Representation of a Hopf algebra

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In mathematics, there is a concept of a representation of a Hopf algebra.

Given a unital associative algebra H, a representation of H is a module.

For some algebra H, suppose an action with

1[X] = X

and

(AB)[X] = A[B[X]]

(this relation is not satisfied, for example, for representations of a Lie algebra)

for all X in the underlying vector space, and any A, B in H.

Then

(A1)[X] = A[1[X]] = A[X] = 1[A[X]] = (1A)[X]

and

((AB)C)[X] = (AB)[C[X]] = A[B[C[X]]] = A[(BC)[X]] = (A(BC))[X]

for all X in the rep and all A, B, C in H.

A sufficient condition for this to hold is if H is unital and associative.

Suppose we wish for any two reps V₁ and V₂ that there is a <math>\otimes</math> operation defined on the reps such that <math>V_1\otimes V_2</math> is also a rep and the underlying vector space of <math>V_1\otimes V_2</math> is just the product of the underlying vector spaces of V₁ and V₂. This would not hold in general for an arbitrary algebra, but let's just say we insist upon it.

Also assume this product operation for reps is functorial. In other words, there is a linear binary operation

<math>\Delta:H\rightarrow H\otimes H\,</math>

such that for any X in <math>V_1\otimes V_2</math> and any A in H,

<math>A[X]=\Delta A[X_{(1)}\otimes X_{(2)}]=A_{(1)}[X_{(1)}]\otimes A_{(2)}[X_{(2)}].\,</math>

Here, we are using Sweedler's notation, which is kind of like an index free form of Einstein's summation convention.

What does it mean when we say <math>V_1\otimes V_2</math> is also a rep?

Obviously,

<math>\Delta 1[X_{(1)}\otimes X_{(2)}]=1[X]=X=X_{(1)}\otimes X_{(2)}=1[X_{(1)}]\otimes 1[X_{(2)}]=(1 \otimes 1)[X_{(1)}\otimes X_{(2)}]\,</math>

and

<math>\Delta(AB)[X_{(1)}\otimes X_{(2)}]=(AB)[X]=A[B[X]]=\Delta A[\Delta B[X_{(1)}\otimes X_{(2)}]]=(\Delta A )(\Delta B)[X_{(1)}\otimes X_{(2)}].\,</math>

So, it would be sufficient if <math>\Delta 1=1\otimes 1</math> and <math>\Delta(AB)=(\Delta A)(\Delta B)</math>.

Suppose we would like to turn the collection of all reps into a monoidal category with respect to <math>\otimes</math>. For the moment, assume it is not weak for the moment; this means, in this stronger form, that <math>V_1\otimes(V_2\otimes V_3)</math> and <math>(V_1\otimes V_2)\otimes V_3</math> are equivalent instead of being merely isomorphic and that there is an identity rep <math>\varepsilon_H</math>, called the trivial rep, such that <math>\varepsilon_H\otimes V</math>, V and <math>V\otimes \varepsilon_H</math> are all equivalent (isomorphic for the corresponding weak version, which is slightly trickier).

It is obvious that ε_H would have to be 1-dimensional. Moreover, because the three reps above are assumed to be equivalent (isomorphic in the weak version), if we assume the equivalence is functorial (over all reps), then there is an element of ε_H, which we will just call 1, such that <math>1\otimes X</math>, X and <math>X \otimes 1</math> are all equal for any X in V. In fact, as a matter of convenience, we can just assume the underlying vector space of ε_H is just the field F that H is over.

Now, we can define a linear map <math>\varepsilon:H\rightarrow F</math> given by ε[A] ≡ A[1].

Obviously, for any X in <math>V_1\otimes(V_2\otimes V_3)=(V_1\otimes V_2)\otimes V_3</math> (<math>\simeq</math> for the weak version),

<math>((\operatorname{id}\otimes \Delta)\Delta A)[X_{(1)}\otimes X_{(2)}\otimes X_{(3)}]=A_{(1)}[X_{(1)}]\otimes A_{(2)(1)}[X_{(2)}]\otimes A_{(2)(2)}[X_{(3)}]=A_{(1)}[X_{(1)}]\otimes A_{(2)}[X_{(2)}\otimes X_{(3)}]=A[X]=((\Delta\otimes \operatorname{id})\Delta A)[X_{(1)}\otimes X_{(2)}\otimes X_{(3)}].</math>

Insisting that <math>(\operatorname{id}\otimes \Delta)\Delta A=(\Delta \otimes \operatorname{id})\Delta A</math> is sufficient.

Obviously,

ε(1) = 1[1] = 1

and

ε(AB) = (AB)[1] = A[B[1]] = A[ε(B)] = ε(A)ε(B).

Also,

<math>(\varepsilon(A_{(1)})A_{(2)})[X]=A_{(1)}[1][A_{(2)}[X]]=A_{(1)}[1]\otimes A_{(2)}[X]=A[1\otimes X]=A[X]=(A_{(1)}\varepsilon(A_{(2)}))[X].</math>

Insisting that

ε(A₍₁₎)A₍₂₎ = A = A₍₁₎ε(A₍₂₎)

is sufficient.

This motivates the concept of a bialgebra, and what it means to have a representation of a bialgebra.

We would like to have a dual representation V^* for every rep V such that the underlying vector spaces of both reps are dual to each other. However, we need not insist V^** = V. Also assume this * is functorial over the monoidal category of reps. In other words, there is a linear map <math>S:H\rightarrow H</math> such that for any A in H, X in V and Y in V^*,

<math>\langle Y, S(A)[X]\rangle = \langle A[Y], X \rangle.</math>

where <math>\langle\bullet,\bullet\rangle</math> is just the contraction for dual vector spaces.

We also require that there is a H-intertwiner (as maps between bialgebra reps, because we have not defined a Hopf algebra rep yet) from ε_H to <math>V^*\otimes V</math> which maps 1 to <math>1_{V^*\otimes V}</math> where <math>1_{V^*\otimes V}</math> is the unique element of <math>V^*\otimes V</math> which satisfies <math>\langle 1_{V^*\otimes V},X \rangle =X</math> and <math>\langle Y,1_{V^*\otimes V} \rangle = Y</math> for all X and Y.

Not so obviously, this implies

<math>A[1_{V^*\otimes V}]=\varepsilon(A)[1_{V^*\otimes V}],</math>

which is satisfied if the sufficient requirement that

<math>S(A_{(1)})A_{(2)}=\varepsilon(A)=A_{(1)}S(A_{(2)}).\,</math>

This is merely one justification of a Hopf algebra. It turns out the same few definitions are motivated in many other topics, which is why the concept of a Hopf algebra is not merely abstract nonsense.

If V is a rep of a Hopf algebra H, then X in V is invariant under H if for all A in H,

A[X] = ε(A)X.

The subset of all invariant elements of V forms a subrep of V.

Algebra representations

A Hopf algebra also has algebra reps with additional structure.

Let H be a Hopf algebra. If A is an algebra with the product operation <math>\mu:A\otimes A\rightarrow A</math>, then a linear map <math>\rho:H\otimes A\rightarrow A</math> is an algebra representation of H if, in addition to being a (vector space) rep of H, μ is an H-intertwiner. Recall that <math>A\otimes A</math> is also a vector space rep of H. If A happens to be unital, we would require that there is an H-intertwiner from ε_H to A such that the 1 of ε_H maps to the unit of A.

Algebra representation of a Lie algebra, algebra representation of a Lie superalgebra and algebra representation of a group are all special cases of this more general concept.

See also Tannaka-Krein reconstruction theorem.