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# The coproduct on

The fact that , defined in this way, admits a non trivial Hopf algebra structure --in particular a coproduct-- is absolutely not obvious at first sight.

Let us remind the reader that it is the existence of a coproduct that makes possible to consider tensor products of representations, exactly as it were a finite group. This is obviously of prime importance if one has in mind to find some physical interpretation for and consider ``many body systems'' (or bound states). For instance, in the case of the rotation group, if denotes the third component of angular momentum, the coproduct reads and this tells us how to calculate the third component of the total angular momentum for a system described by the tensor product of two Hilbert spaces, namely .

We now define directly the coproducts, antipode and counit in .

Coproduct:
We define , , , .

Antipode:
The anti-automorphism S acts as S 1 = 1, , , , . As usual, the square of the antipode is an automorphism (and it is, in this case, a conjugacy by , i.e. ).

Co-unit:
The co-unit is defined by , , , , .

Notice that the resulting Hopf algebra is neither commutative nor cocommutative. We now have to check that all expected properties are indeed satisfied. Here are the main required properties for a Hopf algebra (we do not list the usual algebras axioms involving only the multiplication map and we do not list either the axioms involving the antipode).

Coproduct:

Counit:

Connecting axiom:
where exchanges the second and third factors in .

The reader may check, in an elementary way, that all these properties are indeed satisfied, either by using the above generators and relations or by using the explicit presentation of given before and explicitly performing the tensor products of matrices. For illustration only, we check co-associativity on the generator . We first compute

We then compute

Both expressions are indeed equal.

To illustrate the non triviality of this result (the existence of a Hopf structure on ), let us mention, for instance, that the algebra does not even carry any Hopf structure at all (it is known that the only semi-simple Hopf algebra of dimension 5 is the commutative group algebra defined by the cyclic group on five letters). In our case, the presence of the part is crucial: although can be written as a direct sum of two algebras, namely of and of , the coproduct mixes non trivially the two factors. If one wants to use this algebra (or another non co-commutative Hopf algebra) to characterize ``symmetries'' of some physical system --for instance in elementary particle physics-- one should keep in mind that, in contrast with what is done usually in the case of symmetries described by Lie algebras, the ``quantum numbers'' will not usually be additive.

Our explicit description of the algebra allows one to compute explicitly the coproduct of an arbitrarily chosen element in . One has first to express the chosen element in terms of the generators and K (for that, one may use and ). What is then left to do is a simple calculation using the fundamental property of , namely that it is a homomorphism of algebras : for U and V in . Warning: With the notations given at the end of section 3, we see that, for example, , however is not equal to , etcIn order to appreciate the rather non trivial mixing induced by the coproduct, we give -- part of -- the expression (recall that is an elementary matrix of ``color type'', i.e. , containing only a 1 in position (1,1). We have re-expressed the result in terms of elementary matrices --of color type-- and --of electroweak type.

What is important in this example is not the expression itself (!) but the fact that it involves the and the . In a sense, one can generate a coupling to the part by building ``bound states'' from the ``color part'' alone.

We can also compute the expression of the coproduct for a generator of ``electroweak type'', like with and . A rather long -- but straightforward -- calculation leads (for a two body system, i.e. to a rather lengthy result for . The main feature is that this ``charge'' is not additive : is not equal to and, moreover, it couples non trivially the part together with the part.

Next: The representation theory of Up: No Title Previous: A system of generators

Robert Coquereaux
Tue Nov 5 15:18:21 MET 1996