Monodromy of Fuchsian systems on complex linear spaces

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.     

Isomonodromic deformations and twisted Yangians arising in Teichmüller theory

In this paper we build a link between the Teichmüller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincaré uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra Dn of geodesic length functions is the semiclassical limit of the twisted q-Yangian for the orthogonal Lie algebra on defined by Molev, Ragoucy and Sorba. We give a representation of the braid-group action on Dn in terms of an adjoint matrix action. We characterize two types of finite-dimensional Poissonian reductions and give an explicit expression for the generating function of their central elements. Finally, we interpret the algebra Dn as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semi-simple point.     

Construction of a linear Pfaff system of Fuchsian type with algebraic manifold of singularities

We present a constructive proof of the existence of a two-dimensional completely integrable Fuchsian Pfaff system on CP ⁿ with four singular surfaces forming a pencil, with 2-step solvable monodromy group, and with fundamental solution matrix realizing a given homomorphism.     

Flat Connections and Quantum Groups

We review the Kohno–Drinfeld theorem and a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection _C on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture for the cases when g=sl _n or when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection _C and Cherednik's generalisation of the Knizhnik–Zamolodchikov connection to finite reflection groups.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.     

ON QUOTIENTS OF BRAID GROUPS

ABSTRACT.

Let G be the group ?[t, t ^?1] x ?. By studying the action of the braid group B_n on the set Gⁿ , we obtain representations of B_n into a wreath product of the symmetric group and the general linear group over ?[t, t ^?1]. This in particular recovers the Burau representation of the braid group. Furthermore, some quotients of the braid group are obtained by using the representations found.     

Representations of real-valued forms of the graded analogue of a Lie algebra

Real-valued forms of the ₂ ⁿ -graded analogue of the Lie algebra s(2,C) are described and their irreducible representations are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1518–1524, November, 1992.     

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.     

Integral Formulas for the Minimal Representation of O(p,2)

The minimal representation of O(p,q) (p+q: even) is realized on the Hilbert space of square integrable functions on the conical subvariety of R^p+q–2. This model presents a close resemblance of the Schrödinger model of the Segal–Shale–Weil representation of the metaplectic group. We shall give explicit integral formulas for the inversion together with the analytic continuation to a certain semigroup of O(p+2,C) of the minimal representation of O(p,2) by using Bessel functions.Mathematics Subject Classifications (2000) primary 22E30; secondary 22E46, 20M20, 43A80.     

Geometrische Bedingungen für die Integrabilität von Vektorfeldern auf Teilmengen des ℝn

We give a geometric characterisation for those vectorfields on a subset X ⁿ, wich are locally integrable, that is, which locally have sufficiently many integral curves on X. From this we deduce, that integrable spaces X (where each field of a fixed class of differentiability is locally integrable) are rigid under differentiable deformations in the sense of Kodaira-Kuranishi. We give a general construction for integrable spaces and obtain, that analytic varieties induce integrable spaces for each class of differentiability. Compact analytic varieties are therefore C-rigid, which extends [4], 3,1.     

Pointwise estimation of comonotone approximation

We prove that, for a continuous functionf(x) defined on the interval [–1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP _n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)–P _n (x) ¦C₂(f;n^–2+n ^–11–x ²), whereC is a constant that depends on the length of the smallest interval.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.     

The minimum independence number for designs

Fort=2,3 andk2t–1 we prove the existence oft–(n,k,) designs with independence numberC _,k n ^{(k–t)/(k–1)} (ln n) ^1/(k–1) . This is, up to the constant factor, the best possible.Some other related results are considered.Supported by NSF Grant DMS-9011850     

Irreducible Fuchsian system with reducible monodromy representation

We present an example of the reducible representation χ = χ₁ ? χ₂, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation χ₂ is a counterexample to the Riemann-Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations.     

On the gonality of nodal curves

Here we prove that for every n33 and every t(n ²+3n)/6, the normalization Y of a general plane curve C of degree n and with t nodes has no g _b ¹ with b<n–2 and only g _n–2 ¹ and g _n–1 ¹ induced by a pencil of lines through a point of C. Recently, Coppens and Kato proved stronger results.     

Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation

Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M ⁿ with measure . A construction of matrix equations having a set of attractors in the space of all matrix entries is given.     

A rotation method which gives linear estimates for powers of the Ahlfors–Beurling operator

In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers Tⁿ. Motivated by [T. Iwaniec, G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996) 25–57], we deduce from here that Tⁿ_p are bounded from above by Cnp^*, . We further improve this estimate to obtain the optimal behaviour of the L^p norms in question.