Institute of Math., Scientific researcher

Rooms: 406, 407, 408

Institute of Mathematics of NAS of Ukraine

Tereshchenkivs'ka 3, 01601, Kiev, Ukraine

Hello, I am Alexandre. picture1, picture2 This is my CV (English, French)

I`m interested in Representation Theory of infinite-dimensional Lie groups, Lie algebras, braid groups and quantum groups.

I try to construct and study an analogue of the regular, quasiregular and induced representations for infinite-dimensional groups, using quasi-invariant measures on some completion of the initial group or some completions of the corresponding homogeneous spaces. The question of the ireducibility of the condstructed representations are of the main importance. The "Ismagilov conjecture" explains when the regular and quasiregular representations are ireducible. The principal aim is to veryfie these conjectures for various infinite-dimensional groups. This is basis for developing the "orbit method" for infinite-dimensional groups. Ismagilov's conjecture is true for the following groups and some Gaussian and non-Gaussian measures:

1) The inductive limit of the groups of upper-triangular matrices and more general, the inductive limit of classical matrix groups.

2) The group of the interval and circle diffeomorphisms.

3) The central extension of the group of circle diffeomorphisms.

2) The orbit method for infinite-dimensional groups.

3) Von-Neumann algebras and infinite-dimensional groups.

4) Representations of the braid groups and of the quantum groups.

There is a striking connection (arxiv:[2], see also arxiv:[1]) between the representations of the braid group B_3 and a highest weight modules of the quantum group U_q(sl_2), a one-parameter deformation of the universal enveloping algebra U( sl_2) of the Lie algebra sl_2. We generalize these connection (see ArXiv: [2]) between the representations of the braid group B_n and the highest weight modules of U_q(sl_(n-1)) for arbitrary n.

I was the coordinator (ukrainian) of the Germany-Ukrainian DFG project 436 UKR 113/72 (2003-2006).

Now I am the coordinator (ukrainian) of the Germany-Ukrainian DFG project 436 UKR 113/87 (2006-2008).

Our research is also supported by Ukrainian Foundation for Fundamental Research, Grant 10.01/004 (2004-2006).

## arXiv: |

[1] Kosyak A. and Albeverio S. q-Pascal's triangle and irreducible representations of the braid group B_3 in arbitrary dimension, arXiv:math.QA(RT)/ 0803.2778v2 |

[2] A.V.Kosyak Representations of the braid group $B_n$ and the highest weight modules of $U(\mathfrak{sl}_{n-1})$ and $U_q(\mathfrak{sl}_{n-1})$, arXiv:math.QA(RT)/ 0803.2785v2. |

[3] A.V. Kosyak, Type III_1 factors generated by regular representations of infinite dimensional nilpotent group $B_0^{\mathbb N}$, arXiv:math.RT(OA)/ 0803.3340v1 |

[4] A.V.Kosyak, Irreducibility criterion for the set of two matrices. arXiv:math.RT(GR)/0807.4696. |

## Journals: |

[1] Albeverio, Sergio; Kosyak, Alexandre, Quasiregular representations of the infinite-dimensional nilpotent group, J.Funct. Anal. 236 (2006) 634-681. pdf |

[2] Kosyak, A. V.; Nizhnik, L. P. Generalized translation operators and hypergroups constructed from selfadjoint differential operators. (Russian) Ukr. Mat. Zh. 57 (2005), no. 5, 659--668; translation in Ukrainian Math. J. 57 (2005), no. 5, 782--793. pdf |

[3] Albeverio, Sergio; Kosyak, Alexandre, Group action, quasi-invariant measures and quasiregular representations of the infinite-dimensional nilpotent group. Algebraic and topological dynamics, 259--280, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005. pdf |

[4] Albeverio, Sergio; Kosyak, Alexandre, Quasiregular representations of the infinite-dimensional Borel group. J. Funct. Anal. 218 (2005), no. 2, 445--474. pdf |

[5] Kosyak, A. V. Quasi-invariant measures and irreducible representations of the inductive limit of special linear groups. (Russian) Funktsional. Anal. i Prilozhen. 38 (2004), no. 1, 82--84; translation in Funct. Anal. Appl. 38 (2004), no. 1, 67--68. pdf |

[6] Kosyak, Alexandr; Zekri, Richard, Anti-Wick symbols for infinite products in $K$-homology. $K$-Theory 29 (2003), no. 2, 117--145.MR2029758 (2005d:46149) pdf |

[7] Kosyak, A. V. A critierion for the irreducibility of quasiregular representations of the group of finite upper-triangular matrices. (Russian) Funktsional. Anal. i Prilozhen. 37 (2003), no. 1, 78--81; translation in Funct. Anal. Appl. 37 (2003), no. 1, 65--68. MR1988011 (2004f:22021) pdf |

[8] Kosyak, O. V. Elementary representations of the group $B\sb 0\sp {\Bbb Z}$ of infinite in both directions upper-triangular matrices. I. Ukrai"n. Mat. Zh. 54 (2002), no. 2, 205--215; translation in Ukrainian Math. J. 54 (2002), no. 2, 253--265. MR1952821 (2004a:28028) pdf |

[9] Kosyak, A. V.; Leandr, R. Regular representations of the central extension of the diffeomorphism group of the circle. (Russian) Dokl. Akad. Nauk 385 (2002), no. 4, 453--455. MR1944075 pdf |

[10] Kosyak, A. V. The generalized Ismagilov conjecture for the group $B\sb 0\sp \Bbb N$. II. Methods Funct. Anal. Topology 8 (2002), no. 3, 27--45. MR1926911 (2003k:22027) |

