Ernest Vinberg

Ernest Borisovich Vinberg (Russian Эрнест Борисович Винберг, Ernest Borisovich Vinberg English transcription; born July 26, 1937) is a Russian mathematician who deals with algebra, especially representation theory of groups, and geometry.

Life

Winberg made ​​in 1959 at the Moscow State University graduated and received his PhD in 1962 with Eugene Dynkin and Ilya Pjatetskij - Shapiro. He teaches since 1961 at the Department of Algebra at the Moscow State University (since 1966 with an assistant professor, since 1991 full professor ) and is also a professor at the Independent University of Moscow.

He is in the steering committee of the Moscow Mathematical Society. Vinberg received the Humboldt Research Award. In 1983 he was invited speaker at the International Congress of Mathematicians in Warsaw (Discrete reflection groups in Lobachevsky spaces ).

One of his doctoral Victor Kac (1968) and Boris Weis Feiler.

Work

Vinberg classified in its first working homogeneous spaces of Lie groups with invariant linear relationship forms. Then he dealt with homogeneous convex cones, which he in non- classified -dual case (via connections to Jordan algebras ), where he discovered a new class of non- associative algebras, which is named after him ( Vinberg algebras ). He also gave the first example of a non- self-dual homogeneous convex cone. In the 1980s he studied invariant cones in Lie algebras.

From the 1960s he began a systematic study of discrete crystallographic reflection groups. In 1983 he proved that there are no compact hyperbolic reflection groups in hyperbolic spaces of thirty or more dimensions. He also examined the Arithmetizität hyperbolic reflection groups and proved that with thirty or more dimensions are not discrete arithmetic reflection groups of non- compact type in hyperbolic spaces.

In the invariant theory he classified, for example in 1976 with Victor Kac and VL Popov (which in 1972 he received his doctorate ) the simple continuous irreducible linear algebraic groups with a free algebra of invariants.

In the 1970s, he promoted the study of locally transitive transformations of algebraic groups in algebraic varieties and equivariant embedding of homogeneous spaces. This led among other things to a theory of toric varieties representation in the convex geometry by rational polyhedron and subjects. In 1986, he led the transformation of a reducible group in irreducible algebraic varieties is a measure of the complexity of a.

In the 2000s he studied commutative homogeneous spaces of Lie groups, ie those homogeneous spaces with a commutative algebra of invariant differential operators.

Writings

  • Linear representations of groups. Birkhäuser, 1989
  • A Course in Algebra. American Mathematical Society (AMS ), 2003
  • As editor and co-author: Lie groups and invariant theory. AMS, 2005 ( therein by Vinberg, among other things: simple Construction of the exceptional Lie algebras )
  • By A. L. Onishchik: Lie groups and algebraic groups. Springer, 1990
  • With V. V. Gorbatsevich, A. L. Onishchik: Foundations of Lie groups and Lie transformation groups. Springer, 1997
  • Hyperbolic reflection groups. Russian Mathematical Surveys, Bd.40, 1985, p.31 -75
  • As editor: Geometry II Encyclopedia of Mathematical Sciences, Springer, 1991, ( in Vinberg and others: Geometry of spaces of constant curvature, Discrete groups of motion of spaces of constant curvature )
Ilya Piatetski-Shapiro Humboldt Prize Boris Weisfeiler Mathematics Genealogy Project Name Authority File
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