Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5773089 | Linear Algebra and its Applications | 2017 | 38 Pages |
Abstract
The positive part Uq+ of Uq(slË2) 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 â¡q and show how it is related to both Uq+ and Oq. The algebra â¡q is defined by generators and relations. The generators are {xi}iâZ4 where Z4 is the cyclic group of order 4. For iâZ4 the generators xi,xi+1 satisfy a q-Weyl relation, and xi,xi+2 satisfy the q-Serre relations. We show that â¡q is related to Uq+ in the following way. Let â¡qeven (resp. â¡qodd) denote the subalgebra of â¡q generated by x0,x2 (resp. x1,x3). We show that (i) there exists an algebra isomorphism Uq+ââ¡qeven that sends Xâ¦x0 and Yâ¦x2; (ii) there exists an algebra isomorphism Uq+ââ¡qodd that sends Xâ¦x1 and Yâ¦x3; (iii) the multiplication map â¡qevenââ¡qoddââ¡q, uâvâ¦uv is an isomorphism of vector spaces. We show that â¡q is related to Oq in the following way. For nonzero scalars a,b there exists an injective algebra homomorphism Oqââ¡q that sends Aâ¦ax0+aâ1x1 and Bâ¦bx2+bâ1x3.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Paul Terwilliger,