Hopf algebra

Hopf algebra

touches the specialties

mathematics Abstract algebra Linear Algebra Commutative algebra

is a special case of

Bialgebra

A Hopf algebra - named after the mathematician Heinz Hopf - over a body is a bialgebra with a linear mapping, the so-called "antipode" , so that the following diagram commutes: ${\displaystyle H}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle (H,\nabla ,\eta ,\Delta ,\epsilon )}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle S\colon H\to H}$

Formally written in the Sweedler notation - named after Moss Sweedler - this means: ${\displaystyle S\left(c_{\left(1\right)}\right)c_{\left(2\right)}=c_{\left(1\right)}S\left(c_{\left(2\right)}\right)=\epsilon \left(c\right)1.}$

Convolution and antipode

Be an algebra and a koalgebra . The linear mappings from to form an algebra with a product , called convolution, defined by ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle *}$

${\displaystyle (f*g)(x):=f(x_{(1)})g(x_{(2)})}$.

The neutral element in this algebra is because ${\displaystyle \eta \circ \epsilon }$

${\displaystyle (f*(\eta \circ \epsilon ))(x)=f(x_{(1)})\eta (\epsilon (x_{(2)}))=f(x_{(1)}\epsilon (x_{(2)}))\eta (1)=f(x)}$

and accordingly also

${\displaystyle ((\eta \circ \epsilon )*f)(x)=f(x)}$.

For a bialgebra the linear mappings from to form an algebra in this way. The antipode is the inverse element of the identical mapping in this algebra. This means ${\displaystyle H}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle S}$

${\displaystyle S*\mathrm {id} =\eta \circ \epsilon =\mathrm {id} *S}$.

It can be shown that the antipode of a Hopf algebra is always unique, and at the same time it is an anti-algebra homomorphism and an anticoalgebra homomorphism. With the help of this fact, the value of the antipode on each element of the Hopf algebra can be calculated if the values ​​of the antipode on an algebra-generating system are known.

Examples

Group algebra

An example of a Hopf algebra is group algebra . She is going through ${\displaystyle \mathbb {K} G}$

${\displaystyle \Delta (g):=g\otimes g}$ For ${\displaystyle g\in G}$

and

${\displaystyle \epsilon (g):=1}$ For ${\displaystyle g\in G}$

to a bialgebra , the antipode

${\displaystyle S(g):=g^{-1}}$ For ${\displaystyle g\in G}$

turns it into a Hopf algebra.

Universal enveloping algebra

The universal enveloping algebra of a Lie algebra is naturally a Hopf algebra. For one element , the coproduct is through ${\displaystyle \mathrm {U} ({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle x\in {\mathfrak {g}}}$

${\displaystyle \Delta (x):=1\otimes x+x\otimes 1}$

and the Koeins through

${\displaystyle \epsilon (x):=0}$

Are defined.

${\displaystyle S(x):=-x}$

defines the antipode.

Group-like and primitive elements

An element of a Hopf algebra is called “group-like” if and . Then applies to the antipode . ${\displaystyle g}$ ${\displaystyle \Delta (g)=g\otimes g}$ ${\displaystyle \epsilon (g)=1}$ ${\displaystyle S(g)=g^{-1}}$

An element is called "primitive" if . It follows that and . ${\displaystyle x}$ ${\displaystyle \Delta (x)=1\otimes x+x\otimes 1}$ ${\displaystyle \epsilon (x)=0}$ ${\displaystyle S(x)=-x}$

An element is called "skew-primitive" if with group-like elements and . It follows that and . ${\displaystyle x}$ ${\displaystyle \Delta (x)=g\otimes x+x\otimes h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle \epsilon (x)=0}$ ${\displaystyle S(x)=-g^{-1}xh^{-1}}$

literature

• Christian Kassel : Quantum Groups (= Graduate Texts in Mathematics . 155). Springer, New York NY et al. 1995, ISBN 0-387-94370-6 .
• Moss E. Sweedler : Hopf algebras. Benjamin, New York NY 1969.