Chiral Extensions of the WZNW Phase Space, Poisson-Lie Symmetries and Groupoids

Abstract

The chiral WZNW symplectic form ${\Omega }_{chir}^{\rho }$ is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in ${\Omega }_{chir}^{\rho }$ and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.

40 pages, LaTeX

References in corpus (2)

• Lectures on the dynamical Yang-Baxter equations
• The Chiral WZNW Phase Space and its Poisson-Lie Groupoid

Cited by in corpus (25)

• Deformed integrable $\sigma$-models, classical $R$-matrices and classical exchange algebra on Drinfel'd doubles
• On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras
• Canonical quantization of the boundary Wess-Zumino-Witten model
• Quantum matrix algebra for the SU(n) WZNW model
• Poisson Structure and Moyal Quantisation of the Liouville Theory
• On moment maps associated to a twisted Heisenberg double
• Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras
• A note on a canonical dynamical r-matrix
• A Finite Dimensional Gauge Problem in the WZNW Model
• Chiral zero modes of the SU(n) Wess-Zumino-Novikov-Witten model
• The chiral WZNW phase space as a quasi-Poisson space
• On the Chiral WZNW Phase Space, Exchange r-Matrices and Poisson-Lie Groupoids
• Noncommutative Open strings, D-brane and Duality
• Dynamical r-matrices and the chiral WZNW phase space
• Canonical approach to the WZNW model
• Causal Poisson Brackets of the SL(2,R) WZNW Model and its Coset Theories
• Quantum su(n)_k monodromy matrices
• Dynamical r-matrices and Poisson-Lie symmetries in the chiral WZNW model
• Nested T-duality
• $q\to \infty$ limit of the quasitriangular WZW model
• Classification of Non-Affine Non-Hecke Dynamical R-Matrices
• Zero modes of the SU(2)_k Wess-Zumino-Novikov-Witten model in Euler angles parametrization
• Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists
• u-Deformed WZW Model and Its Gauging
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes