Last update: 28 October 2012
The isomorphism theorem in (1.1) is found (with proof) in [Dri1990] p. 330-331. The proof of this theorem uses several cohomological facts,
The correspondence theorem in (1.3) is also found in [Dri1990] p.331. All of the results in section 2 can be found, with detailed proofs, in [Jan1995] Chapt. 5.
The algebra is just the enveloping algebra of the Lie algebra except over the ring (and then completed) instead over the field It acts exactly like the algebra the only difference is that we have extended coefficients.
The following theorem says that the algebra and the algebra are exactly the same! In fact we have already seen that this must be so, since has no deformations as an algebra (Chapt. III Theorem (2.6)). One might ask: If and are the same then what is the big deal about quantum groups? The answer is: They are the same as algebras but they are not the same when you look at them as Hopf algebras.
Let be a finite dimensional complex simple Lie algebra and let be the Drinfel'd-Jimbo quantum group corresponding to Then there is an isomorphism of algebras
where, in the second condition,
A finite dimensional is a that is a finitely generated free module as a If is a finite dimensional and define the space of to be the subspace
The dimension of the weight space is the number of elements in a basis for it, as a
Theorem (1.1) says that and are the same as algebras. Since the category of finite dimensional modules for an algebra depends only on its algebra structure it follows immediately that the category of finite dimensional modules for is the same as the category of modules for
There is a one to one correspondence between the isomorphism classes of finite dimensional and the isomorphism classes of finite dimensional given by
where the module structure on is defined by the composition
It follows from condition (b) of Theorem (1.1) that, under the correspondence in the Theorem above, weight spaces of are taken to weight spaces of and their dimension remains the same. Furthermore, irreducible finite dimensional correspond taken to indecomposable and vice versa. (Note that is always a of the
The previous theorem combined with Chapt. II Theorem (2.5) gives the following corollary.
Let be the set of dominant integral weights for as in (2.4). For every there is a unique (up to isomorphism) finite dimensional indecomposable corresponding to
The category of finite dimensional modules for the rational form of the quantum group is slightly different from the category of finite dimensional modules for The construction of the finite dimensional irreducible modules for is similar to the construction of these modules in the case of the Lie algebra Let us describe how this is done.
Let be a finite dimensional complex simple Lie algebra and let be the corresponding rational form of the quantum group over a field and with We shall assume that
Let be an element of the weight lattice for The Verma module is be the generated by a single vector where the action of satisfies the relations
is a vector space isomorphism.
The module has a unique maximal proper submodule. For each define
where is the maximal proper submodule of the Verma module
Let be a finite dimensional complex simple Lie algebra and let be the corresponding rational form of the quantum group over a field with Assume that char and that is not a root of unity in Let be an element of the weight lattice of and let be the defined above.
Let be the root lattice corresponding to as give in II (2.6) and let be a group homomorphism. The homomorphism induces an automorphism of defined by
where are the simple roots for Let be an element of the weight lattice and let be the irreducible defined in (2.1). Define a by defining
Let be a finite dimensional complex simple Lie algebra and let be the rational form of the quantum group over a field Assume that and is not a root of unity in Let be the set of dominant integral weights for and let be the root lattice for (see II (2.6)).
Retain the notations and assumptions from (2.3) and let be the inner product on defined in II (2.7). Let be a finite dimensional Let be a group homomorphism and let The of is the vector space
The following proposition if analogous to Chapt. II Proposition (2.4).
Every finite dimensional is a direct sum of its weight spaces.
The following theorem says that the dimensions of the weight spaces of irreducible coincide with the dimensions of the weight space of corresponding irreducible modules for the Lie algebra
Let be a dominant integral weight and let be a group homomorphism Let be the simple indexed by the and let be the irreducible indexed by the pair Then, for all and all group homomorphisms
This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.