Alexander Alexandrovich Kirillov

Alexander Alexandrovich Kirillov (1999)

Alexander Alexandrowitsch Kirillow ( Russian Александр Александрович Кириллов , English transliteration Alexandre Aleksandrovich Kirillov ; born May 9, 1936 ) is a Russian mathematician who deals with representation theory of Lie groups and now teaches in the USA .

Kirillow studied at the Lomonosov University in Moscow , where he received his doctorate in 1962 under Israel Gelfand on Unitary representations of nilpotent Lie groups (Russian Mathematical Surveys Vol. 17, 1962, pp. 57-110). Then he was a professor at Lomonosov University. In 1994 he became Francis J. Carey Professor of Mathematics at the University of Pennsylvania .

Kirillov is known for the orbit method in Lie group representation theory. Kirillov originally only investigated nilpotent Lie groups , where he showed that the irreducible unitary representations (except for unitary equivalence) are classified by the "orbit" of the maps of in the dual Lie algebra (coadjoint orbits) of . The orbit method was extended to solvable Lie groups by Bertram Kostant , Louis Auslander , Lajos Pukánszky and others. From his orbit method, he also derived a formula for the characters of the irreducible representations of the Lie group (Kirillov character formula). ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle G}$

Kirillov also sees a version of geometric quantization in his orbit method.

In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki ( Infinite dimensional groups, their representations, orbits, invariants ), 1966 in Moscow ( Theory of the representation of groups , with Mark Graev ) and 1962 in Stockholm ( Unitary representations of nilpotent Lie groups ) . He is a fellow of the American Mathematical Society .

His son Alexander Kirillow junior is also a mathematician in the USA. He also deals with the representation theory of Lie groups.

As a schoolboy, Kirillov won several mathematical prizes in competitions, including the Mathematics Olympiad. He is the author of several popular science books.

Kirillov had a very active seminar in Moscow for 30 years. Among his students are Victor Ginzburg , the recipient of the Fields Medal Andrei Okunkow and David Kazhdan .

In 1965 he received the Moscow Mathematical Society Prize.

Fonts

Elements of the Theory of Representations. Springer, 1976 (Russian 1972).
Lectures on the orbit method. Graduate Studies in Mathematics, 64th American Mathematical Society, Providence, Rhode Island, 2004.
Geometric quantization. In: Encyclopedia of Mathematical Sciences. Dynamical Systems. Vol. 4, 1990.
Introduction to the theory of representations and noncommutative harmonic analysis. In: Encyclopedia of Mathematical Sciences. Vol. 22, 1994.

Web links

Remarks

↑ originally submitted for the candidate thesis, but with it he immediately received the doctorate, which corresponds to the habilitation in the West
↑ It can also be applied to compact Lie groups, for example, but no longer provides a clear correspondence between irreducible representations and coadjoint orbits. See, for example, Kirillow, Bulletin of the AMS, Vol. 36, 1999, No. 4. Further applications are also discussed there.
↑ Coadjoint orbits correspond to homogeneous symplectic manifolds with a symmetry group , the "classical part", the "quantization" corresponds to the irreducible unitary representations of in a Hilbert space. See Kirillow, Bulletin AMS 1999. ${\ displaystyle G}$ ${\ displaystyle G}$

personal data
SURNAME	Kirillov, Alexander Alexandrovich
ALTERNATIVE NAMES	Кириллов, Александр Александрович (Russian)
BRIEF DESCRIPTION	Russian mathematician
DATE OF BIRTH	May 9, 1936

[1] riginally submitted for the candidate thesis, but with it he immediately received the doctorate, which corresponds to the habilitation in the West

[2] It can also be applied to compact Lie groups, for example, but no longer provides a clear correspondence between irreducible representations and coadjoint orbits. See, for example, Kirillow, Bulletin of the AMS, Vol. 36, 1999, No. 4. Further applications are also discussed there.

[3] Coadjoint orbits correspond to homogeneous symplectic manifolds with a symmetry group , the "classical part", the "quantization" corresponds to the irreducible unitary representations of in a Hilbert space. See Kirillow, Bulletin AMS 1999. ${\ displaystyle G}$ ${\ displaystyle G}$