176 research outputs found
The Gelfand map and symmetric products
If A is an algebra of functions on X, there are many cases when X can be
regarded as included in Hom(A,C) as the set of ring homomorphisms. In this
paper the corresponding results for the symmetric products of X are introduced.
It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the
set of those functions that satisfy equations generalising f(xy)=f(x)f(y).
These equations are related to formulae introduced by Frobenius and, for the
relevant A, they characterise linear maps on A that are the sum of ring
homomorphisms. The main theorem is proved using an identity satisfied by
partitions of finite sets.Comment: 14 pages, Late
-function of the KdV hierarchy
In this paper we construct a family of commuting multidimensional
differential operators of order 3, which is closely related to the KdV
hierarchy. We find a common eigenfunction of this family and an algebraic
relation between these operators. Using these operators we associate a
hyperelliptic curve to any solution of the stationary KdV equation. A basic
generating function of the solutions of stationary KdV equation is introduced
as a special polarization of the equation of the hyperelliptic curve. We also
define and discuss the notion of a -function of a solution of the stationary
-KdV equation
Elliptic Dunkl operators, root systems, and functional equations
We consider generalizations of Dunkl's differential-difference operators
associated with groups generated by reflections. The commutativity condition is
equivalent to certain functional equations. These equations are solved in many
cases. In particular, solutions associated with elliptic curves are
constructed. In the case, we discuss the relation with elliptic
Calogero-Moser integrable -body problems, and discuss the quantization
(-analogue) of our construction.Comment: 30 page
Multi-Dimensional Sigma-Functions
In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of -functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.Comment: 267 pages, 4 figure
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