted torus.

Derivations of the Lie algebras of differential operators

This paper encloses a complete and explicit description of the derivations of the Lie algebra D( M)of all linear differential operators of a smooth manifold M, of its Lie subalgebra D 1( M) of all linear first-order differential operators of M, and of the Poisson algebra S( M) = Pol( T* M) of all polynomial functions on T* M, the symbols of the operators in D( M). It turns out that, in terms of the Chevalley cohomology, H 1( D( M), D( M)) = H DR 1( M), H 1 ( D 1( M), D 1( M)) = H DR 1( M) ⊕ R 2, and H 1( S( M), S( M)) = H DR 1 ( M) ⊕ R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.

Chevalley cohomology for Kontsevich’s graphs

We introduce the Chevalley cohomology for the graded Lie algebra of polyvector ?elds on Rd. This cohomology occurs naturally in the problem of construction and classi?cation of formalities on the space Rd. Considering only graph formalities, that is, formalities de?ned with the help of graphs as in the original construction of Kontsevich, we de?ne (as the ?rst and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing ?rst a noteworthy property for Kontsevich?s explicit formality on Rd, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions for the Chevalley coboundary operator; in particular, we can write this cohomology directly on the space of purely aerial, nonoriented graphs. We also give examples and applications.

The multigraded Nijenhuis–Richardson algebra, its universal property and applications

We define two ( n + 1) graded Lie brackets on spaces of multilinear mappings. The first one is able to recognize n-graded associative algebras and their modules and gives immediately the correct differential for Hochschild cohomology. The second one recognizes n-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.

On the local Chevalley cohomology of the dynamical Lie algebra of a symplectic manifold

We describe the relations between the local Chevalley cohomologies related to the adjoint representation of the Poisson Lie algebra of a symplectic manifold and the Lie algebras of all symplectic or globally Hamiltonian vector fields of the manifold. The proofs are based on the computation of the cohomology of the complex (E, ∂), where E is the space of multilinear local maps from a vector bundle of a manifold M into the space of forms on M and ∂L=d ∘ L.

Local cohomology of the algebra of C ∞ functions on a connected manifold

A multilinear version of Peetre's theorem on local operators is the key to prove the equality between the local and differentiable Hochschild cohomology on the one hand, and on the other hand the equality between the second and third local Chevalley cohomology groups and their differentiable counterpart.