Algebraic loop structures on algebra comultiplications

Dae-Woong Lee

doi:10.1515/math-2019-0060

Abstract

Abstract In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

Highlights

The theory of Lie algebras is an outgrowth of the Lie theory of continuous groups and is playing an important role for many reasons
In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications
We investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications

Summary

Introduction

The theory of Lie algebras is an outgrowth of the Lie theory of continuous groups and is playing an important role for many reasons. This work is licensed under the Creative Commons Attribution alone 4.0 loops and the set of Lie algebra comultiplications which have the inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications We apply these notions to those of the rational cohomology of Hopf spaces. We describe the group structure on the set of Lie algebra comultiplications of the free graded Lie algebras as follows. Let C(L(V)) be the set of all Lie algebra comultiplications of L(V) It has the structure of an abelian group. Qn), respectively, we de ne the operations by φP + φQ = φP+Q and −φ = φ−P, where φP+Q : L(V) → L(V) L(V) is the Lie algebra comultiplication with perturbation P + Q.

Algebraic loop structures

We now compute

Applications to the rational cohomology