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

Abstract

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...

Triangular decomposition of right coideal subalgebras

“…If we apply the found condition to right coideal subalgebras of U q (sl n ), we get precisely the necessary and sufficient condition given in [7,Theorem 11.1]. Hence we have a reason to believe that the found necessary condition is also sufficient for the triangular composition to define a right coideal subalgebra of U q (so 2n+1 ).…”

“…In this paper we are studying when a right coideal subalgebra of U q (g) also has the triangular decomposition. In fact the triangular decomposition holds not only for U q (g), but also for a large class of character Hopf algebras A having positive and negative skew-primitive generators connected by relations of the type [7,Proposition 3.4]. In Theorem 3.2 we show that a right coideal subalgebra U of A containing all E-mail address: vlad@servidor.unam.mx.…”

“…In this section we follow [6] to recall the basic properties of quantum Borel subalgebras U ± q (so 2n+1 ). In what follows we fix a parameter q such that q 2 = ±1,…”

“…These elements are very important since every right coideal subalgebra U ⊇ G of the quantum Borel algebra is generated as an algebra by G and the elements of that form, see [6,Corollary 5.7]. Moreover U is uniquely defined by its root sequence θ = (θ 1 , θ 2 , .…”

“…In Sections 4 and 5 we follow the classification given in [6] to recall the basic properties of right coideal subalgebras of quantum Borel algebras U ± q (so 2n+1 ). In particular we lead out the following "integrability" condition: if all partial derivatives of a homogeneous polynomial f in positive generators of an admissible degree belong to a right coideal subalgebra U ⊇ G of U + q (so 2n+1 ) then f itself belongs to U , see Corollary 5.3.…”