Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups

Abstract

The coadjoint orbits for the series B , C , and D are considered in the case l l l when the base point is a multiple of a fundamental weight. A quantization of the big cell is suggested by means of introducing a )-algebra generated by holomorphic coordinate functions. Starting from this algebraic structure the irreducible representations of the deformed universal enveloping algebra are derived as acting in the vector space of polynomials in quantum coordinate functions.12 12 or equivalently, R y R y1 s q y q y1 P , Ž .12 21the R-matrices for the series B , C , D obey a more complicated relation l l l Ž . below.Unfortunately, this is why the construction of the algebra C C was not clarified entirely and has to some extent a speculative character. However, the prescription for the representation of U U acting in the vector space of h polynomials in quantum coordinate functions z U is derived quite unamjk biguously and verified rigorously. In principle, Section 4, concerned with the quantum parameterization of the big cell, can be skipped. But starting directly from the defining relations for the representation could seem then rather obscure.

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

“…However, as observed already in Ref. [24], it is not necessary to know the structure of the algebra F in full detail for successful derivation of the left module structure. A leftà d -module on F ahol will be defined by prescribing the action on the unit and then extending it to all polynomials in non-commutative variables ζ * st with the help of a recursive rule.…”

“…The module is defined by prescribing the action on the unit and then extending it to all polynomials in non-commutative variables (quantum antiholomorphic coordinate functions) using a recursive rule, an idea utilized already in Ref. [24]. Moreover, we prove that this recursive rule follows from the quantum dressing transformation, making the role of the dressing transformation quite explicit.…”

“…As already mentioned, the idea of defining a module this way was utilized in Ref. [24] (Proposition 5.4) and the proof is quite similar, too. Here we rely heavily on Proposition 3.3 giving a description of F ahol in terms of G (m)…”

“…(3) (3.22) can be rewritten in a way coinciding formally with the classical constraint, 24) and, in fact, it reduces the number of generators. Observe also that it holds trivially true, just by the structure of blocks (cf.…”