A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials

Abstract

The main result of this paper is to calculate the Batalin-Vilkovisky structure of $H{H}^{\ast }(C^{\ast }(\mathbf{K}{P}^n;R);C^{\ast }(\mathbf{K}{P}^n;R))$ for $\mathbf{K}=\mathbb{C}$ and $\mathbb{H}$, and $R=\mathbb{Z}$ and any field; and shows that in the special case when $M=\mathbb{C}{P}^1=S^2$, and $R=\mathbb{Z}$, this structure can not be identified with the BV-structure of ${\mathbb{H}}_{\ast }(L{S}^2;\mathbb{Z})$ computed by Luc Memichi in \cite{menichi2}. However, the induced Gerstenhaber structures are still identified in this case. Moreover, according to a recent work of Y.Felix and J.Thomas \cite{felix--thomas}, the main result of the present paper eventually calculates the BV-structure of the rational loop homology, ${\mathbb{H}}_{\ast }(L\mathbb{C}{P}^n;\mathbb{Q})$ and ${\mathbb{H}}_{\ast }(L\mathbb{H}{P}^n;\mathbb{Q})$, of projective spaces.

References in corpus (4)

• Notes on string topology
• String topology for spheres
• An Algebraic Chain Model of String Topology
• Rational BV-algebra in String Topology

Cited by in corpus (6)

• String Topology for Lie Groups
• String Topology: Background and Present State
• String topology for complex projective spaces
• Poincare duality and commutative differential graded algebras
• Loop homology of quaternionic projective spaces
• On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two