Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Mudrov, Andrey

doi:10.1142/s0129055x1550004x

Cited by 13 publications

(19 citation statements)

References 17 publications

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“…They are the incidence relations. Conditions (10), (20) and (24) define completely the flag supervariety F. We then can state the analog of Theorem 3.2 for the superflag.…”

Section: The Plücker Embedding Of the Super Flag Fl(2|0 2|1; 4|1)mentioning

confidence: 95%

The Segre embedding of the quantum conformal superspace

Fioresi

Latini

Lledó

et al. 2018

Advances in Theoretical and Mathematical Physics

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In this paper study the quantum deformation of the superflag Fl(2|0, 2|1, 4|1), and its big cell, describing the complex conformal and Minkowski superspaces respectively. In particular, we realize their projective embedding via a generalization to the super world of the Segre map and we use it to construct a quantum deformation of the super line bundle realizing this embedding. This strategy allows us to obtain a description of the quantum coordinate superring of the superflag that is then naturally equipped with a coaction of the quantum complex conformal supergroup SL q (4|1).

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“…They are the incidence relations. Conditions (10), (20) and (24) define completely the flag supervariety F. We then can state the analog of Theorem 3.2 for the superflag.…”

Section: The Plücker Embedding Of the Super Flag Fl(2|0 2|1; 4|1)mentioning

confidence: 95%

The Segre embedding of the quantum conformal superspace

Fioresi

Latini

Lledó

et al. 2018

Advances in Theoretical and Mathematical Physics

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“…This result can easily be generalized to the general first order case. It is related to the q-Shapovalov form [5].…”

Section: First Order Intertwinersmentioning

confidence: 99%

Special classes of homomorphisms between generalized Verma modules for ${{\mathscr{U}}}_{q}(su(n,n))$

Jakobsen

2019

J. Phys.: Conf. Ser.

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We study homomorphisms between quantized generalized Verma modules M (VΛ) φ Λ,Λ 1 → M (VΛ 1 ) for Uq(su(n, n)). There is a natural notion of degree for such maps, and if the map is of degree k, we write φ k Λ,Λ 1 . We examine when one can have a series of such homomorphisms φ 1then Λ = (ΛL, ΛR, λ) and Λn = (ΛL, ΛR, λ + 2). The answer is then that Λ must be one-sided in the sense that either ΛL = 0 or ΛR = 0 (non-exclusively). There are further demands on λ if we insist on Uq(g C ) homomorphisms. However, it is also interesting to loosen this to considering only U − q (g C ) homomorphisms, in which case the conditions on λ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of Uq(g C ) homomorphisms φ 1 Λ,Λ 1 .Dedicated to I.E. Segal (1918I.E. Segal ( -1998 in commemoration of the centenary of his birth.

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“…Mudrov [22] describes the Shapovalov basis for the Verma modules of U q (su(3)), and we have adapted his proof and construction to an orthonormal basis for the finite-dimensional unitary representations of U q (su (3)). For completeness, we have sketched the proof in Appendix A.…”

Section: The Finite-dimensional Representations Of U Q (Su(3))mentioning

confidence: 99%

“…However see Oblomkov and Stokman [26] for partial information on the branching rules for the quantum analogue of (gl(2n), gl(n) ⊕ gl(n)). It would be of interest to see whether the results of Mudrov [22] can be used as well in the setting of Oblomkov and Stokman [26] to find precise information on the branching rule for this quantum symmetric pair, or more generally for quantum symmetric pairs involving the quantised universal enveloping algebra of type A.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we review the construction of the quantised universal enveloping algebra U q (su (3)) and its finite dimensional irreducible representations. Then we collect a series of commutation identities for the generators of U q (su(3)) and we introduce an orthogonal basis for finite-dimensional U q (su(3))-representations which is an analogue of Mudrov [22]. We also describe the action of the generators of U q (su(3)) on this basis.…”

Section: Introductionmentioning

confidence: 99%

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Branching Rules for Finite-Dimensional U q ( 𝖘 𝖚 ( 3 ) ) $\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a Right Coideal Subalgebra

Aldenhoven

Koelink

Román

2017

Algebr Represent Theor

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We consider the quantum symmetric pair (U q (su(3)), B) where B is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of B are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of U q (su(3)) to B decomposes multiplicity free into irreducible representations of B. Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.

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Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Cited by 13 publications

References 17 publications

The Segre embedding of the quantum conformal superspace

The Segre embedding of the quantum conformal superspace

Special classes of homomorphisms between generalized Verma modules for ${{\mathscr{U}}}_{q}(su(n,n))$

Branching Rules for Finite-Dimensional U q ( 𝖘 𝖚 ( 3 ) ) $\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a Right Coideal Subalgebra

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