Quantum dynamical Yang-Baxter equation over a nonabelian base

We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an $r$-$n$ structure. We classify, up to a natural equivalence, all ...

The exotic quantum double and its universal R-matrix for quantum Yang-Baxter equation are constructed in terms of Drinfeld's quantum double theory.As a new quasi-triangular Hopf algebra, it is much different from those standard quantum doubles that are the q-deformations for Lie algebras or Lie superalgebras. By studying its representation theory,many-parameter representations of the exotic quantum double are obtained with an explicit example associated with Lie algebra $A_2$...

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson bracket...

We give relations between dynamical Poisson groupoids, classical dynamical Yang--Baxter equations and Lie quasi-bialgebras. We show that there is a correspondance between the class of bidynamical Lie quasi-bialgebras and the class of bidynamical Poisson groupoids. We give an explicit, analytical and canonical equivariant solution of the classical dynamical Yang--Baxter equation (classical dynamical $\ell$-matrices) when there exists a reductive decomposition $\g=\l\oplus\m$, ...

We show that for each semi-Riemannian locally symmetric space the curvature tensor gives rise to a rational solution $r$ of the classical Yang-Baxter equation with spectral parameter. For several Riemannian globally symmetric spaces $M$ such as real, complex and quaternionic Grassmann manifolds we explicitly compute solutions $R$ of the quantum Yang Baxter equations (represented in the tangent spaces of $M$) which generalize the quantum $R$-matrix found by Zamolodchikov and Z...

We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. Our approach is based on the study of a class of Poisson structures on trivial Lie groupoids within the category of biequivariant Poisson manifolds. In the forme...

The purpose of this paper is to establish a connection between various subjects such as dynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures developed in dg-ga/9508013 and dg-ga/9611001. In particular, we give a new method of classifying dynamical r-matrices of simple Lie algebras $\frak g$, and prove that dynamical r-matrices are in one-one correspondence with certain Lagrangian subalgebras of ${\frak g}\oplu...

A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a particular case when the group $SU(2)$ takes place.

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuti...

We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory.