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In this paper, we decided to study properties of without using
any a priori knowledge on quantum groups. Here are nevertheless a
few (non elementary) facts, given without proof, that may interest the reader.
Consider the universal enveloping algebra of , say U.
Let be the corresponding quantum algebra ( is a deformation
parameter) and , K its generators (not the same as before !).
The first way to construct the finite dimensional Hopf algebra
from the infinite dimensional Hopf algebra is to divide
it by an ideal (the ideal I defined by the relations
given in section 3)
and check that it is also a Hopf ideal (which means that , and ).
In the usual --more precisely in the so called adjoint rational form-- and when q is a cubic root of unity,
the center is generated not only by the Casimir operator but also by the elements and .
It is therefore natural to define
new algebras by dividing the `big' object by an ideal generated by relations of the kind we just considered (remember that in ).
Actually, one could as well define new algebras by imposing relations of the
kind and for integers and that are divisible by 3 (the value of the right hand side, namely 0 or 1 is fixed by the existence of a co-unit). The finite dimensional quotient
therefore appears as a ``minimal'' choice.
As a matter of fact,
even at the level of , defined as before, and without any
reference to , it is convenient to introduce an invertible square root
for K, hence .
In this way, one obtains a new algebra of dimension -- just count the number of independent monomials when and (this is a PBW basis).
This algebra is quite interesting because its list of representations contains
not only those of but also ``charge conjugate'' representations.
One can also justify, for the quantum enveloping algebra itself,
and whether q is a root of unity or not, the interest of adding a square root to the generator K. Introducing such a square root at the level of
the so-called simply-connected rational form of the quantum universal enveloping algebra.
The reader should be warned that the algebra or is
sometimes denoted by --with a small u-- and
called the ``restricted quantum universal algebra'' (for a reason that will be explained below), but the terminology is not established yet and one should
always look which of or is equal to 1; for instance, our --see above-- coincide with the K of .
There exists another construction which is more tricky but maybe more
profound. Let us start with the following simple observation. Consider the algebra of polynomials with one unknown x, over the rational numbers;
it can be considered as an algebra
generated by the (to be identified with the
powers of x) with relations . One can make a change of
generators, define the divided powers so that
and define the very same algebra
by using the new generators and the new
relations. The tricky point is that, if we now decide to build an algebra over the finite
field (for instance ) by taking reduction of coefficients modulo p, the two
constructions, with usual powers and divided powers, will lead to two different algebras.
For instance, the relation will be valid in the first algebra (the r.h.s. is
non zero) whereas we obtain if we use divided powers, since 3=0 in . A similar phenomenon appears, in the case of quantum groups when q is a primitive root of unity.
There are indeed two ways of specializing the q of (see for instance ) to a particular complex value . One can use the usual (q-deformed) generators
and relations, but one can also use q-deformed divided powers and corresponding relations
(these relations contain q-deformed factorials on the right hand side).
For generic values of q, both definitions lead to the same algebra, but when q is a
primitive root of unity, the algebras are different. More precisely,
we are interested here in the ``restricted integral form'',
called which is obtained as follows.
We assume that , with ,
and start with considered as an algebra over the cyclotomic field
(it is obtained by adjoining a cubic root of unity to the
field Q of rational numbers).
is then defined as the subalgebra of generated by the elements , where is the q factorial -- so these
elements are q-deformed divided powers-- and by .
we see that !=1, !=-1 and for N>2. This, in turn, implies that,
in the algebra , , is central and .
contains a finite dimensional Hopf subalgebra , of dimension
over , generated by and the for
. Since is central, one can also construct a
quotient (of dimension 27) by dividing by the
relation . If we know take arbitrary complex coefficients
(rather than coefficients belonging to ), we recover .
Before ending this section, we want to comment about a rather beautiful
and mysterious relation with Platonic bodies (actually with the simplest
of them all, the tetrahedron).
A finite Hopf algebra bearing some strong resemblance with
was originally defined by ,  as
the restricted enveloping algebra of a simple Lie algebra over the
finite field with p elements (p, a prime). The construction
goes as follows :
- Start with an algebraic group K. In our case it
will be . Note that this group, of order 24 is isomorphic
with the binary tetrahedral group (the double cover of the SO(3) finite
subgroup preserving a tetrahedron); the tetrahedron group itself is
and is also isomorphic with the alternated group .
- Construct , the so-called Chevalley-Kostant
-form of the universal enveloping algebra U for
the corresponding Lie group G, in our case, . It is
a subring (over ZZ) of U generated by the divided powers
- Build the hyperalgebra of over , namely
- The restricted enveloping algebra is the
subalgebra of the hyperalgebra spanned by the .
The theory of
restricted enveloping algebras goes back to
 (see also
his basic paper on derivations of algebras over a finite field
is, in this way, defined for any Lie algebra as a
subring of the corresponding enveloping algebra generated by the
divided powers of the Chevalley generators. The p-powers of these
generators are zero and the obtained algebra is of
dimension over . For us, is
Lie(SL(2,C) and so that .
The purpose of defining objects like was historically to study the
theory of modular representations of finite
Chevalley groups. Although both , defined as a restricted enveloping
algebra over a finite field, and , defined as the quotient of
by the relation look like very different
objects, (the first is an algebra over , the
second is over or over ), it was shown by
,  that there exists a natural bijection between
representation theory over of the first and usual representation theory
over of the second.
Without entering this deep arithmetical discussion, we want to conclude
this paragraph by a simple description of the theory of modular
representations for the binary tetrahedral group . The table of usual (characteristic 0) characters
of is easy to obtain, for instance from the incidence matrix of
the extended Dynkin diagram of , via the McKay correspondence.
The dimensions of irreducible representations are simply obtained by
taking this diagram as a fusion diagram (tensorialization with the
fundamental of dimension 2). One obtains in this way the seven inequivalent irreducible representations of ; see figure 2.
Figure 2: Irreducible representations and fusion graph for the binary
Modular characters are only interesting in characteristic 2 and 3
(since primes 2 and 3 divide 24). In
characteristic 3, there are only three regular conjugacy classes
(namely the classes of the identity, minus the identity, and the class of the
elements of period 4). Therefore, using Brauer's theory, one can check
that there are also three irreducible inequivalent modular
characters, of respective degrees 1, 2 and 3 (like !).
Next: Other properties of
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Previous: The representation theory of
Tue Nov 5 15:18:21 MET 1996