On Hopf 2-algebras

Abstract

Our main goal in this paper is to translate the diagram below relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans [BaeCra04], another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.

Adjunction between crossed modules of groups and algebras

“…It might be interesting to consider the main results here at the 2-category level and to relate our constructions with the representation theory of 2-groups developed by Elgueta [7]. Clearly, the investigation of possible Hopf algebra structure of our group algebra crossed module in the context of [9] and [11] is another prospective direction.…”

Universal enveloping crossed module of a Lie crossed module

“…Also, it is noted that this does not give a crossed module of associative algebras, but a crossed module of Hopf algebras, presented in [15,Definition 1]. Of course, our definition of the universal enveloping functor XU differs from the definition of the enveloping functor U in [15].…”

“…In the article [15], because of the integration problem of Lie 2-algebras into 2-groups, an enveloping functor U is defined by applying the standard functor U term by term on a Lie crossed module (M, P, µ), i.e., U(M, P, µ) = ( U(M ), U(P ), U(µ) ) . Also, it is noted that this does not give a crossed module of associative algebras, but a crossed module of Hopf algebras, presented in [15,Definition 1]. Of course, our definition of the universal enveloping functor XU differs from the definition of the enveloping functor U in [15].…”

Lie 2-algebras of vector fields

“…A crossed module of Lie algebras determines a Lie 2-algebra: see, for example, the proof of Theorem 3 in [7]. A converse is true as well: any Lie 2-algebra canonically defines a crossed module of Lie 2-algebras.…”

“…A converse is true as well: any Lie 2-algebra canonically defines a crossed module of Lie 2-algebras. In fact more is true: crossed modules form a strict 2-category, and the 2-categories of Lie 2-algebras and of crossed modules are equivalent (see [7,Theorem 3] cited above). We won't need the full strength of this theorem in the present paper.…”