Bicocycle Double Cross Constructions

TL;DR

This paper introduces bicocycle double cross constructions for Lie groups, Lie algebras, and bialgebras, generalizing existing products and sums to broader algebraic structures without requiring substructure conditions.

Contribution

It generalizes the concept of unified products to bicocycle double cross products for Lie groups, Lie algebras, and bialgebras, expanding the scope of algebraic constructions.

Findings

Constructs a Lie group on the product of two pointed manifolds without subgroup requirements.

Defines a Lie algebra on the direct sum of two vector spaces without subalgebra constraints.

Develops a bialgebra on the tensor product of two (co)algebras, not necessarily sub-bialgebras.

Abstract

We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a Lie group on the product space of two pointed manifolds, none of which being necessarily a subgroup. On the level of Lie algebras, similarly, a Lie algebra is obtained on the direct sum of two vector spaces, none of which is required to be a subalgebra. Finally, on the quantum level the theory presents a bialgebra, on the tensor product of two (co)algebras that are not necessarily sub-bialgebras, the semidual of which being a cocycle bicrossproduct bialgebra.