Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Homotopy Dg Algebras Induce Homotopy Bv Algebras

Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential BatalinVilkovisky algebra. Moreover, if A is an A∞ algebra, then TA is a commutative BV∞ algebra. 1. Main Statement Let (A, dA) be a complex over a commutative ring R. Our convention is that dA is of degree +1. The space TA ...

DG (co)algebras, DG Lie algebras and L∞ algebras

Examples of Homotopy Lie Algebras

We look at two examples of homotopy Lie algebras (also known as L∞ algebras) in detail from two points of view. We will exhibit the algebraic point of view in which the generalized Jacobi expressions are verified by using degree arguments and combinatorics. A second approach using the nilpotency of Grassmann-odd differential operators ∆ to verify the homotopy Lie data is shown to produce the sa...

Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...

Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target’ space, B. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G, 1) and also when it is simply c...