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The coproduct on tex2html_wrap_inline1123

The fact that tex2html_wrap_inline1123 , 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 tex2html_wrap_inline1123 and consider ``many body systems'' (or bound states). For instance, in the case of the rotation group, if tex2html_wrap_inline1471 denotes the third component of angular momentum, the coproduct reads tex2html_wrap_inline1473 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 tex2html_wrap_inline1475 .

We now define directly the coproducts, antipode and counit in tex2html_wrap_inline1123 .

We define tex2html_wrap_inline1479 , tex2html_wrap_inline1481 , tex2html_wrap_inline1483 , tex2html_wrap_inline1485 .

The anti-automorphism S acts as S 1 = 1, tex2html_wrap_inline1491 , tex2html_wrap_inline1493 , tex2html_wrap_inline1495 , tex2html_wrap_inline1497 . As usual, the square of the antipode is an automorphism (and it is, in this case, a conjugacy by tex2html_wrap_inline1499 , i.e. tex2html_wrap_inline1501 ).

The co-unit tex2html_wrap_inline1503 is defined by tex2html_wrap_inline1505 , tex2html_wrap_inline1507 , tex2html_wrap_inline1509 , tex2html_wrap_inline1511 , tex2html_wrap_inline1513 .

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 tex2html_wrap_inline1515 and we do not list either the axioms involving the antipode).


tex2html_wrap_inline1519 tex2html_wrap_inline1521

Connecting axiom:
tex2html_wrap_inline1523 where tex2html_wrap_inline1525 exchanges the second and third factors in tex2html_wrap_inline1527 .

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 tex2html_wrap_inline1123 given before and explicitly performing the tensor products of matrices. For illustration only, we check co-associativity on the generator tex2html_wrap_inline1531 . 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 tex2html_wrap_inline1123 ), let us mention, for instance, that the algebra tex2html_wrap_inline1535 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 tex2html_wrap_inline1313 part is crucial: although tex2html_wrap_inline1123 can be written as a direct sum of two algebras, namely of tex2html_wrap_inline1313 and of tex2html_wrap_inline1173 , 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 tex2html_wrap_inline1123 . One has first to express the chosen element in terms of the generators tex2html_wrap_inline1373 and K (for that, one may use tex2html_wrap_inline1445 and tex2html_wrap_inline1447 ). What is then left to do is a simple calculation using the fundamental property of tex2html_wrap_inline1557 , namely that it is a homomorphism of algebras : tex2html_wrap_inline1559 for U and V in tex2html_wrap_inline1123 . Warning: With the notations given at the end of section 3, we see that, for example, tex2html_wrap_inline1569 , however tex2html_wrap_inline1571 is not equal to tex2html_wrap_inline1573 , etcIn order to appreciate the rather non trivial mixing induced by the coproduct, we give -- part of -- the expression tex2html_wrap_inline1575 (recall that tex2html_wrap_inline1577 is an elementary matrix of ``color type'', i.e. tex2html_wrap_inline1171 , containing only a 1 in position (1,1). We have re-expressed the result in terms of elementary matrices tex2html_wrap_inline1311 --of color type-- and tex2html_wrap_inline1315 --of electroweak type.


What is important in this example is not the expression itself (!) but the fact that it involves the tex2html_wrap_inline1311 and the tex2html_wrap_inline1315 . In a sense, one can generate a coupling to the tex2html_wrap_inline1593 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 tex2html_wrap_inline1595 with tex2html_wrap_inline1597 and tex2html_wrap_inline1599 . A rather long -- but straightforward -- calculation leads (for a two body system, i.e. tex2html_wrap_inline1601 to a rather lengthy result for tex2html_wrap_inline1603 . The main feature is that this ``charge'' is not additive : tex2html_wrap_inline1605 is not equal to tex2html_wrap_inline1607 and, moreover, it couples non trivially the tex2html_wrap_inline1215 part together with the tex2html_wrap_inline1169 part.

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

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