Homogeneous right coideal subalgebras of quantized enveloping algebras

Heckenberger, I.; Kolb, Stefan

doi:10.1112/blms/bds027

Cited by 19 publications

(10 citation statements)

References 12 publications

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“…By [HK11a] Prop. 2.1 every homogeneous right coideal subalgebra B is triangular and together with the classification of homogeneous right coideal subalgebras of U ± q in [HS09] Thm 7.3 they are of the form:…”

Section: Examplementioning

confidence: 97%

“…The main motivation for studying Borel subalgebras of quantum groups is the more general goal to understand the set of all coideal subalgebras, which is considered a main problem in the area of quantum groups, with considerable progress made in [Let99,KS08,HS09,HK11a,HK11b]. Main examples are the quantum symmetric pairs, which are constructed in close analogy to the Lie algebra case, starting with [NS95,Let97].…”

mentioning

confidence: 99%

“…We illustrate it by treating a first example in type A 2 . A technical complication is that in general gr(C) 0 is not a Hopf algebra (just a semigroup in the root lattice) so it need not be graded and [HK11a] does not apply. This is why we formulate our conjecture about the localization of gr(C) on gr(C) 0 , which captures the essential information.…”

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confidence: 99%

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A Family of New Borel Subalgebras of Quantum Groups

Lentner

Vocke

2020

Algebr Represent Theor

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For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra.Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We then give structural results using the graded algebra, which in particular leads to a conjectural formula for all triangular Borel subalgebras, which we partly prove. As examples, we determine all Borel subalgebras of U q (sl 2 ) and U q (sl 3 ) and discuss the induced modules.

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Section: Examplementioning

confidence: 97%

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confidence: 99%

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A Family of New Borel Subalgebras of Quantum Groups

Lentner

Vocke

2020

Algebr Represent Theor

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“…In [34], Vocke discusses right coideal subalgebras, equivalently subgroups, of U q (g), and in particular finds all the right coideal subalgebras of U q (sl 2 ). Her work builds upon work done in [6] on right coideal subalgebras of U q (g) containing U 0 q (g)…”

Section: Subgroups Of U Q (Sl 2 )mentioning

confidence: 99%

A Categorical Approach to Subgroups of Quantum Groups and Their Crystal Bases

Savage

2019

Preprint

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Suppose that we have a semisimple, connected, simply connected algebraic group G with corresponding Lie algebra g. There is a Hopf pairing between the universal enveloping algebra U (g) and the coordinate ring O(G). By introducing a parameter q, we can consider quantum deformations U q (g) and O q (G) respectively, between which there again exists a Hopf pairing. We show that the category of crystals associated with U q (g) is a monoidal category. We define subgroups of U q (g) to be right coideal subalgebras, and subgroups of O q (G) to be quotient left O q (G)-module coalgebras. Furthermore, we discuss a categorical approach to subgroups of quantum groups which we hope will provide us with a link to crystal basis theory.• We occasionally omit the structure maps for algebras, Hopf algebras etc. , and simply refer to them as A, H etc.• id X denotes the identity map on X.• We refer to (co)unital, (co)associative (co)algebras as '(co)algebras'.

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“…[BK1, BK2, FLLW1, FLLW2, FL2]), which appeared for the first time in the work of S. Kolb [Ko1]. See also [HK,ES,BKLW,FL1]. Important examples of coideal subalgebras of quantized enveloping algebras appeared even before the aforementioned work of G. Letzter: around 1990, G. Olshanskii introduced in [Ol] the twisted Yangians of type AI and AII (that is, those associated to the symmetric pairs (gl N , so N ) and (gl N , sp N )), which are coideal subalgebras of the Yangian of gl N , the later being one of the two families of quantized enveloping algebras of affine type A (1) .…”

Section: Introductionmentioning

confidence: 99%

Representations of Twisted Yangians of Types B, C, D: Ii

Guay

Regelskis

Wendlandt

2019

Transformation Groups

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We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of these types, the highest weight of such a module necessarily satisfies a certain set of relations, we classify the finite-dimensional irreducible representations of twisted Yangians for the pairs (soN , soN−2 ⊕ so2) and (so2n+1, so2n).

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Homogeneous right coideal subalgebras of quantized enveloping algebras

Cited by 19 publications

References 12 publications

A Family of New Borel Subalgebras of Quantum Groups

A Family of New Borel Subalgebras of Quantum Groups

A Categorical Approach to Subgroups of Quantum Groups and Their Crystal Bases

Representations of Twisted Yangians of Types B, C, D: Ii

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