14 research outputs found
Quadratic Poisson brackets and Drinfeld theory for associative algebras
The paper is devoted to the Poisson brackets compatible with multiplication
in associative algebras. These brackets are shown to be quadratic and their
relations with the classical Yang--Baxter equation are revealed. The paper also
contains a description of Poisson Lie structures on Lie groups whose Lie
algebras are adjacent to an associative structure.Comment: 16 pages, latex, no figure
Quadratic Poisson brackets and Drinfel'd theory for associative algebras
Quadratic Poisson brackets on associative algebras are studied. Such a
bracket compatible with the multiplication is related to a differentiation in
tensor square of the underlying algebra. Jacobi identity means that this
differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie
groups are canonically equipped with a Poisson Lie structure. A way to quantize
such structures is suggested.Comment: latex, no figures
Poisson homology of r-matrix type orbits I: example of computation
In this paper we consider the Poisson algebraic structure associated with a
classical -matrix, i.e. with a solution of the modified classical
Yang--Baxter equation. In Section 1 we recall the concept and basic facts of
the -matrix type Poisson orbits. Then we describe the -matrix Poisson
pencil (i.e the pair of compatible Poisson structures) of rank 1 or -type
orbits of . Here we calculate symplectic leaves and the integrable
foliation associated with the pencil. We also describe the algebra of functions
on -type orbits. In Section 2 we calculate the Poisson homology of
Drinfeld--Sklyanin Poisson brackets which belong to the -matrix Poisson
family
Physical phase space of lattice Yang-Mills theory and the moduli space of flat connections on a Riemann surface
It is shown that the physical phase space of \g-deformed Hamiltonian
lattice Yang-Mills theory, which was recently proposed in refs.[1,2], coincides
as a Poisson manifold with the moduli space of flat connections on a Riemann
surface with handles and therefore with the physical phase space of
the corresponding -dimensional Chern-Simons model, where and are
correspondingly a total number of links and vertices of the lattice. The
deformation parameter \g is identified with and is an
integer entering the Chern-Simons action.Comment: 12 pages, latex, no figure
Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
In these notes we review and further explore the Lie enveloping algebra of a
post-Lie algebra. From a Hopf algebra point of view, one of the central
results, which will be recalled in detail, is the existence of a second Hopf
algebra structure. By comparing group-like elements in suitable completions of
these two Hopf algebras, we derive a particular map which we dub post-Lie
Magnus expansion. These results are then considered in the case of
Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined
in terms of solutions of modified classical Yang-Baxter equation. In this
context, we prove a factorization theorem for group-like elements. An explicit
exponential solution of the corresponding Lie bracket flow is presented, which
is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl
Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
We give a concise summary of the impressive recent development unifying a
number of different fundamental subjects. The quiver Nekrasov functions
(generalized hypergeometric series) form a full basis for all conformal blocks
of the Virasoro algebra and are sufficient to provide the same for some
(special) conformal blocks of W-algebras. They can be described in terms of
Seiberg-Witten theory, with the SW differential given by the 1-point resolvent
in the DV phase of the quiver (discrete or conformal) matrix model
(\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p}
\rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS
parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for
conformal blocks in terms of analytically continued contour integrals and
resolves the old puzzle of the free-field description of generic conformal
blocks through the Dotsenko-Fateev integrals. Most important, this completes
the GKMMM description of SW theory in terms of integrability theory with the
help of exact BS integrals, and provides an extended manifestation of the basic
principle which states that the effective actions are the tau-functions of
integrable hierarchies.Comment: 14 page