Algebra Temperli – Liba – algebra, with the help of which some transfer matrices are constructed. Discovered by Neville Temperley and Elliot Leeb. Algebra is used in statistical mechanics, in the theory of integrable models, it is related to knot theory and braid groups, quantum groups and subfactors of von Neumann algebras.

Definition

Let R displaystyle R be a commutative ring (most often the field of real numbers), in which an element _ _ R displaystyle delta in R is fixed. The Temperley – Lieb algebra TL n (_) displaystyle TL_ n (delta) is the R displaystyle R -algebra formed by the generators U 1, U 2,…, U n – 1 displaystyle U_ 1, U_ 2, ldots, U_ n-1, subordinate Jones relations:


U i 2 = _ U i displaystyle U_ i ^ 2 = delta U_ i for 1 _ i _ n – 1 displaystyle 1leq ileq n-1

U i U i + 1 U i = U i displaystyle U_ i U_ i + 1 U_ i = U_ i for 1 _ i _ n – 2 displaystyle 1leq ileq n-2

U i U i – 1 U i = U i displaystyle U_ i U_ i-1 U_ i = U_ i for 2 _ i _ n – 1 displaystyle 2leq ileq n-1

U i U j = U j U i displaystyle U_ i U_ j = U_ j U_ i for 1 _ i, j _ n – 1 displaystyle 1leq i, jleq n-1, such that | i – j | _ 1 displaystyle


TL n (_) displaystyle TL_ n (delta) can be thought of as a vector space, with basis vectors, each of which is a diagram in the form of a square, on two opposite sides of which there are n displaystyle n points. The points form n pairs, each pair is connected by a curve, and no two curves intersect. The five basis vectors TL 3 (_) displaystyle TL_ 3 (delta) look like this:

…

Multiplication of two basic elements occurs by joining two squares end-to-end, after each formed cycle gives a multiplier _… For instance,

_ = = _.

The unit element is a diagram with n horizontal lines, and the generator U i displaystyle U_ i is a diagram in which i-th vertex is connected to i + 1-Oh, 2n – i + 1-th point – with 2n – i-th point, and all other points are connected to the opposite ones. For example, the generators for TL 5 (_) displaystyle TL_ 5 (delta) are:

From left to right: identity element (one) and generators U1, U2, U3, U4.

Jones’ ratios can be depicted graphically:

= _

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