Quantum group

In mathematical group theory, a quantum group is a specific type of Hopf algebras , namely quantizations (i.e. non-trivial deformations) of the enveloping Hopf algebras of semi-simple Lie algebras . Alternatively, one can consider quantum groups as deformations from the algebra of regular functions on algebraic groups .

The term was coined by the Ukrainian-US mathematician Vladimir Drinfeld at the International Congress of Mathematicians in 1986 in Berkeley . Independently of him, they were found around the same time by the Japanese mathematician Michio Jimbō .

example

The simplest quantum group is . This is the algebra of the variables , , and is produced and in which the relations ${\displaystyle U_{q}({\mathfrak {sl}}(2))}$ ${\displaystyle K}$ ${\displaystyle K^{-1}}$ ${\displaystyle E}$ ${\displaystyle F}$

${\displaystyle KK^{-1}=K^{-1}K=1}$,

${\displaystyle KEK^{-1}=q^{2}E}$,

${\displaystyle KFK^{-1}=q^{-2}F}$,

${\displaystyle [E,F]={\frac {K-K^{-1}}{q-q^{-1}}}}$

be valid.

The Hopf algebra structure is given by

${\displaystyle \Delta (E)=1\otimes E+E\otimes K}$,

${\displaystyle \Delta (F)=K^{-1}\otimes F+F\otimes 1}$,

${\displaystyle \Delta (K)=K\otimes K}$,

${\displaystyle \Delta (K^{-1})=K^{-1}\otimes K^{-1}}$,

${\displaystyle \epsilon (E)=\epsilon (F)=0}$,

${\displaystyle \epsilon (K)=\epsilon (K^{-1})=1}$,

${\displaystyle S(E)=-EK^{-1}}$,

${\displaystyle S(F)=-KF}$,

${\displaystyle S(K)=K^{-1}}$,

${\displaystyle S(K^{-1})=K}$.

${\displaystyle E}$and are therefore skew-primitive, and and are group-like. ${\displaystyle F}$ ${\displaystyle K}$ ${\displaystyle K^{-1}}$

Universal enveloping algebra ${\displaystyle U({\mathfrak {sl}}(2))}$

${\displaystyle U_{1}({\mathfrak {sl}}(2))}$is not defined in this form, as one would have to divide by 0. However, it is possible to use another variable to formulate the definition in such a way that this is possible. ${\displaystyle L}$

${\displaystyle KK^{-1}=K^{-1}K=1}$,

${\displaystyle KEK^{-1}=q^{2}E}$,

${\displaystyle KFK^{-1}=q^{-2}F}$,

${\displaystyle [E,F]=L}$

${\displaystyle (q-q^{-1})L=K-K^{-1}}$

${\displaystyle [L,E]=q(EK+K^{-1}E)}$

${\displaystyle [L,F]=-q^{-1}(FK+K^{-1}F)}$

In this form is well-defined and is closely related to universal enveloping algebra . It is true ${\displaystyle U_{1}({\mathfrak {sl}}(2))}$ ${\displaystyle U({\mathfrak {sl}}(2))}$

${\displaystyle U_{1}({\mathfrak {sl}}(2))/(K-1)\cong U({\mathfrak {sl}}(2))}$,

where up , up and up is shown. ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle F}$ ${\displaystyle Y}$ ${\displaystyle L}$ ${\displaystyle H}$

literature

• Christian Kassel: Quantum Groups (Graduate Texts in Mathematics). Springer-Verlag 1998, ISBN 0-387-94370-6 (English)