Celebratio Mathematica

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Usp. Mat. Nauk 29 : 1(175) (1974), pp. 195–223. An English translation was published in Russ. Math. Surv. 29:1 (1974). MR 386977 Zbl 0288.01022 article

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Russ. Math. Surv. 29 : 1 (February 1974), pp. 5–35. English translation of Russian original published in Usp. Mat. Nauk 29:1(175) (1974). Zbl 0294.01031 article

This survey is timed to coincide with the sixtieth birthday of I. M. Gel’fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different.

Gel’fand’s research in the theory of group representations, which has lasted for thirty years, falls into several cycles; the majority of his results are widely known and were dealt with in the survey [Vishik et al. 1964] on the occasion of his fiftieth birthday. For this reason we deal here only with the results of the last ten years.

His first articles on integral geometry appeared more than ten years ago, but this branch of mathematics is still in a formative phase. Because of this we include in the survey an outline of Gel’fand’s basic research in integral geometry, not excluding some that is comparatively old.

Topology is a new branch of his scientific activity; all the work in this field was carried out between 1968 and 1973.

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

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
Albert S. Schwarz	Related	Volume
Ernest Borisovich Vinberg	Related

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Dmitry Fuchs

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