Right coideal subalgebras of the quantum Borel algebra of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

Pogorelsky, Barbara

doi:10.1016/j.jalgebra.2009.06.019

Cited by 12 publications

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References 16 publications

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“…In the second and third sections, following [3] and [5], we introduce main concepts and general results about hard super-letters and PBW-generators. In sections 4, 5 and 6 we present the considered quantum algebras U + q (g) and explicitly calculate their hard super-letters.…”

Section: Introductionmentioning

confidence: 99%

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A simple method for obtaining PBW-basis for some small quantum algebras

Dylewski¹,

Pogorelsky²,

Renz³

2018

ija

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Section: Introductionmentioning

confidence: 99%

“…5.2) from the right by x 2 , while relation(5.3) from the left by x 1 . The leading term of the difference equals:(−p 12 (1 + p 22 + p 222 ) + p 12 (1 + p 11 ))BC Therefore BC is also a linear combination of smaller words and the superletter is not hard by Proposition 2.9.…”

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A simple method for obtaining PBW-basis for some small quantum algebras

Dylewski¹,

Pogorelsky²,

Renz³

2018

ija

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Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

Heckenberger

Schneider

2013

Isr. J. Math.

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We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko.1. Right coideal subalgebras of quantized enveloping algebras U ≥0 . Let g be a semisimple complex Lie algebra, Π a basis of its root system with respect to a fixed Cartan subalgebra, and U = U q (g) the quantized enveloping algebra of g in the sense of [Jan96, Ch. 4]. We assume that q is not a root of unity. Let U + and U 0 be the subalgebras of U generated by the sets {E α | α ∈ Π} and {K α , K −1 α | α ∈ Π}, respectively, and let U ≥0 = U + U 0 . For any element w of the Weyl group W of g let U + [w] ⊂ U + be the subspace defined in [Jan96, 8.24] in terms of root vectors constructed via Lusztig's automorphisms. We prove in Thm. 7.3, see also Cor. 6.13, the following:The map w → U + [w]U 0 defines an order preserving bijection between W and the set of all right coideal subalgebras of U ≥0 containing U 0 , where right coideal subalgebras are ordered by inclusion and W is ordered by the Duflo order. If 2000 Mathematics Subject Classification. 17B37,16W30;20F55.

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Triangular decomposition of right coideal subalgebras

Kharchenko

2010

Journal of Algebra

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Let $\mathfrak g$ be a Kac-Moody algebra. We show that every homogeneous right coideal subalgebra $U$ of the multiparameter version of the quantized universal enveloping algebra $U_q(\mathfrak{g}),$ $q^m\neq 1$ containing all group-like elements has a triangular decomposition $U=U^-\otimes_{{\bf k}[F]} {\bf k}[H] \otimes_{{\bf k}[G]} U^+$, where $U^-$ and $ U^+$ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if $ U_1$ and $ U_2$ are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular composition $ U_2\otimes_{{\bf k}[F]} {\bf k}[H]\otimes_{{\bf k}[G]} U_1$ is a right coideal but not necessary a subalgebra. Using a recent combinatorial classification of right coideal subalgebras of the quantum Borel algebra $U_q^+(\mathfrak{so}_{2n+1}),$ we find a necessary condition for the triangular composition to be a right coideal subalgebra of $U_q(\mathfrak{so}_{2n+1}).$ If $q$ has a finite multiplicative order $t>4,$ similar results remain valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum groups $u_q({\frak g}),$ $u_q(\frak{so}_{2n+1}).

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Right coideal subalgebras of the quantum Borel algebra of type G2

Cited by 12 publications

References 16 publications

A simple method for obtaining PBW-basis for some small quantum algebras

A simple method for obtaining PBW-basis for some small quantum algebras

Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

Triangular decomposition of right coideal subalgebras

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