Celebratio Mathematica — Schwarz

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Works connected to Michael Penkava

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M. Penkava and A. Schwarz: “On some algebraic structures arising in string theory,” pp. 219–227 in Perspectives in mathematical physics (Taiwan, summer 1992 and Los Angeles, winter 1992). Edited by R. C. Penner and S.-T. Yau. Conference Proceedings and Lecture Notes in Geometry and Topology 3. International Press (Boston), 1994. MR 1314668 Zbl 0871.17021 ArXiv hep-th/9212072 incollection

Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; i.e., one can introduce a multiplication, an odd bracket, and an odd operator \( \Delta \) having the same properties as the corresponding operations in Batalin–Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator \( \Delta \): If \( \mathcal{A} \) is a supercommutative, associative algebra and \( \Delta \) is an odd second order derivation on \( \mathcal{A} \) satisfying \( \Delta^2 = 0 \) , one can provide \( \mathcal{A} \) with the structure of a BV-algebra.

Michael Penkava	Related
Shing-Tung Yau	Related
Robert Clark Penner	Related

M. Penkava and A. Schwarz: “\( A_{\infty} \) algebras and the cohomology of moduli spaces,” pp. 91–107 in Lie groups and Lie algebras: E. B. Dynkin’s seminar. Edited by S. G. Gindikin and E. B. Vinberg. American Mathematical Society Translations. Series 2 169. American Mathematical Society (Providence, RI), 1995. This book is also no. 26 in the Advances in Mathematics series. MR 1364455 Zbl 0863.17017 ArXiv hep-th/9408064 incollection

Let us consider an \( A_{\infty} \) algebra with an invariant inner product. The main goal of this paper is to classify the infinitesimal deformations of this \( A_{\infty} \) algebra preserving the inner product and to apply this result to the construction of homology classes on the moduli spaces of algebraic curves. With this aim, we define cyclic cohomology of an \( A_{\infty} \) algebra and show that it classifies the deformations we are interested in. To make the reading of our paper more independent of other works, we include a short review of Hochschild and cyclic cohomology of associative algebras, and explain the definition of \( A_{\infty} \) algebras.

Evgeniĭ Borisovich Dynkin	Related
Simon Grigorevich Gindikin	Related
Michael Penkava	Related
Ernest Borisovich Vinberg	Related