Volume 3 (2003), Number 4. Abstracts

V. Arnold. Frequent Representations [PDF]

Given a unitary representation T of a finite group G in Cⁿ, write M for the variety of such representations which are unitary equivalent to T. The representation T is said to be frequent if the dimension of the variety M is maximal (among all representations of G in the same complex space). We prove that the irreducible representations are distributed, in the frequent representation (of large dimension), asymptotically in the same way as in the fundamental representation in the space of functions on G: the frequencies of the irreducible components are proportional to their dimensions.

Keywords. Representations of finite groups, unitary representations, frequent representations.

2000 Mathematics Subject Classification. Primary: 20Cxx; Secondary: 11N37.

S. Bogatyi, D. Gonçalves, E. Kudryavtseva, and H. Zieschang. On the Wecken Property for the Root Problem of Mappings between Surfaces [PDF]

Let M₁ and M₂ be two closed (not necessarily orientable) surfaces, f: M₁ → M₂ be a continuous map, and c be a point in M₂. By definition, the map f has the Wecken property for the root problem if f can be deformed into a map f^˜ such that the number |f^˜-1(c)| of roots of f^˜ coincides with the number NR[f] of the essential Nielsen root classes of f, that is, MR[f] = NR[f]. We characterize the pairs of surfaces M₁, M₂ for which all continuous mappings f: M₁ → M₂ have the Wecken property for the root problem. The criterion is formulated in terms of the Euler characteristics of the surfaces and their orientability properties.

Keywords. Coincidence points, roots of maps, Nielsen classes, branched covering.

2000 Mathematics Subject Classification. 54H25, 57M12, 55M20.

B. Feigin, J. Hong, and T. Miwa. A Construction of Level 1 Irreducible Modules for U_q(sp₄^ˆ) using Level 2 Intertwiners for U_q(sl₂^ˆ) [PDF]

We bosonize certain components of level l U_q(sl₂^ˆ)-intertwiners of (l+1)-dimensions. For l=2, these intertwiners, after certain modification by bosonic vertex operators, are added to the algebra U_q(sl₂^ˆ) at level 2 to construct all irreducible highest weight representations of level 1 for the quantum affine algebra U_q(sp₄^ˆ).

Keywords. Quantized enveloping algebra, sp₄^ˆ, irreducible representation, extended vertex operator.

2000 Mathematics Subject Classification. Primary: 17B37; Secondary: 17B69.

G. Felder and A. Veselov. Action of Coxeter Groups on m-Harmonic Polynomials and Knizhnik—Zamolodchikov Equations [PDF]

The Matsuo—Cherednik correspondence is an isomorphism from solutions of Knizhnik—Zamolodchikov equations to eigenfunctions of generalized Calogero—Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply a version of this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials. We compute the Poincaré polynomials for the space H_m and for its isotypical components corresponding to each irreducible representation of the group G. We also give an explicit formula for m-harmonic polynomials of lowest positive degree in the S_n case.

Keywords. Coxeter groups, m-harmonic polynomials, Knizhnik—Zamolodchikov equation.

2000 Mathematics Subject Classification. 13A50, 20F55.

A. Mikhailov and V. Novikov. Classification of Integrable Benjamin—Ono-Type Equations [PDF]

Integrable generalisations of the Benjamin—Ono equation are constructed. The integrable equations of this type are classified by using the perturbative symmetry approach.

Keywords. Soliton theory, integrable equations, Benjamin—Ono-type equations, symmetry approach.

2000 Mathematics Subject Classification. 35Q58.

M. Olshanetsky. The Large N limits of integrable models [PDF]

We consider the large N limits of Hitchin-type integrable systems. The first system is the elliptic rotator on GL_N that corresponds to the Higgs bundle of degree 1 over an elliptic curve with a marked point. This system is gauge equivalent to the N-body elliptic Calogero—Moser system, which is obtained from the Higgs bundle of degree zero over the same curve. The large N limit of the former system is the integrable rotator on the group of the non-commutative torus. Its classical limit leads to an integrable modification of 2D hydrodynamics on the two-dimensional torus. We also consider the elliptic Calogero—Moser system on the group of the non-commutative torus and consider the systems that arise after the reduction to the loop group.

Keywords. Classical integrable systems, noncommutative geometry, 2D hydrodynamics.

2000 Mathematics Subject Classification. 37KXX, 70EXX.

A. Ranicki. Blanchfield and Seifert Algebra in High-Dimensional Knot Theory [PDF]

The Blanchfield and Seifert forms of knot theory have algebraic analogues over arbitrary rings with involution. The covering Blanchfield form of a Seifert form is an algebraic analogue of the expression of the infinite cyclic cover of a knot complement as the infinite union of copies of a cobordism between two copies of a Seifert surface. The inverse construction of the Seifert forms of a Blanchfield form is an algebraic analogue of the transversality construction of the Seifert surfaces of a knot as codimension 1 submanifolds of the knot complement.

Keywords. Blanchfield form, Seifert form, algebraic transversality.

2000 Mathematics Subject Classification. 19J25, 57C45.

J. Sanders and J. Wang. Integrable Systems in n-Dimensional Riemannian Geometry [PDF]

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and a hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in o_n+1 with the vector modified Korteweg—De Vries equation (vmKDV)

u_t = u_xxx + 3/2 ||u||² u_x

as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the natural or parallel frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) Frenêt frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame.

Keywords. Hamiltonian pair, Riemanian geometry, Cartan connection, moving frame, generalized Hasimoto transformation.

2000 Mathematics Subject Classification. Primary: 37K; Secondary: 53A55.

M. Schlichenmaier. Higher Genus Affine Algebras of Krichever—Novikov Type [PDF]

For higher genus multi-point current algebras of Krichever—Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are studied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a simple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the currents algebras.

Keywords. Krichever—Novikov algebras, central extensions, almost-grading, conformal field theory, infinite-dimensional Lie algebras, affine algebras, differential operator algebras, local cocycles.

2000 Mathematics Subject Classification. 17B67, 17B56, 17B66, 14H55, 17B65, 30F30, 81R10, 81T40.

Ya. Sinai. Uniform Distribution in the (3x+1)-Problem [PDF]

Structure theorem of the (3x+1)-problem claims that the images under Tⁿ of arithmetic progressions with step 2^k are arithmetic progressions with step 3^m. Here T is the basic underlying map and a given 3^m progression can be the image of many different 2^k progressions. This gives rise to a probability distribution on the space of 3^m progressions. In this paper it is shown that this distribution is in a sense close to the uniform law.

Keywords. (3x+1)-problem, uniform distribution, characteristic function.

2000 Mathematics Subject Classification. 60c05.

A. Vershik. Strange Factor Representations of Type II₁ and Pairs of Dual Dynamical Systems [PDF]

Given a pair of dynamical systems, we construct a pair of commuting factors of type II₁. This construction is a generalization of the classical von Neumann—Murray construction of factors as crossed products and of the groupoid construction. The suggested construction provides natural examples of factors with non-unity coupling constant. First examples of this kind, related to actions of abelian groups and to the theory of quantum tori, were given by Connes and Rieffel and by Faddeev; our generalization includes these examples as well as new examples of factorizations related to lattices in Lie groups, the infinite symmetric group, etc.

Keywords. Coupling constant, dynamical system, factor representation, Heisenberg group, pseudogroupoid, infinite symmetric group.

2000 Mathematics Subject Classification. Primary 46L10; Secondary 22E47, 22E65.

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