# TTT41: Manchester

Department of Mathematics, University of Manchester.

## Date

20 October 2003

## Speakers

Alastair Hamilton (Bristol)

Deformations of A-infinity algebras

Revaz Kurdiani (Aberdeen)

The Leibniz algebra structure on the second tensor power of Lie algebra

Andy Tonks (UNL)

The associahedron diagonal approximation

## Abstracts

### Deformations of A-infinity algebras

Deformations of A-infinity algebras was considered, focusing in particular on formal one-parameter deformations. A-infinity algebras was defined via the cobar construction and the resulting effect on the calculations discussed. Hochschild cohomology of an A-infinity algebra was defined and its relation to deformation theory explained.

A type of A-infinity algebra called a Moore algebra was introduced. These were closely related to the one dimensional A-infinity algebras considered by Kontsevich which generate the Miller-Morita-Mumford classes in the cohomology of moduli spaces of complex algebraic curves.

### The Leibniz algebra structure on the second tensor power of Lie algebra

The notion of Leibniz algebra was introduced not very long time ago, which is a generalisation of the notion of Lie algebra. It turns out that the second tensor power of Lie algebra is a Leibniz algebra. The talk concentrated on this Leibniz algebra. Namely the elementary properties of this algebra was stated. Revaz also described the non-abelian second tensor power of Lie algebra constructed by Ellis in J. Pure Appl. Algebra 46 (1987), 111–115 and Glasgow Math. J. 33 (1991), 101–120.

Non-abelian tensor product of groups was introduced by Brown and Loday in Topology 26 (1987), 311–335. Moreover, the maximal Lie algebra quotient of studied Leibniz algebra is an abelian extension of the non-abelian second tensor power of Lie algebra. For finite dimensional semi-simple Lie algebra the kernel of this extension was described explicitly.

### The associahedron diagonal approximation

The associahedra, or "Stasheff polytopes", were introduced by Jim Stasheff in 1963 for the study of homotopy associativity of H-spaces. Combinatorially their cells correspond to bracketings or to planar rooted trees, and their vertices are counted by the Catalan numbers.

The associahedron diagonal approximation, recently introduced in preprints of Saneblidze and Umble, can be seen as a generalising both the classical (simplicial) Alexander-Whitney diagonal approximation and the obvious cubical diagonal approximation. This talk covered some geometric and combinatorial aspects of these diagonal approximations, and their relation with loop spaces, cobar constructions, and the Baues conjecture on the homotopy type of certain order complexes.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.