∞-Lie theory (higher geometry)
Formal Lie groupoids
A Lie 2-algebra is to a Lie 2-group as a Lie algebra is to a Lie group. Thus, it is a vertical categorification of a Lie algebra.
A (“semistrict”) Lie 2-algebra is an L-∞-algebra with generators concentrated in the lowest two degrees.
This means that it is
a pair of vector spaces
equipped with linear functions as follows:
a unary bracket encoding a differential
and a binary bracket , whose component on elements in degree 0 is a Lie bracket
and whose component on elements in degree 0 and degree 1 is a weak action
and a trinary bracket
called the Jacobiator;
and are skew-symmetric in their arguments, as indicated;
the differential respects the brackets: for all and we have
the Jacobi identity of holds up to the image under of the Jacobiator : for all we have
as does the action property:
the Jacobiator is coherent:
If the trinary bracket in a Lie 2-algebra is trivial, one speaks of a strict Lie 2-algebra. Strict Lie 2-algebras are equivalently differential crossed modules (see there for details).
John Baez, Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras Theory and Applications of Categories, Vol. 12, (2004) No. 15, pp 492-528. (TAC)
Dmitry Roytenberg, On weak Lie 2-algebras, (arXiv)
Daniel Berwick-Evans, Eugene Lerman, Lie 2-algebras of vector fields, arxiv/1609.03944
Revised on September 18, 2016 17:01:15
by Urs Schreiber