Andrei Minchenko

Visiting Assistant Professor of Mathematics

Web Site www.math.cornell.edu/~andreim/ Contact Information Office:  408 Malott Hall Phone:  (607) 255-4640 Fax:  (607) 255-7149 Email:  andreim@math.cornell.edu Courses & Office Hours

Education

Ph.D. (2008) Lomonosov Moscow State University

Research Area: Semisimple Lie groups and algebras, their representations

My general interests are in algebra, especially in Lie groups and algebras. My master thesis contains a solution of a problem posed by my adviser, Professor Vinberg: to describe all classes of conjugate semisimple subalgebras in a complex semisimple Lie algebra. This was done by Maltsev for classical algebras and by Dynkin for exceptional algebras. However, Dynkin described only classes of linearly conjugate subalgebras which can contain more than one class of conjugate subalgebras. In my master dissertation I established that this split is not trivial only in four cases, namely these are types A₂, B₂ and G₂ which consists of two conjugacy classes.

The subject of my Ph. D. thesis is the classification of conjugacy classes of real semisimple subalgebras. Every real subalgebra r is a real form of a complex subalgebra g. F. I. Karpelevich has classified r up to quasi-conjugacy. [If r ⊂ g, then s₁, s₂ ⊂ g are quasi-conjugate if there is an inner automorphism of g which transforms s₁ to s₂ preserving r.] However the question how the class of quasy-conjugate s ⊂ r is partitioned into classes of conjugate s ⊂ r remained open. We developed a method for solution of this problem.

In parallel to my work on dissertation I collaborated with E. B. Dynkin in his project of description of Weyl orbits in the space of subsets of a root system. By using tools developed in our paper we describe a natural partial order between the classes of conjugate subalgebras of a semisimple Lie algebra. We hope that these tools can be useful also in other parts of Lie theory, for instance, in the theory of nillpotent orbits (playing an important role in the representation theory).

Selected Publications

The semisimple subalgebras of exceptional Lie algebras, Trudy MMO 67 (2006), 256–293 (Russian); English transl.: Trans. Moscow Math. Soc. 67 (2006), 225–259.

Triads and short SO₃-subgroups of compact Lie groups, UMN 62 no. 5 (2007), 159–160 (in Russian).

On semisimple subalgebras of real exceptional Lie algebras, Trudy MMO (to appear), approx. 40 pages.

Projective root systems, enhanced Dynkin diagrams and Weyl orbits (with E. B. Dynkin), in preparation.

Last modified: August 17, 2009