Quantum deformations of certain prehomogeneous vector spaces. III

Kamita, Atsushi

doi:10.32917/hmj/1206124769

Cited by 12 publications

(17 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…• If g = sl n and V q = V q 2ω 1 , then S σ (V q ) is isomorphic to the algebra of quantum symmetric matrices introduced by Nouri in [20,Theorem 4 [16].…”

Section: Remark 425mentioning

confidence: 97%

R-matrix Poisson algebras and their deformations

Zwicknagl

2009

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…• If g = sl n and V q = V q 2ω 1 , then S σ (V q ) is isomorphic to the algebra of quantum symmetric matrices introduced by Nouri in [20,Theorem 4 [16].…”

Section: Remark 425mentioning

confidence: 97%

R-matrix Poisson algebras and their deformations

Zwicknagl

2009

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…, where S q (S 2 V ) and S q (Λ 2 V ) are respectively the algebras of quantum symmetric and quantum exterior matrices studied in [9,19,14,22]. Due to this and the canonical identification S(V ⊗V ) = S(Λ 2 V ⊕S 2 V ) = S(Λ 2 V )⊗S(S 2 V ), we can view S q (V ⊗V ) as a deformation of the braided (in the category of U q (sl n )-modules) tensor product S q (Λ 2 V ) ⊗ S q (S 2 V ) (see also Remark 1.16 for the Poisson version of this discussion).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Quantum Folding

Berenstein

Greenstein

2010

International Mathematics Research Notices

View full text Add to dashboard Cite

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.

show abstract

“…• If g = sl n and V = V 2ω 1 , then one obtains the algebra of quantum symmetric matrices introduced by Noumi in [24,Theorem 4.3 and Proposition 4.4] and by Kamita [20].…”

Section: Examples Of Quantized Symmetric Algebrasmentioning

confidence: 98%

Equivariant quantizations of symmetric algebras

Zwicknagl

2009

Journal of Algebra

View full text Add to dashboard Cite

Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra U h (g), resp. U q (g). We investigate, in particular, such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V . We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce a more general notion, coPoisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language.

show abstract

Quantum deformations of certain prehomogeneous vector spaces. III

Cited by 12 publications

References 13 publications

R-matrix Poisson algebras and their deformations

R-matrix Poisson algebras and their deformations

Quantum Folding

Equivariant quantizations of symmetric algebras

Contact Info

Product

Resources

About