3,855 research outputs found
q and q,t-Analogs of Non-commutative Symmetric Functions
We introduce two families of non-commutative symmetric functions that have
analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a
definition in terms of triangularity/scalar product relations - to be
submitted FPSAC'0
From Reflection Equation Algebra to Braided Yangians
In general, quantum matrix algebras are associated with a couple of
compatible braidings. A particular example of such an algebra is the so-called
Reflection Equation algebra. In this paper we analyse its specific properties,
which distinguish it from other quantum matrix algebras (in first turn, from
the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity
for its generating matrix, which in a limit turns into the Cayley-Hamilton
identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we
consider some specific properties of the braided Yangians, recently introduced
by the authors. In particular, we establish an analog of the Cayley-Hamilton
identity for the generating matrix of such a braided Yangian. Besides, by
passing to a limit of the braided Yangian, we get a Lie algebra similar to that
entering the construction of the rational Gaudin model. In its enveloping
algebra we construct a Bethe subalgebra by the method due to D.Talalaev
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
Examples of noncommutative manifolds: complex tori and spherical manifolds
We survey some aspects of the theory of noncommutative manifolds focusing on
the noncommutative analogs of two-dimensional tori and low-dimensional spheres.
We are particularly interested in those aspects of the theory that link the
differential geometry and the algebraic geometry of these spaces.Comment: Survey article. Final version. To appear in the proceedings volume of
the "International Workshop on Noncommutative Geometry", IPM, Tehran 200
Braided Weyl algebras and differential calculus on U(u(2))
On any Reflection Equation algebra corresponding to a skew-invertible Hecke
symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we
define analogs of the partial derivatives. Together with elements of the
initial Reflection Equation algebra they generate a "braided analog" of the
Weyl algebra. When , the braided Weyl algebra corresponding to the
Quantum Group goes to the Weyl algebra defined on the algebra
\Sym((u(2)) or that depending on the way of passing to the limit.
Thus, we define partial derivatives on the algebra , find their
"eigenfunctions", and introduce an analog of the Laplace operator on this
algebra. Also, we define the "radial part" of this operator, express it in
terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on
the algebra . Eventual applications of our approach are discussed.Comment: LaTex, 18 pages. Accepted in Journal of Geometry and Physic
- …