Graded Lie Algebras Research Articles

Cohomology and duality for coalgebras over a quadratic operad

The cohomology of a finite-dimensional coalgebra over a finitely generated quadratic operad, with coefficients in itself, is defined and is shown to have the structure of a graded Lie algebra. The cohomology of such a coalgebra is isomorphic to the cohomology of its linear dual as graded Lie algebras.

The recognition theorem for graded Lie algebras in prime characteristic

The recognition theorem for graded Lie algebras in prime characteristic

Deformations of Lie brackets: cohomological aspects

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.

Generalized flag geometries and manifolds associated to short [formula omitted]-graded Lie algebras in arbitrary dimension

The object of this Note is to define the generalized flag geometry of a graded Lie algebra which corresponds to the generalized projective geometry in the case of 3-gradings. Then we construct a structure of manifold on this generalized flag geometry. This result generalizes a result known for 3-graded Lie algebras to the more general case of ( 2 k + 1 ) -graded Lie algebras. To cite this article: J. Chenal, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Lax operator algebras and integrable hierarchies

We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type.

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by \({R^{(\varepsilon)}}\), is defined by a complex n × n-matrix \({\varepsilon=(\varepsilon_{ij})}\), i, j = 1, . . . , n. In order for \({R^{(\varepsilon)}}\) to be a Lie algebra, \({\varepsilon_{ij}}\) should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X R . We compute the first homology group of an irreducible component M of X R , under some assumptions on M. We look into algebraic properties of graded Lie algebras \({R^{(\varepsilon)}}\) where \({\varepsilon\in X_R}\).

Deformation of a smooth Deligne–Mumford stack via differential graded Lie algebra

For a smooth Deligne–Mumford stack over C , we define its associated Kodaira–Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira–Spencer algebra if the stack is proper over C .

Multiplicity matrices for the affine graded Hecke algebra

In this paper we are looking at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan–Lusztig type polynomials. We review the algorithms of [G. Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv. Math. 112 (1995) 147–217; G. Lusztig, Graded Lie algebras and intersection cohomology, preprint], and use them in particular to compute, at every real central character which admits tempered modules, the geometric parameterization, the Kazhdan–Lusztig polynomials, the composition series, and the Iwahori–Matsumoto involution for the representations with Iwahori fixed vectors of the split p-adic groups of type G 2 and F 4 (and by the nature of the algorithms, for their Levi subgroups).

Smoothness theorem for differential BV algebras

Given a differential Batalin–Vilkovisky algebra (V, Q, Δ.) the associated odd differential graded Lie algebra (V, Q, + Δ, [,].) is always smooth formal. The quantum differential graded Lie algebra L ℏ : = ( V [ [ ℏ ] ] , Q + ℏ Δ , [ , ] ) is not always smooth formal, but when it is — for example, when a Q-Δ version of the ∂-∂ Lemma holds — there is a weak Frobenius manifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, we prove that Lħ is smooth formal if and only if the spectral sequence associated to the filtration F p : = ℏ p V [ [ ℏ ] ] on the complex ( V [ [ ℏ ] ] , Q + ℏ Δ ) degenerates at E1. A priori, this degeneration means that a collection of first-order obstructions vanish and we prove that it follows that all obstructions vanish. For those differential BV algebras that arise from the Hochschild complex of a Calabi–Yau category, the degeneration of this spectral sequence is an expression of the noncommutative Hodge to deRham degeneration, conjectured by Kontsevich and Soibelman and proved to hold in certain cases by Kaledin. The results in this paper imply that the noncommutative Hodge to de Rham degeneration conjecture is equivalent to the existence of a versal solution to the quantum master equation. At the end of the paper, some physical considerations are mentioned.

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (ℤ/p ℤ)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L 0 = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.

Einstein solvmanifolds with a simple Einstein derivation

The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).

Cohomology and deformations of the infinite-dimensional filiform Lie algebra [formula omitted

Denote m 2 the infinite-dimensional N -graded Lie algebra defined by the basis e i for i ⩾ 1 and by relations [ e 1 , e i ] = e i + 1 for all i ⩾ 2 , [ e 2 , e j ] = e j + 2 for all j ⩾ 3 . We compute in this article the bracket structure on H 1 ( m 2 , m 2 ) , H 2 ( m 2 , m 2 ) and in relation to this, we establish that there are only finitely many true deformations of m 2 in each weight by constructing them explicitly. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.

On the Lie-formality of Poisson manifolds

AbstractIn the present note we prove formality of the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold.

