2015/06/29 by Paul Terwilliger, Terwilliger, Paul
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Primary: 17B37 #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1506.08666
openalex publication_date 2015/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The positive part U+q of Uq( widehat mathfraksl2) has a\npresentation by two generators X,Y that satisfy the q-Serre relations. The\nq-Onsager algebra mathcal Oq has a presentation by two generators A,B\nthat satisfy the q-Dolan/Grady relations. We give two results that describe\nhow U+q and mathcal Oq are related. First, we consider the filtration\nof mathcal Oq whose nth component is spanned by the products of at most\nn generators. We show that the associated graded algebra is isomorphic to\nU+q. Second, we introduce an algebra squareq and show how it is related\nto both U+q and mathcal Oq. The algebra squareq is defined by\ngenerators and relations. The generators are lbrace xi rbracei \∈\n mathbb Z4 where mathbb Z4 is the cyclic group of order 4. For i \∈\n mathbb Z4 the generators xi, xi+1 satisfy a q-Weyl relation, and\nxi,xi+2 satisfy the q-Serre relations. We show that squareq is\nrelated to U+q in the following way. Let square rm evenq (resp. \n square rm oddq) denote the subalgebra of squareq generated by x0,\nx2 (resp. x1, x3). We show that (i) there exists an algebra isomorphism\nU+q \→ square rm evenq that sends X\↦ x0 and Y\↦ x2;\n(ii) there exists an algebra isomorphism U+q \→ square rm oddq that\nsends X\↦ x1 and Y\↦ x3; (iii) the multiplication map\n square rm evenq \⊗ square rm oddq \→ squareq, u \⊗ v\n\↦ uv is an isomorphism of vector spaces. We show that squareq is\nrelated to mathcal Oq in the following way. For nonzero scalars a,b there\nexists an injective algebra homomorphism mathcal Oq \→ squareq that\nsends A \↦ a x0+ a-1 x1 and B \↦ b x2+ b-1 x3.\n