Hopf algebra

 Hopf algebra touches the specialties mathematics 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}}$