Quantum Folding

Abstract

In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g σ along a Dynkin diagram automorphism σ of g. For each quantum folding we replace g σ by its Langlands dual g σ ∨ and construct a nilpotent Lie algebra n which interpolates between the nilpotnent parts of g and g σ∨ , together with its quantized enveloping algebra U q (n) and a Poisson structure on S(n). Remarkably, for the pair (g, g σ ∨ ) = (so 2n+2 , sp 2n ), the algebra U q (n) admits an action of the Artin braid group Br n and contains a new algebra of quantum n×n matrices with an adjoint action of U q (sl n ), which generalizes the algebras constructed by K. Goodearl and M. Yakimov in [10]. The hardest case of quantum folding is, quite expectably, the pair (so 8 , G 2 ) for which the PBW presentation of U q (n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.

Positive representations of non-simply-laced split real quantum groups

“…It is known that the philosophy of folding in the setting of quantum groups is more complicated [1]. Nevertheless, with the mixture of classical construction, Mellin transformation and quantization described in this paper, we can still obtain the description of the positive representations in the non-simply-laced case from the corresponding unfolded simply-laced type by means of the folding of pinning [13, 1.5].…”

Weak Splittings of Quotients of Drinfeld and Heisenberg Doubles

“…We expect that a quantum version of our weak splittings of surjective Poisson submersions will be helpful in understanding the spectra of the quantizations of these families of varieties on the basis of the works on spectra of quantum groups [13,17]. It appears that such quantum weak splittings should be also closely related to the notion of quantum folding of Berenstein and Greenstein [3].…”

“…By the general facts in Sect. 3 with fibers F w g := R g p − Adẇ(u − + h ⊥ ) + L g (b + ) for g ∈ (U + ∩ wU − w −1 )ẇH. Here h ⊥ denotes the orthogonal complement to h := Lie H in t with respect to ., .…”

Koszul Duality for Semidirect Products and Generalized Takiff Algebras

“…All simple modules with this property where classified in [19], and it should be noted that the adjoint representation of g is not among them. The first non-simple example of V for which S q (V q ) is flat was constructed in [3] where the corresponding semidirect product is one of the key ingredients of the construction. The relationship between generalized Takiff algebras and semidirect products U q (g) ⋉ S q (V q ) was studied in [20].…”