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Register a new userHomotopy DG algebras induce homotopy BV algebras
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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 Batalin-Vilkovisky algebra. Moreover, if A is an A-infinity algebra, then TA is a commutative BV-infinity algebra.
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We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped with $k$ Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the {em 2-oriented} generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck-Teichmuller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multi-oriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension $dgeq 4$ generalizing to higher dimensions Drinfeld-Etingof-Kazhdans quantizations of Lie bialgebras (the case $d=3$) and Kontsevichs quantizations of Poisson structures (the case $d=2$).
We determine the emph{$L_infty$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to emph{pre-Lie$_infty$-algebras}.
We present $PL_{infty}$-algebras in the form of composition of maps and show that a $PL_{infty}$-algebra $V$ can be described by a nilpotent coderivation on coalgebra $P^*V$ of degree $-1$. Using coalgebra maps among $T^*V$, $wedge^*V$, $P^*V$, we show that every $A_{infty}$-algebra carries a $PL_{infty}$-algebra structure and every $PL_{infty}$-algebra carries an $L_{infty}$-algebra structure. In particular, we obtain a pre Lie $n$-algebra structure on an arbitrary partially associative $n$-algebra and deduce pre Lie $n$-algebras are $n$-Lie admissible.
We describe $L_infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronovs higher derived brackets construction which is of independent interest.
We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Delta 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 brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the big bracket of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.
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