Abstract
We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.
Highlights
Lie groups are ubiquitous in mathematics and theoretical physics as the structures formalizing the notion of continuous symmetries
We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity
We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator
Summary
Lie groups are ubiquitous in mathematics and theoretical physics as the structures formalizing the notion of continuous symmetries Their infinitesimal objects are Lie algebras: vector spaces equipped with an antisymmetric bracket satisfying the Jacobi identity. . ., satisfying generalized Jacobi identities involving all brackets Such structures, referred to as L∞ or strongly homotopy Lie algebras, first appeared in the physics literature in closed string field theory [1] and in the mathematics literature in topology [2,3,4]. At first sight the above theorem may shed doubt on the usefulness of L∞ algebras, since it states that any generally non-Lie algebra can be extended to an L∞ algebra It should be emphasized, that for a generic bracket the resulting structure is quite degenerate in that the 2-term L∞ algebra may not be extendable further in a nontrivial way, say by including a vector space X−1.
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