Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of Type<i>G</i><sub>2</sub>*

Pogorelsky, Barbara

doi:10.1080/00927871003623570

Cited by 7 publications

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“…It is based on Lemma 3.7 and the first equality of (3.7) in the same way as the proof of the above lemma is based on Lemma 3.8 and the second equality of (3.7). 17) where 3.13) and (3.15) with [v, u], which proves the second equality. To prove the first, we apply Lemma 3.9 if m < ψ(k), and otherwise we apply Lemma 3.10.…”

Section: Relations In the Quantum Borel Algebramentioning

confidence: 51%

“…We note that in [17], B. Pogorelsky found a similar lattice for the quantum groups U q (g), u q (g), where g is the simple Lie algebra of type G 2 .…”

Section: Examplesmentioning

confidence: 99%

“…We define the bracketing of u(k, m), k ≤ m, as follows: 16) where β = −p(u(n + 1, m), u(k, n)) −1 normalizes the coefficient of u(k, m). The conditional identity (2.9) shows that the value of u[k, m] in U + q (so 2n+1 ) is independent of the arrangement of brackets provided that m ≤ n or k > n.…”

Section: Relations In the Quantum Borel Algebramentioning

confidence: 99%

“…In [13], we proved that the number of different right coideal subalgebras that contain the coradical of U + q (sl n+1 ) equals (n + 1)!, the order of the Weyl group for the root system of type A n . Recently, B. Pogorelsky [16] proved that the quantum Borel algebra U + q (g) for the simple Lie algebra of type G 2 has 12 different right coideal subalgebras over the coradical. This number also coincides with the order of the Weyl group of type G 2 .…”

Section: Introductionmentioning

confidence: 99%

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Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Kharchenko¹

2011

J. Eur. Math. Soc.

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Abstract. We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group U + q (so 2n+1 ), provided that q is not a root of 1. If q has a finite multiplicative order t > 4, this classification remains valid for homogeneous right coideal subalgebras of the Frobenius-Lusztig kernel u + q (so 2n+1 ). In particular, the total number of right coideal subalgebras that contain the coradical equals (2n)!!, the order of the Weyl group defined by the root system of type B n .Keywords. Coideal subalgebra, Hopf algebra, PBW-basis IntroductionIn the present paper, we continue the classification of right coideal subalgebras in quantised enveloping algebras begun in [13]. We offer a complete classification of right coideal subalgebras that contain all grouplike elements for the multiparameter version of the quantum group U + q (so 2n+1 ), provided that the main parameter q is not a root of 1. If q has a finite multiplicative order t > 4, this classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the Frobenius-Lusztig kernel u + q (so 2n+1 ). The main result of the paper is the establishment of a bijection between all sequences (θ 1 , . . . , θ n ) such that 0 ≤ θ k ≤ 2n − 2k + 1, 1 ≤ k ≤ n, and the set of all (homogeneous if q t = 1, t > 4) right coideal subalgebras of U + q (so 2n+1 ), q t = 1 (respectively of u + q (so 2n+1 )) that contain the coradical. (Recall that in a pointed Hopf algebra, the grouplike elements span the coradical.) In particular, there are (2n)!! different right coideal subalgebras that contain the coradical. Interestingly, this number coincides with the order of the Weyl group for the root system of type B n . In [13], we proved that the number of different right coideal subalgebras that contain the coradical of U + q (sl n+1 ) equals (n + 1)!, the order of the Weyl group for the root system of type A n . Recently, B. Pogorelsky [16] proved that the quantum Borel algebra U + q (g) for the simple Lie algebra of type G 2 has 12 different right coideal subalgebras over the coradical. This Conjecture. Let g be a simple Lie algebra defined by a finite root system R. The number of different right coideal subalgebras that contain the coradical in a quantum Borel algebra U + q (g) equals the order of the Weyl group defined by the root system R, provided that q is not a root of 1. 1In Section 2, following [13], we introduce the main concepts of the paper and we formulate known results that are useful for classification. In the third section, we prove auxiliary relations in a multiparameter version of U + q (so 2n+1 ). In the fourth section, we note that the Weyl basisin place of the Lie operation is a set of PBW-generators for U + q (so 2n+1 ) and u + q (so 2n+1 ). By means of the shuffle representation, in Theorem 4.3, we prove an explicit formula for the coproduct of these PBW-generators, which is the key result for further considerations:where τ i = 1 with only one exception, τ n = q, while g ki are su...

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Section: Relations In the Quantum Borel Algebramentioning

confidence: 51%

“…We note that in [17], B. Pogorelsky found a similar lattice for the quantum groups U q (g), u q (g), where g is the simple Lie algebra of type G 2 .…”

Section: Examplesmentioning

confidence: 99%

Section: Relations In the Quantum Borel Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Kharchenko¹

2011

J. Eur. Math. Soc.

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“…If g is the simple Lie algebra of type G 2 then the probability equals 60/144 = 41.7%, see B. Pogorelsky [9,10].…”

Section: Introductionmentioning

confidence: 99%

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 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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Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of TypeG₂*

Cited by 7 publications

References 7 publications

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Triangular decomposition of right coideal subalgebras

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

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Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of TypeG2*

Cited by 7 publications

References 7 publications

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Triangular decomposition of right coideal subalgebras

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

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Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of TypeG₂*