161 research outputs found

### Integral formulas for wave functions of quantum many-body problems and representations of gl(n)

We derive explicit integral formulas for eigenfunctions of quantum integrals
of the Calogero-Sutherland-Moser operator with trigonometric interaction
potential. In particular, we derive explicit formulas for Jack's symmetric
functions. To obtain such formulas, we use the representation of these
eigenfunctions by means of traces of intertwining operators between certain
modules over the Lie algebra $\frak gl_n$, and the realization of these modules
on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and
an introduction have been adde

### On the spectrum of S=1/2 XXX Heisenberg chain with elliptic exchange

It is found that the Hamiltonian of S=1/2 isotropic Heisenberg chain with $N$
sites and elliptic non-nearest-neighbor exchange is diagonalized in each sector
of the Hilbert space with magnetization $N/2-M$, $1<M\leq[N/2]$, by means of
double quasiperiodic meromorphic solutions to the $M$-particle quantum
Calogero-Moser problem on a line. The spectrum and highest-weight states are
determined by the solutions of the systems of transcendental equations of the
Bethe-ansatz type which arise as restrictions to particle pseudomomenta.Comment: 9 pages, Late

### Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation

We construct solutions of an infinite Toda system and an analogue of the
Painlev'e II equation over noncommutative differential division rings in terms
of quasideterminants of Hankel matrices.Comment: 16 pp; final revised version, will appear in J.Phys. A, minor changes
(typos corrected following the Referee's List, aknowledgements and a new
reference added

### Elliptic quantum groups and Ruijsenaars models

We construct symmetric and exterior powers of the vector representation of
the elliptic quantum groups $E_{\tau,\eta}(gl_N)$. The corresponding transfer
matrices give rise to various integrable difference equations which could be
solved in principle by the nested Bethe ansatz method. In special cases we
recover the Ruijsenaars systems of commuting difference operators.Comment: 15 pages, late

### Logarithmic corrections to finite size spectrum of SU(N) symmetric quantum chains

We consider SU(N) symmetric one dimensional quantum chains at finite
temperature. For such systems the correlation lengths, ground state energy, and
excited state energies are investigated in the framework of conformal field
theory. The possibility of different types of excited states are discussed.
Logarithmic corrections to the ground state energy and different types of
excited states in the presence of a marginal opeartor, are calculated. Known
results for SU(2) and SU(4) symmetric systems follow from our general formula.Comment: 5 pages, 1 figure; Typos corrected and minor changes made for clarit

### Parametrization of semi-dynamical quantum reflection algebra

We construct sets of structure matrices for the semi-dynamical reflection
algebra, solving the Yang-Baxter type consistency equations extended by the
action of an automorphism of the auxiliary space. These solutions are
parametrized by dynamical conjugation matrices, Drinfel'd twist representations
and quantum non-dynamical $R$-matrices. They yield factorized forms for the
monodromy matrices.Comment: LaTeX, 24 pages. Misprints corrected, comments added in Conclusion on
construction of Hamiltonian

### Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

For the rational Baker-Akhiezer functions associated with special
arrangements of hyperplanes with multiplicities we establish an integral
identity, which may be viewed as a generalisation of the self-duality property
of the usual Gaussian function with respect to the Fourier transformation. We
show that the value of properly normalised Baker-Akhiezer function at the
origin can be given by an integral of Macdonald-Mehta type and explicitly
compute these integrals for all known Baker-Akhiezer arrangements. We use the
Dotsenko-Fateev integrals to extend this calculation to all deformed root
systems, related to the non-exceptional basic classical Lie superalgebras.Comment: 26 pages; slightly revised version with minor correction

### On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf
algebra H. When H is semisimple over C, these invariants are respectively, the
trace of the map induced by the antipode on the endomorphism ring of a
self-dual simple module, and the higher Frobenius-Schur indicators of the
regular representation. We further study the values of these higher indicators
in the context of complex semisimple quasi-Hopf algebras H. We prove that these
indicators are non-negative provided the module category over H is modular, and
that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a
factor of dim H. As an application, we show the existence of a non-trivial
self-dual simple H-module with bounded dimension which is determined by the
value of the second indicator.Comment: additional references, fixed some typos, minor additions including a
questions and some remark

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