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 $HH^*(C^*(\mathbf{K}P^n;R);C^*(\mathbf{K}P^n;R))$ 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 $\mathbb{H}_*(LS^2;\mathbb{Z})$ 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, $\mathbb{H}_*(L\mathbb{C}P^n;\mathbb{Q})$ and $\mathbb{H}_*(L\mathbb{H}P^n;\mathbb{Q})$, of projective spaces.