Generalized symmetries as homotopy Lie algebras

Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the description of generalized gauge symmetries using two specific examples. The first example deals with the non-commutative gauge symmetry obtained using Drinfel’d twist of the symmetry Hopf algebra. The homotopy Lie algebra compatible with the twisted gauge symmetry turns out to be the recently proposed braided L${}_{\mathrm{\infty }}$-algebra. In the second example we focus on the generalized gauge symmetry of the double field theory. The symmetry includes both diffeomorphisms and gauge transformation and can consistently be defined using a curved L${}_{\mathrm{\infty }}$-algebra.