Formal definition Hopf algebra

1 formal definition

1.1 structure constants
1.2 properties of antipode
1.3 hopf subalgebras
1.4 hopf orders
1.5 group-like elements

formal definition

formally, hopf algebra (associative , coassociative) bialgebra h on field k k-linear map s: h → h (called antipode) such following diagram commutes:

here Δ comultiplication of bialgebra, ∇ multiplication, η unit , ε counit. in sumless sweedler notation, property can expressed as

s
(

c

(
1
)


)

c

(
2
)


=

c

(
1
)


s
(

c

(
2
)


)
=
ϵ
(
c
)
1


 for all 

c
∈
h
.


{\displaystyle s(c_{(1)})c_{(2)}=c_{(1)}s(c_{(2)})=\epsilon (c)1\qquad {\mbox{ }}c\in h.}

as algebras, 1 can replace underlying field k commutative ring r in above definition.

the definition of hopf algebra self-dual (as reflected in symmetry of above diagram), if 1 can define dual of h (which possible if h finite-dimensional), automatically hopf algebra.

structure constants

fixing basis



{

e

k


}


{\displaystyle \{e_{k}\}}

underlying vector space, 1 may define algebra in terms of structure constants multiplication:

e

i


∇

e

j


=

∑

k



μ


i
j


k



e

k




{\displaystyle e_{i}\nabla e_{j}=\sum _{k}\mu _{\;ij}^{k}e_{k}}

for co-multiplication:

Δ

e

i


=

∑

j
,
k



ν

i



j
k



e

j


⊗

e

k




{\displaystyle \delta e_{i}=\sum _{j,k}\nu _{i}^{\;jk}e_{j}\otimes e_{k}}

and antipode:

s

e

i


=

∑

j



τ

i



j



e

j




{\displaystyle se_{i}=\sum _{j}\tau _{i}^{\;j}e_{j}}

associativity requires that

μ


i
j


k



μ


k
n


m


=

μ


j
n


k



μ


i
k


m




{\displaystyle \mu _{\;ij}^{k}\mu _{\;kn}^{m}=\mu _{\;jn}^{k}\mu _{\;ik}^{m}}

while co-associativity requires that

ν

k



i
j



ν

i



m
n


=

ν

k



m
i



ν

i



n
j




{\displaystyle \nu _{k}^{\;ij}\nu _{i}^{\;mn}=\nu _{k}^{\;mi}\nu _{i}^{\;nj}}

the connecting axiom requires that

ν

k



i
j



τ

j



m



μ


p
m


n


=

ν

k



j
m



τ

j




i



μ


p
m


n




{\displaystyle \nu _{k}^{\;ij}\tau _{j}^{\;m}\mu _{\;pm}^{n}=\nu _{k}^{\;jm}\tau _{j}^{\,\;i}\mu _{\;pm}^{n}}



properties of antipode

the antipode s required have k-linear inverse, automatic in finite-dimensional case, or if h commutative or cocommutative (or more quasitriangular).

in general, s antihomomorphism, s homomorphism, therefore automorphism if s invertible (as may required).

if s = idh, hopf algebra said involutive (and underlying algebra involution *-algebra). if h finite-dimensional semisimple on field of characteristic zero, commutative, or cocommutative, involutive.

if bialgebra b admits antipode s, s unique ( bialgebra admits @ 1 hopf algebra structure ).

the antipode analog inversion map on group sends g g.

hopf subalgebras

a subalgebra of hopf algebra h hopf subalgebra if subcoalgebra of h , antipode s maps a. in other words, hopf subalgebra hopf algebra in own right when multiplication, comultiplication, counit , antipode of h restricted (and additionally identity 1 of h required in a). nichols–zoeller freeness theorem established (in 1989) natural a-module h free of finite rank if h finite-dimensional: generalization of lagrange s theorem subgroups. corollary of , integral theory, hopf subalgebra of semisimple finite-dimensional hopf algebra automatically semisimple.

a hopf subalgebra said right normal in hopf algebra h if satisfies condition of stability, adr(h)(a) ⊆ h in h, right adjoint mapping adr defined adr(h)(a) = s(h(1))ah(2) in a, h in h. similarly, hopf subalgebra left normal in h if stable under left adjoint mapping defined adl(h)(a) = h(1)as(h(2)). 2 conditions of normality equivalent if antipode s bijective, in case said normal hopf subalgebra.

a normal hopf subalgebra in h satisfies condition (of equality of subsets of h): ha = ah denotes kernel of counit on k. normality condition implies ha hopf ideal of h (i.e. algebra ideal in kernel of counit, coalgebra coideal , stable under antipode). consequence 1 has quotient hopf algebra h/ha , epimorphism h → h/ah, theory analogous of normal subgroups , quotient groups in group theory.

hopf orders

a hopf order o on integral domain r field of fractions k order in hopf algebra h on k closed under algebra , coalgebra operations: in particular, comultiplication Δ maps o o⊗o.

group-like elements

a group-like element nonzero element x such Δ(x) = x⊗x. group-like elements form group inverse given antipode. primitive element x satisfies Δ(x) = x⊗1 + 1⊗x.