2008/06/24 by G. I. Lehrer R. B. Zhang, G.I. Lehrer R.B. Zhang, Zhang, G. I. Lehrer R. B.
Mathematics · Physics and Astronomy · #20G42 #81R50 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT #msc:20G42 #msc:81R50
paper · pdf · doi:10.48550/arxiv.0806.3807
14 pages
arxiv created 2008/06/24 · openalex publication_date 2008/06/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a presentation of the endomorphism algebra \End\cUq(\fsl2)(V⊗ r), where V is the 3-dimensional irreducible module for quantum \fsl2 over the function field \C(q1/2). This will be as a quotient of the Birman-Wenzl-Murakami algebra BMWr(q):=BMWr(q-4,q2-q-2) by an ideal generated by a single idempotent Φq. Our presentation is in analogy with the case where V is replaced by the 2- dimensional irreducible \cUq(\fsl2)-module, the BMW algebra is replaced by the Hecke algebra Hr(q) of type Ar-1, Φq is replaced by the quantum alternator in H3(q), and the endomorphism algebra is the classical realisation of the Temperley-Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on V⊗ r are consequences of relations among the three R-matrices acting on V⊗ 4. The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.