On BV-Algebras

In this paper, we introduce the notion of a BV-algebra, and we show that a BV-algebra is logically equivalent to several algebras, i.e., BM-algebras, BT-algebras, BO-algebras and 0-commutative B-algebras. Moreover, we show that a BV-algebra with (F) is logically equivalent to several algebras, and we show some relationships between a BV-algebra with (F) and several related algebras.

Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

The purpose of this article is to embed the string topology bracket developed by Chas–Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Völcsey–Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincaré duality is given. For negative cyclic cohomology, this in particular leads to a Batalin–Vilkoviskiĭ (BV) algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an $e_3$-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.

BV-GENERATORS AND LIE ALGEBROIDS

Let [Formula: see text] be a Gerstenhaber algebra generated by [Formula: see text] and [Formula: see text]. Given a degree -1 operator D on [Formula: see text], we find the condition on D that makes [Formula: see text] a BV-algebra. Subsequently, we apply it to the Gerstenhaber or BV algebra associated to a Lie algebroid and obtain a global proof of the correspondence between BV-generators and flat connections.