The q-Onsager algebra and the positive part of Uq(slˆ2)

The positive part Uq+ of Uq(slˆ2) has a presentation by two generators X,Y that satisfy the q-Serre relations. The q-Onsager algebra Oq has a presentation by two generators A,B that satisfy the q-Dolan/Grady relations. We give two results that describe how Uq+ and Oq are related. First, we consider the filtration of Oq whose nth component is spanned by the products of at most n generators. We show that the associated graded algebra is isomorphic to Uq+. Second, we introduce an algebra □q and show how it is related to both Uq+ and Oq. The algebra □q is defined by generators and relations. The generators are {xi}i∈Z4 where Z4 is the cyclic group of order 4. For i∈Z4 the generators xi,xi+1 satisfy a q-Weyl relation, and xi,xi+2 satisfy the q-Serre relations. We show that □q is related to Uq+ in the following way. Let □qeven (resp. □qodd) denote the subalgebra of □q generated by x0,x2 (resp. x1,x3). We show that (i) there exists an algebra isomorphism Uq+→□qeven that sends X↦x0 and Y↦x2; (ii) there exists an algebra isomorphism Uq+→□qodd that sends X↦x1 and Y↦x3; (iii) the multiplication map □qeven⊗□qodd→□q, u⊗v↦uv is an isomorphism of vector spaces. We show that □q is related to Oq in the following way. For nonzero scalars a,b there exists an injective algebra homomorphism Oq→□q that sends A↦ax0+a−1x1 and B↦bx2+b−1x3.