A Combinatorial approach to the quantification of Lie algebras

Kharchenko, V. K.

doi:10.2140/pjm.2002.203.191

Cited by 38 publications

(29 citation statements)

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“…Lusztig's theorem [154] provides linear canonical bases of these algebras. Different approaches were developed by Ringel [191,192], Green [110], and Kharchenko [131][132][133][134][135]. GS bases of quantum enveloping algebras are unknown except for the case A n , see [55,86,195,220] [151]).…”

Section: ((U)(v)) > (V)mentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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In this survey we give an exposition of the theory of Gröbner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, -algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.Keywords Gröbner basis · Gröbner-Shirshov basis · Composition-Diamond lemma · Congruence · Normal form · Braid group · Free semigroup · Chinese monoid · Plactic monoid · Associative algebra · Lie algebra · Lyndon-Shirshov basis · Lyndon-Shirshov word · PBW theorem · -algebra · Dialgebra · Semiring · Pre-Lie algebra · Rota-Baxter algebra · Category · ModuleSupported by the NNSF of China (11171118) The set of all associative Lyndon-Shirshov words in X NLSW(X)The set of all non-associative Lyndon-Shirshov words in X PBW theoremThe Poincare-Birkhoff-Witt theorem X * The free monoid generated byThe free commutative monoid generated by X X * *The set of all non-associative words (u) in X gp X |S The group generated by X with defining relations S sgp X |S The semigroup generated by X with defining relations S k A field K A commutative algebra over k with unity k X The free associative algebra over k generated by X k X |S The associative algebra over k with generators X and defining relations S S c A Gröbner-Shirshov completion of S I d(S)The ideal generated by a set S sThe maximal word of a polynomial s with respect to some ordering < Irr(S)The set of all monomials avoiding the subwords for all s ∈ S k [X ] The polynomial algebra over k generated by X Lie(X )The free Lie algebra over k generated by X Lie K (X )The free Lie algebra generated by X over a commutative algebra K

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Section: ((U)(v)) > (V)mentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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“…This modification is possible due to the following general relation in k X (see [9,Corollary 4.10]): 5) provided that p ij p j i = p 1−n jj .…”

Section: Relations In the Quantum Borel Algebramentioning

confidence: 99%

“…If q is not a root of 1, then the fourth statement of [9,Theorem B n ,p. 211] shows that each skew-primitive element in U + q (so 2n+1 ) is proportional to either…”

Section: Relations In the Quantum Borel Algebramentioning

confidence: 99%

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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“…Kang, and D. Melville [2], M. Costantini and M. Varagnolo [4], J. Towber [42]), the bosonizations of quantum symmetric (bitensor, external, Nichols) algebras related to diagonal braidings, and so on. To some extent this class of Hopf algebras can be treated as the abstractly defined class of all quantum universal enveloping algebras (see [13,Theorem 6.1], [14]). We prove the following general statement on the structure of the right coideal subalgebras.…”

Section: Introductionmentioning

confidence: 99%

PBW-bases of coideal subalgebras and a freeness theorem

Kharchenko

2008

Trans. Amer. Math. Soc.

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Abstract. Let H be a character Hopf algebra. Every right coideal subalgebra U that contains the coradical has a PBW-basis which can be extended up to a PBW-basis of H. If additionally U is a bosonization of an invariant with respect to the left adjoint action subalgebra, then H is a free left (and right) U-module with a free PBW-basis over U. These results remain valid if H is a braided Hopf algebra generated by a categorically ordered subset of primitive elements. If the ground field is algebraically closed, the results are still true provided that H is a pointed Hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite-dimensional YetterDrinfeld submodules of skew primitive elements.

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A Combinatorial approach to the quantification of Lie algebras

Cited by 38 publications

References 39 publications

Gröbner–Shirshov bases and their calculation

Gröbner–Shirshov bases and their calculation

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

PBW-bases of coideal subalgebras and a freeness theorem

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