Showing papers on "Differential graded Lie algebra published in 1996"

The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces

Abstract: In this paper we develop the relative deformation theory of representations and flat connections and apply our theory to the local deformation theory of linkages in spaces of constant curvature.

Drinfel´d algebra deformations, homotopy comodules and the associahedra

Abstract: The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra A and thus finish the program which began in (13), (14). The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involve a new, highly non- trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfeld algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in (12).

Massey products and deformations

Abstract: The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given, depending on an arbitrary graded commutative, associative algebra. In terms of these products, the above condition of extendibility is generalized to deformations with arbitrary local bases. Dually, a construction of generalized Massey products on the cohomology of a differential graded commutative associative algebra depends on a nilpotent graded Lie algebra. For example, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while the matric Massey products correspond to the Lie algebra of block strictly upper triangular matrices.

Massey products and deformations

Abstract: The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given, depending on an arbitrary graded commutative, associative algebra. In terms of these products, the above condition of extendibility is generalized to deformations with arbitrary local bases. Dually, a construction of generalized Massey products on the cohomology of a differential graded commutative associative algebra depends on a nilpotent graded Lie algebra. For example, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while the matric Massey products correspond to the Lie algebra of block strictly upper triangular matrices.

The center of a graded connected Lie algebra is a nice ideal

Abstract: Let (L(V),d) be a free graded connected differential Lie algebra over the field Q of rational numbers. An ideal I in the Lie algebra H(L(V),d) is called nice if, for every cycle alpha is an element of L(V) such that [alpha] belongs to I, the kernel of the map H(L(V),d) --> H(L(V + Qx),d), d(x) = alpha, is contained in I. We show that the center of H(L(V),d) is a nice ideal and we give in that case some informations on the structure of the Lie algebra H(L(V + Qx),d). We apply this computation for the determination of the rational homotopy Lie algebra L(X) = pi*(Omega X) X Q of a simply connected space X. We deduce that the kernel of the map L(X) --> L(Y) induced by the attachment of a cell along an element in the center is contained in the center.