Representation theory
The canonical basis for the irreducible representations of a quantized enveloping algebra of type
and also for the plus part of that algebra was introduced by Lusztig [2] by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter
to
yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter
to
yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara; [3] it is sometimes called the crystal basis. The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara [4] (by an algebraic method) and by Lusztig [5] (by a topological method).
There is a general concept underlying these bases:
Consider the ring of integral Laurent polynomials
with its two subrings
and the automorphism
defined by
.
A precanonical structure on a free
-module
consists of
- A standard basis
of
, - An interval finite partial order on
, that is,
is finite for all
, - A dualization operation, that is, a bijection
of order two that is
-semilinear and will be denoted by
as well.
If a precanonical structure is given, then one can define the
submodule
of
.
A canonical basis of the precanonical structure is then a
-basis
of
that satisfies:
and
for all
.
One can show that there exists at most one canonical basis for each precanonical structure. [6] A sufficient condition for existence is that the polynomials
defined by
satisfy
and
.
A canonical basis induces an isomorphism from
to
.
Linear algebra
If we are given an n × n matrix
and wish to find a matrix
in Jordan normal form, similar to
, we are interested only in sets of linearly independent generalized eigenvectors. A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible. A diagonal matrix
is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector.
Every n × n matrix
possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If
is an eigenvalue of
of algebraic multiplicity
, then
will have
linearly independent generalized eigenvectors corresponding to
.
For any given n × n matrix
, there are infinitely many ways to pick the n linearly independent generalized eigenvectors. If they are chosen in a particularly judicious manner, we can use these vectors to show that
is similar to a matrix in Jordan normal form. In particular,
Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors
that are in the Jordan chain generated by
are also in the canonical basis. [7]
Computation
Let
be an eigenvalue of
of algebraic multiplicity
. First, find the ranks (matrix ranks) of the matrices
. The integer
is determined to be the first integer for which
has rank
(n being the number of rows or columns of
, that is,
is n × n).
Now define

The variable
designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue
that will appear in a canonical basis for
. Note that

Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector). [8]
Example
This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order. [9] The matrix

has eigenvalues
and
with algebraic multiplicities
and
, but geometric multiplicities
and
.
For
we have 
has rank 5,
has rank 4,
has rank 3,
has rank 2.
Therefore 




Thus, a canonical basis for
will have, corresponding to
one generalized eigenvector each of ranks 4, 3, 2 and 1.
For
we have 
has rank 5,
has rank 4.
Therefore 


Thus, a canonical basis for
will have, corresponding to
one generalized eigenvector each of ranks 2 and 1.
A canonical basis for
is

is the ordinary eigenvector associated with
.
and
are generalized eigenvectors associated with
.
is the ordinary eigenvector associated with
.
is a generalized eigenvector associated with
.
A matrix
in Jordan normal form, similar to
is obtained as follows:


where the matrix
is a generalized modal matrix for
and
. [10]