Showing papers by "Martin Markl published in 2004"

Homotopy algebras are homotopy algebras

Abstract: We prove that strongly homotopy algebras (such as $A_\infty$, $C_\infty$, sh Lie, $B_\infty$, $G_\infty$,...) are homotopically invariant in the category of chain complexes. An important consequence is a rigorous proof that `strongly homotopy structures transfer over chain homotopy equivalences.'

Algebras with one operation including Poisson and other Lie-admissible algebras

Abstract: We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality, responsible for the existence of the dihedral cohomology. The main trick, which we call the polarization, will be to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operation. We will try to convince the reader that this change of perspective might sometimes lead to new insights and results. This point of view was used by Livernet and Loday to introduce a one-parametric family of operads whose specialization at 0 is the operad for Poisson algebras, while at a generic point it equals the operad for associative algebras. We study this family and explain how it can be used to interpret the deformation quantization in a neat and elegant way.

Intrinsic brackets and the $L_\infty$-deformation theory of bialgebras

Abstract: We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a PROP, induced by a strongly homotopy Lie structure on the defining cochain complex, such that the associated "quantum" master equation captures deformations. This in particular implies the existence of a Lie bracket on the Gerstenhaber-Schack cohomology of a bialgebra that extends the classical intrinsic bracket on the Hochschild cohomology, giving an affirmative answer to an old question about the existence of such a bracket.

Transferring $A_\infty$ (strongly homotopy associative) structures

Abstract: We give explicit formulas for transfers of $A_\infty$-structures and related maps and homotopies in the most easy situation in which these transfers exist. One half of our formulas was already known to Kontsevich-Soibelman and to Merkulov who derived them, without explicit signs, under slightly stronger assumptions than those made in this note.

Free Homotopy Algebras

Abstract: This note was originated many years ago as my reaction to questions of several people how free strongly homotopy algebras can be described and what can be said about the structure of the universal enveloping A(m)-algebra of an L(m)-algebra, and then circulated as a "personal communication." I must honestly admit that it contains no really deep result and that everything I did was that I expanded definitions and formulated a couple of statements with more or less obvious proofs.