Yang-baxter Equation

PARAMETER-DEPENDENT YANG-BAXTER EQUATION

Let

A

be a Unital Associative Algebra . The parameter-dependent Yang-Baxter equation is an equation for

R(u)

, a parameter-dependent Invertible element of the tensor product

A \otimes A

(here,

u

is the parameter, which usually ranges over all real numbers in the case of an additive parameter, or over all positive real numbers in the case of a multiplicative parameter). The Yang-Baxter equation is

:

R_{12}(u) \ R_{13}(u+v) \ R_{23}(v) = R_{23}(v) \ R_{13}(u+v) \ R_{12}(u),

for all values of

u

and

v

, in the case of an additive parameter, and

:

R_{12}(u) \ R_{13}(uv) \ R_{23}(v) = R_{23}(v) \ R_{13}(uv) \ R_{12}(u),

for all values of

u

and

v

, in the case of a multiplicative parameter, where

R_{12}(w) = \phi_{12}(R(w))

R_{13}(w) = \phi_{13}(R(w))

, and

R_{23}(w) = \phi_{23}(R(w))

, for all values of the parameter

w

, and

\phi_{12} : A \otimes A 	o A \otimes A \otimes A

\phi_{13} : A \otimes A 	o A \otimes A \otimes A

, and

\phi_{23} : A \otimes A 	o A \otimes A \otimes A

, are algebra morphisms determined by

:

\phi_{12}(a \otimes b) = a \otimes b \otimes 1

,

:

\phi_{13}(a \otimes b) = a \otimes 1 \otimes b

,

:

\phi_{23}(a \otimes b) = 1 \otimes a \otimes b

.

PARAMETER-INDEPENDENT YANG-BAXTER EQUATION

Let

A

be a unital associative algebra. The parameter-independent Yang-Baxter equation is an equation for

R

, an invertible element of the tensor product

A \otimes A

. The Yang-Baxter equation is

:

R_{12} \ R_{13} \ R_{23} = R_{23} \ R_{13} \ R_{12},

where

R_{12} = \phi_{12}(R)

R_{13} = \phi_{13}(R)

, and

R_{23} = \phi_{23}(R)

.

Let

V

be a of the Braid Group ,

B_n

, can be constructed on

V^{\otimes n}

\sigma_i = 1^{\otimes i-1} \otimes \check{R} \otimes 1^{\otimes n-i-1}

for

i = 1,\dots,n-1

, where

\check{R} = T \circ R

V \otimes V

. This representation can be used to determine quasi-invariants of Braids , Knots and Links .

SEE ALSO

Lie Bialgebra

REFERENCES

H.-D. Doebner, J.-D. Hennig, eds, ''Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989'', Springer-Verlag Berlin, ISBN 3-540-53503-9.

Vyjayanthi Chari and Andrew Pressley, ''A Guide to Quantum Groups'', (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.

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Yang-baxter Equation