Algebra Temperli - Liba

The Temperley – Lieb algebra is an algebra with which some are built . Opened by Neville Temperley and Elliot Leeb. The algebra is applied in statistical mechanics , in the theory of , it is related to knot theory and braid groups , quantum groups and subfactors of von Neumann algebras .

Definition

Let be $R$ ${\ displaystyle R}$ - commutative ring (most often - the field of real numbers ), in which the element is fixed $\delta \in R$ ${\ displaystyle \ delta \ in R}$ . Algebra Temperli - Liba $TL_{n}(\delta )$ ${\ displaystyle TL_ {n} (\ delta)}$ called $R$ ${\ displaystyle R}$ - algebra formed by generators $U_{1},U_{2},\ldots ,U_{n-1}$ ${\ displaystyle U_ {1}, U_ {2}, \ ldots, U_ {n-1}}$ Subject to the Jones ratios :

$U_{i}^{2}=\delta U_{i}$ ${\ displaystyle U_ {i} ^ {2} = \ delta U_ {i}}$ at $1\leq i\leq n-1$ ${\ displaystyle 1 \ leq i \ leq n-1}$
$U_{i}U_{i+1}U_{i}=U_{i}$ ${\ displaystyle U_ {i} U_ {i + 1} U_ {i} = U_ {i}}$ at $1\leq i\leq n-2$ ${\ displaystyle 1 \ leq i \ leq n-2}$
$U_{i}U_{i-1}U_{i}=U_{i}$ ${\ displaystyle U_ {i} U_ {i-1} U_ {i} = U_ {i}}$ at $2\leq i\leq n-1$ ${\ displaystyle 2 \ leq i \ leq n-1}$
$U_{i}U_{j}=U_{j}U_{i}$ ${\ displaystyle U_ {i} U_ {j} = U_ {j} U_ {i}}$ at $1\leq i,j\leq n-1$ ${\ displaystyle 1 \ leq i, j \ leq n-1}$ such that $|i-j|\neq 1$ ${\ displaystyle | ij | \ neq 1}$

$TL_{n}(\delta )$ ${\ displaystyle TL_ {n} (\ delta)}$ can be represented 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 joined by a curve, and no two curves intersect. Five basic vectors $TL_{3}(\delta )$ ${\ displaystyle TL_ {3} (\ delta)}$ look like this:

.

The multiplication of two basic elements occurs by the junction of two squares junction-in-junction, after each formed cycle gives the factor δ . For example,

× = = δ .

The unit element is a diagram with n horizontal lines, and a generator $U_{i}$ ${\ displaystyle U_ {i}}$ - a diagram in which the i -th vertex is connected to the i + 1 -th, 2n - i + 1- th point - with 2n - i -th point, and all other points are connected to opposite to themselves. For example, generators $TL_{5}(\delta )$ ${\ displaystyle TL_ {5} (\ delta)}$ are:

From left to right: the identical element (unit) and the generators U ₁ , U ₂ , U ₃ , U ₄ .

Jones ratios can be represented graphically:

= δ

=

Links

Louis H. Kauffman , State Models and the Jones Polynomial . Topology, 26 (3): 395-407, 1987.
RJ Baxter , Exactly solved models in mechanical mechanics Academic Press Inc., 1982.
Other Templars, E.C. Proceedings of the Royal Society Series A 322 (1971), 251-280.

Algebra Temperli - Liba

Definition

Links

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