E. B. Vinberg

Ernest Borisovich Vinberg

This issue is dedicated to Ernest Borisovich Vinberg, outstanding Russian mathematician, professor at Moscow State University and at the Independent University of Moscow, who celebrated his 70th anniversary in July 2007.

Professor Vinberg made significant contributions to many fields of mathematics related to algebra and geometry. He expressed his mathematical credo in the preface to his beautiful manual Course of algebra: “In accordance with my view of mathematics, I aimed at replacing calculations and intricate arguments by ideas.” Ernest Vinberg follows this principle in all his works, where he avoids long routine proofs and always presents short, elegant, and at the same time absolutely rigorous arguments. For several generations of mathematicians, Vinberg's works serve as samples of deep content, mathematical taste and style of exposition. The same principle is reflected in his lectures, which are characterized by diverse and substantive content and richness of mathematical ideas together with accurate and well-prepared exposition. Attending the lectures of Vinberg, one realizes that mathematics is very natural and in fact simple in its best manifestations.

E.B. Vinberg graduated from the Faculty of Mechanics and Mathematics of Moscow State Lomonosov University in 1959 and defended his PhD thesis in 1962. From 1961 to the present day he works at the Algebra Chair of Moscow State University (as an associate professor from 1966, as a full professor from 1991).

The first scientific work of Vinberg was devoted to the description of invariant linear connections on a homogeneous space G/H of a Lie group G. He proved that for semisimple G, any flat invariant linear connection on G/H determines a realization of G/H as an open orbit M in a linear G-action on a vector space V and is a restriction to M of a standard flat connection on V. The homogeneous spaces admitting a flat invariant linear connection were classified.

The next cycle of Vinberg's works develops the theory of homogeneous convex cones. This theory is closely related to the theory of homogeneous Siegel domains, which was developed by É. Cartan and I.I. Pyatetski-Shapiro. Vinberg constructed the first example of a non-self-dual homogeneous convex cone and obtained a complete classification of self-dual homogeneous convex cones based on an unexpected connection of such cones with compact Jordan algebras. In this context, Vinberg discovered another class of non-associative algebras. The author called them compact left-symmetric normal algebras (shortly, clans), but in the modern literature these algebras are called Vinberg algebras. In the same period Vinberg wrote well-known joint papers with S.G. Gindikin and I.I. Pyatetski-Shapiro on homogeneous Kählerian manifolds.

From 1980, Vinberg revisited convex cones in a different context. He studies invariant cones in Lie algebras, finds a criterion for the existence of an invariant convex cone in the representation space of a finite-dimensional irreducible real representation of a semisimple group. He introduces and studies in detail the concept of invariant continuous ordering on a Lie group. It is worth noting that invariant cones and orderings are closely related to subsemigroups in Lie groups. The results and ideas of Vinberg stimulated essential progress in this area, which is now in the focus of active research.

E.B. Vinberg initiated the systematic study of discrete crystallographic reflection groups. From the end of the 60's, he is the recognized world leader of this field. In 1983 Vinberg proved the following difficult theorem: in hyperbolic spaces of dimension ≥30 there are no hyperbolic reflection groups of compact type. There are other important results of Vinberg in this area: the theorem asserting that an arithmetic hyperbolic reflection group is an arithmetic group of simplest type, i.e., is commensurable with the unit group of an appropriate hyperbolic quadratic lattice; a criterion of arithmeticity of hyperbolic reflection groups; the non-existence of arithmetic discrete reflection groups of non-compact type in hyperbolic spaces of dimension ≥30. The research of Vinberg is focused on unit groups of hyperbolic lattices, discrete linear groups generated by finitely many reflections, and arithmetic discrete groups.

It is hard to overestimate the contribution of Vinberg to Invariant Theory or, more generally, to the theory of algebraic transformation groups. He initiated and fulfilled in certain aspects the program of study and classification of those finite-dimensional representations of reductive algebraic groups which have certain “good” properties. In the seminal paper of 1976 (joint with V.G. Kac and V.L. Popov) the connected simple irreducible linear groups with a free algebra of invariants were classified. The approach based on the study of slice representations, used in this paper, is now a standard tool in modern Invariant Theory. The theory of theta-groups, developed by Vinberg, allows to prove a number of remarkable properties (the fact that the algebra of invariants and the module of covariants is free, analogues of Cartan subspaces and Weyl groups, Jordan decomposition, finite number of orbits in the fibres of the quotient map) for a wide class of representations related to periodic gradings of complex semisimple Lie algebras. This theory generalizes the results on the adjoint representation of a semisimple group and the isotropy representation of a symmetric space (B. Kostant and S. Rallis).

The works of Vinberg on locally transitive algebraic group actions stimulated the development of the theory of equivariant embeddings of homogeneous spaces, which is now a very popular direction of research. This theory includes the theory of toric varieties, which gives an effective description of an important class of algebraic varieties in terms of convex geometry of rational polyhedral cones and fans. In 1972 Vinberg published two papers (the second one jointly with V.L. Popov), which initiated the use of such a combinatorial language for the description of algebraic group actions with an open orbit. In the theory of equivariant embeddings of homogeneous spaces, the central rôle is played by the notion of complexity c(X,G) for an action of a reductive group G on an irreducible algebraic variety X, which was introduced by Vinberg. In 1986 Vinberg proved that the complexity of a reductive group action coincides with the modality of the action of its Borel subgroup. Further development of the theory showed that c(X,G) is an adequate measure for the complexity of an action, which separates “tame” actions from “wild” ones. Vinberg obtained important results on the structure and classification of affine algebraic monoids. His works extended this theory by a number of beautiful results, which were essentially developed further. For instance, Vinberg's construction of the enveloping semigroup yields the Cox realization for the wonderful completion of an adjoint semisimple group.

The scientific interests of E.B. Vinberg in recent years are connected with problems in equivariant symplectic geometry. Vinberg studied commutative homogeneous spaces of Lie groups, i.e., those homogeneous spaces possessing commutative algebra of invariant differential operators. The theory of commutative spaces, which was developed and is actively popularized by Vinberg, demonstrates the effectivity of interaction between differential geometry, representation theory, harmonic calculus, and algebraic theory of spherical varieties.

E.B. Vinberg is the editor and member of editorial boards of several leading mathematical journals. He is a member of the Executive Committee of the Moscow Mathematical Society. His diligence and responsibility give rise to the deep respect of his colleagues.

Ernest Vinberg combines both mathematical and pedagogical talents. Lectures and courses given by Vinberg during more than 40 years attract new and new generations of disciples. As a scientific advisor, Vinberg is distinguished by a special talent for formulating interesting scientific problems which stimulate young researchers to make progress in their studies and find their own way in mathematics. On this way, they can always count on his support.

The human traits of Ernest Borisovich Vinberg, such as his kindness and benevolence, objectivity and adherence to principles, win the love and respect of his colleagues and numerous disciples. With all our hearts we wish Ernest Borisovich good health and further success in his creative work!

I. Arzhantsev, S. Gusein-Zade, Yu. Ilyashenko,
A. Onishchik, A. Sossinsky, D. Timashev, M. Tsfasman

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