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

Abstract

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.

Distinguished Pre-Nichols Algebras

“…as the braided vector space with associated bicharacter H1], so (9) holds, and [HS1,Theorems 6.2,6.9] completes the proof.…”

“…First we show that each element of Z(V ) commutes up to scalars with each homogeneous element of U (V ), so U (V ) is a free Z(V )-module. Next we obtain a formula relating the coproduct on B(V ) with Lusztig isomorphisms, close to [HS1,Theorem 4.2]. We present a recursive formula for the coproduct of powers of root vectors in B(V ), with a view towards the computation of the liftings of B(V ), as proposed in [AAGMV].…”

Virasoro Central Charges for Nichols Algebras

“…But passing from a Nichols algebras to screenings involves various ambiguities. Nevertheless, the central charges associated with Nichols algebras in what follows have the nice property of being invariant under the Weyl groupoid action-the natural "symmetry" up to which Nichols algebras are classified [10,19,20].…”

“…On the Nichols algebra side, the Weyl groupoid action is defined as follows [10,19,20,23]. There exists a generalized Cartan matrix (a i, j ) 1≤i, j≤θ such that a i,i = 2 and…”

“…Withx j = x j + y j , this system becomes and yet another use of (19) shows that this is where the last factor vanishes by virtue of (20). The central charge is invariant.…”