49 research outputs found
Fingerprint databases for theorems
We discuss the advantages of searchable, collaborative, language-independent
databases of mathematical results, indexed by "fingerprints" of small and
canonical data. Our motivating example is Neil Sloane's massively influential
On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater
mathematical community to search for the appropriate fingerprints within each
discipline, and to compile fingerprint databases of results wherever possible.
The benefits of these databases are broad - advancing the state of knowledge,
enhancing experimental mathematics, enabling researchers to discover unexpected
connections between areas, and even improving the refereeing process for
journal publication.Comment: to appear in Notices of the AM
Observable algebra for the rational and trigonometric Euler Calogero Moser models
We construct polynomial Poisson algebras of observables for the classical
Euler-Calogero-Moser (ECM) models. The conserved Hamiltonians and symmetry
algebras derived in a previous work are subsets of these algebras. We define
their linear, limits, realizing \w_{\infty} type
algebras coupled to current algebras.Comment: 11 pages; Latex; PAR LPTHE 94-16 Misprints and minor mistakes
corrected; references update
A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations
We consider the universal solution of the Gervais-Neveu-Felder equation in
the case. We show that it has a quasi-Hopf algebra
interpretation. We also recall its relation to quantum 3-j and 6-j symbols.
Finally, we use this solution to build a q-deformation of the trigonometric
Lam\'e equation.Comment: 9 pages, 4 figure
The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems
We quantize the spin Calogero-Moser model in the -matrix formalism. The
quantum -matrix of the model is dynamical. This -matrix has already
appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's
quantization of the Knizhnik-Zamolodchikov-Bernard equation.Comment: Comments and References adde
The R-matrix structure of the Euler-Calogero-Moser model
We construct the -matrix for the generalization of the Calogero-Moser
system introduced by Gibbons and Hermsen. By reduction procedures we obtain the
-matrix for the Euler-Calogero-Moser model and for the standard
Calogero-Moser model.Comment: 7 page
Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras
In previous work, we introduced a class of integrable spin Calogero-Moser
systems associated with the classical dynamical r-matrices with spectral
parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here
the main purpose is to establish the Liouville integrability of these systems
by a uniform method
Combinatorics of -orbits and Bruhat--Chevalley order on involutions
Let be the group of invertible upper-triangular complex
matrices, the space of upper-triangular complex matrices with
zeroes on the diagonal and its dual space. The group acts
on by , , ,
.
To each involution in , the symmetric group on letters, one
can assign the -orbit . We present a
combinatorial description of the partial order on the set of involutions
induced by the orbit closures. The answer is given in terms of rook placements
and is dual to A. Melnikov's results on -orbits on .
Using results of F. Incitti, we also prove that this partial order coincides
with the restriction of the Bruhat--Chevalley order to the set of involutions.Comment: 27 page