[11] Kosyak, A. V. The generalized Ismagilov conjecture for the group $B\sb 0\sp {\Bbb N}$. I. Methods Funct. Anal. Topology 8 (2002), no. 2, 33--49. MR1922683 (2003f:22025) |

[12] Kosyak, A. V. Elementary representations of the group $B\sb 0\sp {\Bbb N}$ of finite upper-triangular matrices. I. Methods Funct. Anal. Topology 7 (2001), no. 1, 34--44. MR1909906 (2003e:22011) dvi |

[13] Kosyak, A. V. Irreducibility of regular Gaussian representations of the group $B\sb 0\sp {\Bbb Z}$. Methods Funct. Anal. Topology 7 (2001), no. 2, 42--51. MR1909687 (2003e:22012) dvi |

[14] Kosyak, A. V.; Zekri, R. Regular representations of infinite-dimensional group $B\sp {\Bbb Z}\sb 0$ and factors. Methods Funct. Anal. Topology 7 (2001), no. 4, 43--48. MR1879484 (2002k:22005) |

[15] Kosyak, A. V. Regular representations of the group of finite upper-triangular matrices, corresponding to product measures, and criteria for their irreducibility. Methods Funct. Anal. Topology 6 (2000), no. 4, 43--55. MR1819617 (2002a:22025) dvi |

[16] Kosyak, A. V.; Zekri, R. Regular representations of infinite-dimensional groups and factors. I. Methods Funct. Anal. Topology 6 (2000), no. 2, 50--59. MR1783774 (2001h:46116) dvi |

[17] Kosyak, A. V. Gaussian measures, quasi-invariant with respect to inverse transformations, on the group of upper-triangular matrices of infinite order. (Russian) Funktsional. Anal. i Prilozhen. 34 (2000), no. 1, 86--90; translation in Funct. Anal. Appl. 34 (2000), no. 1, 70--72. MR1756739 (2001j:28013) pdf |

[18] Kosyak, Alexandr; Zekri, Richard Anti-Wick symbols on infinite tensor product spaces. Methods Funct. Anal. Topology 5 (1999), no. 2, 29--39. MR1771898 (2001e:46121) |

[19] Kosyak, A. V. Measures on infinite-dimensional groups, quasi-invariant with respect to inverse mapping and the commutant theorem. Analysis on infinite-dimensional Lie groups and algebras (Marseille, 1997), 182--196, World Sci. Publishing, River Edge, NJ, 1998. MR1743167 (2001f:22062) www |

[20] Kosyak, A. V. Irreducible regular Gaussian representations of the groups of the interval and circle diffeomorphisms. J. Funct. Anal. 125 (1994), no. 2, 493--547. MR1297679 (95j:22026) pdf |

[21] Kosyak, A. V. Criteria for irreducibility and equivalence of regular Gaussian representations of groups of finite upper-triangular matrices of infinite order. Selected translations. Selecta Math. Soviet. 11 (1992), no. 3, 241--291.MR1181263 (93i:22024) pdf |

[22] Kosyak, A. V.; Samoilenko, Yu. S. Quasi-invariant measures on "large" groups [translation of The spectral theory of operator-differential equations (Russian), 64--69, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1986; MR0893011 (88g:60024)]. Selected translations. Selecta Math. Soviet. 10 (1991), no. 1, 1--6. MR1099431 |

[23] Kosyak, A. V. A criterion for the irreducibility and equivalence of regular Gaussian representations of the group of finite upper-triangular matrices of infinite order. (Russian) Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 1990, no. 50, 55 pp. MR1124490 |

[24] Kosyak, A. V. A criterion for irreducibility of regular Gaussian representations of the group of finite upper-triangular matrices. (Russian) Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 82--83; translation in Funct. Anal. Appl. 24 (1990), no. 3, 243--245 (1991). MR1082042 (91m:20016) pdf |

[25] Kosyak, A. V. Extension of unitary representations of inductive limits of finite-dimensional Lie groups. Rep. Math. Phys. 26 (1988), no. 2, 285--302. MR0991725 (90d:22025) pdf |

[26] Kosyak, A. V.; Samoilenko, Yu. S. Quasi-invariant measures on "large" groups. (Russian) The spectral theory of operator-differential equations (Russian), 64--69, iv, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1986. MR0893011 (88g:60024) |

[27] Kosyak, A. V. Extension of unitary representations of a group of finite upper-triangular matrices of infinite order. (Russian) Spectral theory of operators and infinite-dimensional analysis, 102--111, iv, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984. MR0817223 (87g:22022) |

[28] Kosyak, A. V. The Garding domain for representations of canonical commutation relations. (Russian) Ukrain. Mat. Zh. 36 (1984), no. 6, 709--715. MR0772544 (87h:81080) |

[29] Kosyak, A. V.; Samoilenko, Yu. S. The Garding domain and entire vectors for inductive limits of commutative locally compact groups. (Russian) Ukrain. Mat. Zh. 35 (1983), no. 4, 427--434. MR0712461 (85g:22009) pdf |

[30] Kosyak, A. V. Analytic and entire vectors for families of operators. (Russian) Spectral analysis of differential operators, pp. 3--18, 132, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1980. MR0642526 (83f:47021) |

[31] Kosjak, A. V.; Samoilenko, Ju. S. Families of commuting selfadjoint operators. (Russian) Ukrain. Mat. Zh. 31 (1979), no. 5, 555--558. MR0552491 (80m:47023) pdf |

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