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Register a new userThe BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products
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Authors
Thomas Tradler
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We define a BV-structure on the Hochschild-cohomology of a unital, associative algebra A with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhabers original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital A-infinity-algebra with a symmetric and non-degenerate infinity-inner product.
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Consider a smooth affine algebraic variety $X$ over an algebraically closed field, and let a finite group $G$ act on it. We assume that the characteristic of the field is greater than the dimension of $X$ and the order of $G$. An explicit formula for multiplication on the Hochschild cohomology of a crossed product of $k[G]$ and $k[X]$ is given in terms of multivector fields on $X$ and $g$-invariant subvarieties of $X$ for $gin G$.
We summarize some recent results obtained in collaboration with J. McCarthy on the spectrum of physical states in $W_3$ gravity coupled to $c=2$ matter. We show that the space of physical states, defined as a semi-infinite (or BRST) cohomology of the $W_3$ algebra, carries the structure of a BV-algebra. This BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector fields on the base affine space of $SL(3,C)$. Details have appeared elsewhere. [Published in the proceedings of Gursey Memorial Conference I: Strings and Symmetries, Istanbul, June 1994, eds. G. Aktas et al., Lect. Notes in Phys. 447, (Springer Verlag, Berlin, 1995)]
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