Abstract
We give a description of the center of the affine nilTemperley–Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley–Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley–Lieb algebra on N generators into the affine nilTemperley–Lieb algebra on $$N+1$$ generators.
Highlights
The main goal of this work is to describe the center of the affine nilTemperley–Lieb algebra nTLN over any ground field
We provide an alternative proof of that fact by constructing a basis for nTLN that is especially adapted to the problem
The affine nilTemperley–Lieb algebra appears in many different settings, which we describe
Summary
The main goal of this work is to describe the center of the affine nilTemperley–Lieb algebra nTLN over any ground field. The affine Temperley–Lieb algebra TLN (δ) has the usual commuting relations and the relations ai ai±1ai = ai and ai2 = δai for some parameter δ ∈ k instead of the nil relations (where again all indices are mod N ) It is a filtered algebra with its th filtration space generated by all monomials of length ≤. Our basis is reminiscent of the Jones normal form for reduced expressions of monomials in the Temperley–Lieb algebra, as discussed in [20], and is characterised in Theorem 8.6 as follows: (See Theorem 7.5 which gives a different description.). For simplicity we have chosen to assume k is a field throughout the article
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