Celebratio Mathematica — Schwarz

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Works connected to Evgeniĭ Borisovich Dynkin

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D. Fuchs and A. Schwarz: “Matrix Vieta theorem,” pp. 15–22 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 1364450 Zbl 0837.15011 ArXiv math/9410207 incollection

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-\( k \) matrices.

Specifically, we prove that if \( X_1,\dots \), \( X_n \) are solutions of an algebraic matrix equation \[ X^n + A_1 X^{n-1} + \cdots + A_n = 0 ,\] independent in the sense that they determine the coefficients \( A_1,\dots \), \( A_n \), then the trace of \( A_1 \) is the sum of the traces of the \( X_i \), and the determinant of \( A_n \) is, up to a sign, the product of the determinants of the \( X_i \). We generalize this to arbitrary rings with appropriate structures.

This result is related to and motivated by some constructions in non-commutative geometry.

Evgeniĭ Borisovich Dynkin	Related
Simon Grigorevich Gindikin	Related
Dmitry Fuchs	Related	Volume
Ernest Borisovich Vinberg	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