Separation of Variables for Quantized Enveloping Algebras

Joseph, Anthony; Letzter, Gail

doi:10.2307/2374984

Cited by 73 publications

(153 citation statements)

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“…Hence the center Z(B) ∼ = Z(Ǔ q (g )) is well known by [JL94]. Here we obtain the same result, it should, however, be borne in mind that our calculations heavily depended on the previous knowledge of the structure of F r (Ǔ ).…”

Section: Determining P Z(b)supporting

confidence: 82%

The center of quantum symmetric pair coideal subalgebras

Kolb

Letzter

2008

Represent. Theory

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Section: Determining P Z(b)supporting

confidence: 82%

The center of quantum symmetric pair coideal subalgebras

Kolb

Letzter

2008

Represent. Theory

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“…We are in the framework of [Appendix, Proposition A4] with T=q − 1 and R=Z A . From the quantum and classical theorems of separation of variables [21,27], the hypotheses of Proposition A4 are verified with…”

Section: Proof Remark That F(u a ) Is The Direct Sum Of Its Isotypicmentioning

confidence: 88%

“…We use the Joseph-Letzter's decomposition theorem as a basic tool [21]. It asserts that, see Theorem 1.4,…”

Section: 4mentioning

confidence: 99%

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On Harmonic Elements for Semi-simple Lie Algebras

Caldero

2002

Advances in Mathematics

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Let g be a semi-simple complex Lie algebra and g=n − À h À n its triangular decomposition. Let U(g), resp. U q (g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math. 116, 127-177). On the one hand, we give specialization results concerning harmonic elements, central elements of U q (g), and the decomposition of Joseph and Letzter (cited above). For g=sl n+1 , we describe the specialization of quantum harmonic space in the N-filtered algebra U(sl n+1 ) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys. 35, 359-374). This enables us to study a Joseph-Letzter decomposition in the algebra U(sl n+1 ). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C 0 of integral points of a convex cone. © 2002 Elsevier Science (USA) Key Words: quantum groups; enveloping algebras; canonical basis; harmonic space.0. INTRODUCTION 0.1. Let V be a complex finite dimensional space and G a Lie subgroup of GL(V). Assume that V is completely reducible as a G-module. Then G acts semi-simply on the symmetric C-algebra S(V). Let J be the ideal generated by the non-constant homogeneous G-invariant elements of S(V). Let V g be the dual of V. There exists a natural non-degenerate duality between S(V) and S(V) be the orthogonal of J and H g be the space of G-harmonic polynomials of V.

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“…There is the Verma module M λ := U q ⊗ Uq(b) C λ for U q with highest weight λ ∈ T Λ , where C λ is the 1-dimensional representation of U q (b) defined by λ. For q generic we have the quantum Duflo formula (see [JL94] and [BK11])…”

Section: Integrability Of Modules a (Say Right) Umentioning

confidence: 99%