Pinczon Algebras

TL;DR

This paper explores Pinczon algebras, establishing a graded Lie bracket on cyclic multilinear forms that induces a quadratic associative algebra structure up to homotopy, and relates it to classical cohomologies.

Contribution

It generalizes the construction of Pinczon algebras to quadratic associative, commutative, and Lie algebras, linking them to known cohomology theories.

Findings

Constructs a graded Lie bracket on cyclic multilinear forms.

Shows the algebra structure up to homotopy for various algebra types.

Connects the construction to Hochschild, Harrison, and Chevalley cohomologies.

Abstract

After recalling the construction of a graded Lie bracket on the space of cyclic multilinear forms on a vector space V, due to Georges Pinczon and Rosane Ushirobira, we prove this construction gives a structure of quadratic associative algebra, up to homotopy, on V. In the associative case, it is easy to refind the associated usual Hochshild cohomology. By considering restriction to a subspace or a quotient space of forms, we can present in a completely similar way the cases of quadratic commutative and quadratic Lie algebras, up to homotopy, and the corresponding Harrison and Chevalley cohomologies.