SU3 coherent state operators and invariant correlation functions and their quantum group counterparts

Abstract

Coherent state operators (CSO) are defined as operator valued functions on G = SL(n, C) being homogeneous with respect to right multiplication by lower triangular matrices.They act on a model space containing all holomorphic finite dimensional representations of G with multiplicity 1. CSO provide an analytic tool for studying G invaraiant 2and 3-point functions, which are written down in the case of SU 3 . The quantum group deformation of the construction gives rise to a non-commutative coset space. We introduce a "standard" polynomial basis in this space (related to but not identical with the Lusztig canonical basis) which is appropriate for writing down U q (sℓ 3 ) invariant 2-point functions for representations of the type (λ, 0) and (0, λ). General invariant 2-point functions are written down in a mixed Poncaré-Birkhoff-Witt type basis.

Coherent States for Hopf Algebras

“…The generators of G coB λ are the analogues of the local coordinates on a big open cell in the coset space, and the coherent vector C λ may be viewed as a parametrization of an open set in projective orbit by points in a coset space. In similar spirit, in the case of O(SL q (3)), the reference [33] views G coB λ as an analogue of the (algebra of functions on) unipotent group parametrizes quantum orbit (though they note this algebra is not a bialgebra, unlike the classical case). Here we clarify that, as in the classical case, this should be understood as a parametrization of an open dense subset of orbit, the latter being a noncommutative space.…”

“…Computations of coherent states in selected local coordinates in concrete examples O(SU q (n)) with n = 2, 3, appeared in [13,15,33], though without full geometric justification, and sometimes with nongeometric factors. Rudiments of another picture involving quantum group coherent states, related to geometric quantization and orbit method, are discussed in [42].…”

“…Earlier studies of coherent states for quantum groups ( [17,33]) used computations in a single local chart. One of our goals was to show that states locally computed cohzoki4.tex; 24/12/2017; 12:04; p.2 in ( [33]) may be defined a priori, regardless coordinate choices. The main goal was to find a resolution of unity in terms of coherent states of compact quantum groups.…”

Representations of 𝒰h (𝔰𝔲(N) ) derived from quantum flag manifolds

“…At the same time, we introduce some shorthand notation, also only for the purpose of this proof. Set 18) and by Lemma 4.2,…”

“…The particular case of SU(3) was treated in an analogous way in Ref. [18]. Our goal is to derive a similar description but taking a different approach and presenting some additional results, too.…”