Papers by Alexandre Kosyak
arXiv (Cornell University), Aug 23, 2023
We show that Γ(f 0 ,f 1 ,...,fm) Γ(f 1 ,...,fm) = ∞ for m + 1 vectors having the properties that ... more We show that Γ(f 0 ,f 1 ,...,fm) Γ(f 1 ,...,fm) = ∞ for m + 1 vectors having the properties that no non-trivial linear combination of them belongs to l 2 (N). This property is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.
arXiv (Cornell University), Oct 25, 2023
We introduced previously the generalized characteristic polynomial defined byP C (λ) = det C(λ), ... more We introduced previously the generalized characteristic polynomial defined byP C (λ) = det C(λ), where C(λ) = C + diag λ 1 ,. .. , λ n for C ∈ Mat(n, C) and λ = (λ k) n k=1 ∈ C n and gave the explicit formula for P C (λ). In this article we define an analogue of the resolvent C(λ) −1 , calculate it and the expression (C(λ) −1 a, a) for a ∈ C n explicitly. The obtained formulas and their variants were applied to the proof of the irreducibility of unitary representations of some infinite-dimensional groups.
Type III1 factors generated by regular representations of infinite dimensional nilpotent group B N 0
Mathematika, 2021
Given any positive integer n, let A(n) denote the height of the n th cyclotomic polynomial, that ... more Given any positive integer n, let A(n) denote the height of the n th cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A(n) is unbounded. We conjecture that every natural number can arise as value of A(n) and prove this assuming that for every pair of consecutive primes p and p ′ with p ≥ 127 we have p ′ −p < √ p+1. We also conjecture that every natural number occurs as the maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture holds, i.e., that √ p ′ − √ p < 1 always holds. This is the first time, as far as the authors know, that a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number m ≤ x occurs as A(n) value with at most O ǫ (x 3/5+ǫ) exceptions. On the Lindelöf Hypothesis we show there are at most O ǫ (x 1/2+ǫ) exceptions and study them further by using deep work of Bombieri-Friedlander-Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier.
We study the von Neumann algebra, generated by the unitary representations of infinite-dimensiona... more We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group B_0^ N. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type I_∞ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,14]). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group B_0^ N is type III_1 hyperfinite factor. The case of the nilpotent group B_0^ Z of infinite in both directions matrices will be studied in [6].
We study the von Neumann algebra, generated by the regular representations of the infinite-dimens... more We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group B_0^ Z. In [14] a condition have been found on the measure for the right von Neumann algebra to be the commutant of the left one. In the present article, we prove that, in this case, the von Neumann algebra generated by the regular representations of group B_0^ Z is the type III_1 hyperfinite factor. We use a technique, developed in [20] where a similar result was proved for the group B_0^ N. The crossed product allows us to remove some technical condition on the measure used in [20]. [14] A.V. Kosyak, Inversion-quasi-invariant Gaussian measures on the group of infinite-order upper-triangular matrices, Funct. Anal. i Priloz. 34, issue 1 (2000) 86--90. [20] A.V. Kosyak, Type III_1 factors generated by regular representations of infinite dimensional nilpotent group B_0^ N, arXiv:0803.3340v1.
In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Br... more In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n+1/2] parameters (and also use the classical Pascal's triangle). The latter authors also gave the complete classification of all simple representations of B_3 for dimension n≤ 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and equivalence of the constructed representations. In the present article we show that all representatio...
Our aim is to find the irreducibility criteria for the Koopman representation, when the group act... more Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group GL_0(2∞, R) = _n GL(2n-1, R), the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian non-centered measures. The corresponding G-space X_m is a subspace of the space Mat(2∞, R) of infinite in both directions real matrices. In fact, X_m is a collection of m infinite in both directions rows. This result was announced in [20]. We give the proof only for m≤ 2. The general case will be studied later.
We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability ... more We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two n× n matrices Λ_n and A_n in terms of the subalgebra associated with the "support" of the matrix A_n, where Λ_n is a diagonal matrix with different non zeros eigenvalues and A_n is an arbitrary one. The list of all maximal subalgebras of the algebra Mat(n, C) and the list of the corresponding invariant subspaces connected with these two matrices is also given. The properties of the corresponding subalgebras are expressed in terms of the graphs associated with the support of the second matrix. For arbitrary n we describe all minimal subsets of the elementary matrices E_km that generate the algebra Mat(n, C).
We construct the so-called quasiregular representations of the group B_0^ N( F_p) of infinite upp... more We construct the so-called quasiregular representations of the group B_0^ N( F_p) of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field F_p is compact some new operators in the commutant emerges. Therefore, the Ismagilov conjecture in the case of the finite field should be corrected.
We study the von Neumann algebra, generated by the unitary representations of infinite-dimensiona... more We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group B N 0 . The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see ). In this case the corresponding von Neumann algebra is type I ∞ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see ). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group B N 0 is type III 1 hyperfinite factor. The case of the nilpotent group B Z 0 of infinite in both directions matrices will be studied in .
We study the von Neumann algebra, generated by the unitary representations of infinite-dimensiona... more We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group B 0 . The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9,10,11]). In this case the corresponding von Neumann algebra is type I∞ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,14]). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group B 0 is type III1 hyperfinite factor. The case of the nilpotent group B 0 of infinite in both directions matrices will be studied in [6].
arXiv: Representation Theory, 2016
Our aim is to find the irreducibility criteria for the Koopman representation, when the group act... more Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibilty of this representation are established. In the particular case of the group ${\rm GL}_0(2\infty,{\mathbb R})$ $= \varinjlim_{n}{\rm GL}(2n-1,{\mathbb R})$, the inductive limit of the general linear groups we prove that these conditions are also the nessesary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian non-centered measures. The corresponding $G$-space $X_m$ is a subspace of the space ${\rm Mat}(2\infty,{\mathbb R})$ of infinite in both directions real matrices. In fact, $X_m$ is a collection of $m$ infinite in both directions rows. This result was announced in [20]. We give the proof only for $m\leq 2$. The general case will be studied later.
Representations of the braid group B 3 and the highest weight modules of U (sl 2) and U q (sl 2) ... more Representations of the braid group B 3 and the highest weight modules of U (sl 2) and U q (sl 2) Abstract In [1] we have constructed a n+1 2 + 1 parameters family of irre-ducible representations of the Braid group B 3 in arbitrary dimension n ∈ N, using a q−deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B 3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B 3 by considering two special operators in the space C n [X]. Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [ n+1 2 ] parameters (and also use the classical Pascal's triangle). The latter authors also gave the complete classification of all simple representations of B 3 for dimension n ≤ 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the ir...
In we have constructed a n+1 2 * The author would like to thank the Max-Planck-Institute of Mathe... more In we have constructed a n+1 2 * The author would like to thank the Max-Planck-Institute of Mathematics and the Institute of Applied Mathematics, University of Bonn for the hospitality. The partial financial support by the DFG project 436 UKR 113/87 is gratefully acknowledged.
arXiv: Representation Theory, 2016
We show that the Lawrence--Krammer representation can be obtained as the quantization of the symm... more We show that the Lawrence--Krammer representation can be obtained as the quantization of the symmetric square of the Burau representation. This connection allows us to construct new representations of braid groups
We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_... more We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also use the classical Pascal triangle). The latter authors also gave the complete classification of all simple representations of B_3 for dimension n\leq 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and the equivalence of the representations. In [17] we establish the connection between the constructed representation of ...
arXiv: Quantum Algebra, 2008
We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_... more We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also use the classical Pascal triangle). The latter authors also gave the complete classification of all simple representations of B_3 for dimension n\leq 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and the equivalence of the representations. In [17] we establish the connection between the constructed representation of ...
arXiv: Representation Theory, 2016
We construct the so-called quasiregular representations of the group $B_0^{\mathbb N}({\mathbb F}... more We construct the so-called quasiregular representations of the group $B_0^{\mathbb N}({\mathbb F}_p)$ of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field ${\mathbb F}_p$ is compact some new operators in the commutant emerges. Therefore, the Ismagilov conjecture in the case of the finite field should be corrected.
Journal of Functional Analysis
Our aim is to find the irreducibility criteria for the Koopman representation, when the group act... more Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group GL 0 (2∞, R) = lim − →n GL(2n − 1, R), the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian noncentered measures. The corresponding G-space X m is a subspace of the space Mat(2∞, R) of infinite in both directions real matrices. In fact, X m is a collection of m infinite in both directions rows. This result was announced in [20]. We give the proof only for m ≤ 2. The general case will be studied later.
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Papers by Alexandre Kosyak