Enveloping algebras of Hom-Lie algebras. (English) Zbl 1214.17001

Yau, Donald

J. Gen. Lie Theory Appl. 2, No. 2, 95-108 (2008).

A nonassociative algebra \(A=(A,\mu,\alpha)\) over a field \(\mathbb K\) (of characteristic 0 in the paper) is called Hom-nonassociative if it has a binary multiplication \(\mu\) and is equipped with a \(\mathbb K\)-linear map \(\alpha:A\to A\). The algebra is Hom-associative if it satisfies the \(\alpha\)-twisted associativity law \(\alpha(x)(yz)=(xy)\alpha(z)\), \(x,y,z\in A\). It is Hom-Lie if the multiplication (usually denoted by brackets) is skew-symmetric: \([x,y]=-[y,x]\) and satisfies the Hom-Jacobi identity \([\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0\). The commutator \([x,y]=xy-yx\), \(x,y\in A\), turns every Hom-associative algebra into a Hom-Lie algebra. This gives rise to a functor \(HLie\) from the category HomAs of Hom-associative algebras to the category HomLie of Hom-Lie algebras.
In the paper under review the author constructs free Hom-nonassociative algebras involving weighted planar binary trees in the construction. Then he proves the existence and presents an explicit construction of the enveloping Hom-associative algebra of a Hom-Lie algebra. This defines a functor \(U_{\mathbf{HLie}}:{\mathbf{HomLie}}\to {\mathbf{HomAs}}\) which is left adjoint to the functor \(HLie\). Further, the author constructs Hom-dialgebras and enveloping Hom-dialgebras of Hom-Leibniz algebras.

Reviewer: Vesselin Drensky (Sofia)

MSC:

17A30	Nonassociative algebras satisfying other identities
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
05C05	Trees
17A32	Leibniz algebras
17A50	Free nonassociative algebras
17B35	Universal enveloping (super)algebras

Keywords:

Hom-Lie algebras; Hom-Leibniz algebras; enveloping algebras; left adjoint functor