On endomorphisms of quantum tensor space
G.I. Lehrer R.B. Zhang

TL;DR
This paper presents a detailed algebraic description of endomorphisms of tensor powers of a 3-dimensional quantum sl2 module, connecting it to BMW algebras and cellular algebra theory, with implications for tensor decomposition at roots of unity.
Contribution
It provides a new presentation of the endomorphism algebra as a quotient of BMW algebra, extending known results from the 2-dimensional case to the 3-dimensional module.
Findings
Endomorphism algebra is a quotient of BMW algebra by a single idempotent.
Relations among R-matrices are generated by those acting on four tensor factors.
Cellular algebra theory is used to establish the algebraic structure.
Abstract
We give a presentation of the endomorphism algebra , where is the 3-dimensional irreducible module for quantum over the function field . This will be as a quotient of the Birman-Wenzl-Murakami algebra by an ideal generated by a single idempotent . Our presentation is in analogy with the case where is replaced by the 2- dimensional irreducible -module, the BMW algebra is replaced by the Hecke algebra of type , is replaced by the quantum alternator in , 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 -matrices on are consequences of relations among the three -matrices…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Finite Group Theory Research