Formality Theorem with Coefficients in a Module

In this paper, X will denote a $ {\user1{\mathcal{C}}}^{\infty } $ manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, $\Gamma \left( {X,^E D_{\text{poly}}^* } \right)$ and showed that it is formal. Denote by $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ the DGLA of E-polyvector fields. Considering M, a module over E, we define $\Gamma \left( {X,^E T_{\text{poly}}^* \left( M \right)} \right)$ the $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ -module of E-polyvector fields with values in M. Similarly, we define the $\Gamma \left( {X,^E D_{\text{poly}}^* } \right)$ -module of E-polydifferential operators with values in M, $\Gamma \left( {X,^E D_{\text{poly}}^* \left( M \right)} \right)$ . We show that there is a quasi-isomorphism of L ∞-modules over $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ from $\Gamma \left( {X,^E T_{\text{poly}}^* \left( M \right)} \right)$ to $\Gamma \left( {X,^E D_{\text{poly}}^* \left( M \right)} \right)$ . Our result extends Calaque’s (and Kontsevich’s) result.

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism \(B_n^1 \rightarrow ({\mathbb{Z}}/N{\mathbb{Z}})^n \rtimes {\mathfrak{S}}_n\), where Bn1 is a braid group of type B. The formality isomorphism depends algebraically on a series ΨKZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded \(\mathfrak{grtm}\)(N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmuller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue \(\mathfrak{grtmd}\)(N, k) of \(\mathfrak{grtm}\)(N, k); it is a graded Lie algebra with an action of \(({\mathbb{Z}}/N{\mathbb{Z}})^{\times}\), and we give a lower bound for the character of its space of generators.

A DOMAIN TEST FOR LIE COLOR ALGEBRAS

Lie color algebras are generalizations of Lie superalgebras and graded Lie algebras. The properties of a Lie color algebra can often be related directly to the ring structure of its universal enveloping algebra. We study the effects of torsion elements and torsion subspaces. Let [Formula: see text] denote a Lie color algebra. If [Formula: see text] is homogeneous and torsion then x2 = 0 in [Formula: see text]. If no homogeneous element of [Formula: see text] is torsion, then [Formula: see text] so [Formula: see text] is semiprime. In this case we can give a test which uses Gröbner basis methods to determine when [Formula: see text] is a domain. This is applied in an example to show [Formula: see text] may be a domain even if [Formula: see text] contains torsion elements and torsion subspaces.

A Unified Model of Phantom Energy and Dark Matter

To explain the acceleration of the cosmological expansion researchers have con- sidered an unusual form of mass-energy generically called dark energy. Dark energy has a ratio of pressure over mass density which obeys w = p/ < 1/3. This form of mass- energy leads to accelerated expansion. An extreme form of dark energy, called phantom energy, has been proposed which has w = p/ < 1. This possibility is favored by the observational data. The simplest model for phantom energy involves the introduction of a scalar field with a negative kinetic energy term. Here we show that theories based on graded Lie algebras naturally have such a negative kinetic energy and thus give a model for phantom energy in a less ad hoc manner. We find that the model also contains ordi- nary scalar fields and anti-commuting (Grassmann) vector fields which act as a form of two component dark matter. Thus from a gauge theory based on a graded algebra we naturally obtained both phantom energy and dark matter.

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n ⊕ ⋯ ⊕ n ( s times, s ≥ 2 ) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types ( 5 , 3 ) and ( 3 , 3 , 2 ) are not possible for Anosov Lie algebras.

Integral homology $3$-spheres and the Johnson filtration

The mapping class group of an oriented surface Σg, 1 of genus g with one boundary component has a natural decreasing filtration Mg,1 ⊃ Mg,i(⊃) 3 Me , i(⊃) D M g,1 (3) ⊃..., where M g,1 (k) is the kernel of the action of M g,1 on the k th nilpotent quotient of π 1 (Σ g,1 ). Using a tree Lie algebra approximating the graded Lie algebra ⊕ k M g,1 (k)/M g,1 (k + 1) we prove that any integral homology sphere of dimension 3 has for some g a Heegaard decomposition of the form M = Hg? lgΦ -H g , where Φ ∈ Μ g,1 (3) and lg is such that H g ∏ lg -H g = S 3 . This proves a conjecture due to S. Morita and shows that the core of the Casson invariant is indeed the Casson invariant.

Complex structures, totally real and totally geodesic submanifolds of compact 3-symmetric spaces, and affine symmetric spaces

We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.