218 research outputs found

### R-matrices in Rime

We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by
a weaker condition which we call "rime". Rime solutions include the standard
Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime
Ansatz which are maximally different from the standard one we call "strict
rime". A strict rime non-unitary solution is parameterized by a projective
vector. We show that this solution transforms to the Cremmer-Gervais R-matrix
by a change of basis with a matrix containing symmetric functions in the
components of the parameterizing vector. A strict unitary solution (the rime
Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to
a quantization of a classical "boundary" r-matrix of Gerstenhaber and
Giaquinto. We analyze the structure of the elementary rime blocks and find, as
a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly
described in a rime form.
We discuss then connections of the classical rime solutions with the Bezout
operators. The Bezout operators satisfy the (non-)homogeneous associative
classical Yang--Baxter equation which is related to the Rota-Baxter operators.
We classify the rime Poisson brackets: they form a 3-dimensional pencil. A
normal form of each individual member of the pencil depends on the discriminant
of a certain quadratic polynomial. We also classify orderable quadratic rime
associative algebras.
For the standard Drinfeld-Jimbo solution, there is a choice of the
multiparameters, for which it can be non-trivially rimed. However, not every
Belavin-Drinfeld triple admits a choice of the multiparameters for which it can
be rimed. We give a minimal example.Comment: 50 pages, typos correcte

### Reality in the Differential Calculus on q-euclidean Spaces

The nonlinear reality structure of the derivatives and the differentials for
the euclidean q-spaces are found. A real Laplacian is constructed and reality
properties of the exterior derivative are given.Comment: 10 page

### Cyclotomic shuffles

Analogues of 1-shuffle elements for complex reflection groups of type
$G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms
of rotational permutations of polygonal cards is given. We compute the
eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra
of the group $G(m,1,n)$. Considering shuffling as a random walk on the group
$G(m,1,n)$, we estimate the rate of convergence to randomness of the
corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue
in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$

### Realization of $U_q(so(N))$ within the differntial algebra on ${\bf R}_q^N$

We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential
operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are
suitable q-deformed analogs of the angular momentum components on ordinary
${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$
splits into a direct sum of irreducible vector representations of
$U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight
representations.Comment: 26 pages, 1 figur

### Braidings of Tensor Spaces

Let $V$ be a braided vector space, that is, a vector space together with a
solution $\hat{R}\in {\text{End}}(V\otimes V)$ of the Yang--Baxter equation.
Denote $T(V):=\bigoplus_k V^{\otimes k}$. We associate to $\hat{R}$ a solution
$T(\hat{R})\in {\text{End}}(T(V)\otimes T(V))$ of the Yang--Baxter equation on
the tensor space $T(V)$. The correspondence $\hat{R}\rightsquigarrow
T(\hat{R})$ is functorial with respect to $V$.Comment: 10 pages, no figure

### On representations of complex reflection groups G(m,1,n)

An inductive approach to the representation theory of the chain of the
complex reflection groups G(m,1,n) is presented. We obtain the Jucys-Murphy
elements of G(m,1,n) from the Jucys--Murphy elements of the cyclotomic Hecke
algebra, and study their common spectrum using representations of a degenerate
cyclotomic affine Hecke algebra. Representations of G(m,1,n) are constructed
with the help of a new associative algebra whose underlying vector space is the
tensor product of the group ring of G(m,1,n) with a free associative algebra
generated by the standard m-tableaux.Comment: 18 page

### BRST operator for quantum Lie algebras and differential calculus on quantum groups

For a Hopf algebra A, we define the structures of differential complexes on
two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an
exterior extension of the dual algebra A^*. The Heisenberg double of these two
exterior Hopf algebras defines the differential algebra for the Cartan
differential calculus on A. The first differential complex is an analog of the
de Rham complex. In the situation when A^* is a universal enveloping of a Lie
(super)algebra the second complex coincides with the standard complex. The
differential is realized as an (anti)commutator with a BRST- operator Q. A
recurrent relation which defines uniquely the operator Q is given. The BRST and
anti-BRST operators are constructed explicitly and the Hodge decomposition
theorem is formulated for the case of the quantum Lie algebra U_q(gl(N)).Comment: 20 pages, LaTeX, Lecture given at the Workshop on "Classical and
Quantum Integrable Systems", 8 - 11 January, Protvino, Russia; corrected some
typo

- …