An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$

Abstract

Let F denote a field, and pick a nonzero q ∈ F that is not a root of unity. Let Z 4 = Z/4Z denote the cyclic group of order 4. Define a unital associative F-algebra q by generators {x i } i∈Z 4 and relationswhere [3] q = (q 3 − q −3 )/(q − q −1 ). Let V denote a q -module. A vector ξ ∈ V is called NIL whenever x 1 ξ = 0 and x 3 ξ = 0 and ξ = 0. The q -module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL q -module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the q-shuffle algebra for affine sl 2 .

Using a $q$-shuffle algebra to describe the basic module $V(Λ_0)$ for the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$

“…In this section we recall from [32] some maps A ℓ , B ℓ , A r , B r in End(V) that will be used in our main results.…”

“…Among the things to check, is that qA r K −1 − q −1 A ℓ and B r K − B ℓ satisfy the q-Serre relations. This can be checked easily using [32,Lemma 10.3,Corollary 10.4].…”

“…In this appendix we list some relations satisfied by the maps [32,Proposition 9.3].) The following relations hold on U:…”

“…The inclusion ⊇ in (32) holds by Lemma 18.3. We next obtain the inclusion ⊆ in (32). Let v ∈ ker(S n+1 ).…”

Using Catalan words and a q-shuffle algebra to describe a PBW basis for the positive part of Uq(slˆ2Terwilliger1 2019Journal of AlgebraSelf Cite9050View full textAdd to dashboardCite

“…Therefore ♮ : U + q → U is an algebra isomorphism. See [10,19,21,22] for more information about the q-shuffle algebra V and its relationship to U + q . Earlier we mentioned a grading for both U + q and the q-shuffle algebra V. These gradings are related as follows.…”