Basis of a type of algebraic structure
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
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![{\displaystyle c_{i}\in \sum _{j\leq i}{\mathcal {Z}}^{+}t_{j}{\text{ and }}c_{i}\equiv t_{i}\mod vF^{+}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
.
Hecke algebras
Let
be a Coxeter group. The corresponding Iwahori-Hecke algebra
has the standard basis
, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by
. This is a precanonical structure on
that satisfies the sufficient condition above and the corresponding canonical basis of
is the Kazhdan–Lusztig basis
![{\displaystyle C_{w}'=\sum _{y\leq w}P_{y,w}(v^{2})T_{w}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
being the Kazhdan–Lusztig polynomials.
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
![{\displaystyle \rho _{k}=\operatorname {rank} (A-\lambda _{i}I)^{k-1}-\operatorname {rank} (A-\lambda _{i}I)^{k}\qquad (k=1,2,\ldots ,m_{i}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle \operatorname {rank} (A-\lambda _{i}I)^{0}=\operatorname {rank} (I)=n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle A={\begin{pmatrix}4&1&1&0&0&-1\\0&4&2&0&0&1\\0&0&4&1&0&0\\0&0&0&5&1&0\\0&0&0&0&5&2\\0&0&0&0&0&4\end{pmatrix}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
has eigenvalues
and
with algebraic multiplicities
and
, but geometric multiplicities
and
.
For
we have ![{\displaystyle n-\mu _{1}=6-4=2,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
has rank 5,
has rank 4,
has rank 3,
has rank 2.
Therefore ![{\displaystyle m_{1}=4.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{4}=\operatorname {rank} (A-4I)^{3}-\operatorname {rank} (A-4I)^{4}=3-2=1,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{3}=\operatorname {rank} (A-4I)^{2}-\operatorname {rank} (A-4I)^{3}=4-3=1,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{2}=\operatorname {rank} (A-4I)^{1}-\operatorname {rank} (A-4I)^{2}=5-4=1,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{1}=\operatorname {rank} (A-4I)^{0}-\operatorname {rank} (A-4I)^{1}=6-5=1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Thus, a canonical basis for
will have, corresponding to
one generalized eigenvector each of ranks 4, 3, 2 and 1.
For
we have ![{\displaystyle n-\mu _{2}=6-2=4,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
has rank 5,
has rank 4.
Therefore ![{\displaystyle m_{2}=2.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{2}=\operatorname {rank} (A-5I)^{1}-\operatorname {rank} (A-5I)^{2}=5-4=1,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \rho _{1}=\operatorname {rank} (A-5I)^{0}-\operatorname {rank} (A-5I)^{1}=6-5=1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Thus, a canonical basis for
will have, corresponding to
one generalized eigenvector each of ranks 2 and 1.
A canonical basis for
is
![{\displaystyle \left\{\mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},\mathbf {x} _{4},\mathbf {y} _{1},\mathbf {y} _{2}\right\}=\left\{{\begin{pmatrix}-4\\0\\0\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}-27\\-4\\0\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}25\\-25\\-2\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\36\\-12\\-2\\2\\-1\end{pmatrix}},{\begin{pmatrix}3\\2\\1\\1\\0\\0\end{pmatrix}},{\begin{pmatrix}-8\\-4\\-1\\0\\1\\0\end{pmatrix}}\right\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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:
![{\displaystyle M={\begin{pmatrix}\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {x} _{4}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix}}={\begin{pmatrix}-4&-27&25&0&3&-8\\0&-4&-25&36&2&-4\\0&0&-2&-12&1&-1\\0&0&0&-2&1&0\\0&0&0&2&0&1\\0&0&0&-1&0&0\end{pmatrix}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle J={\begin{pmatrix}4&1&0&0&0&0\\0&4&1&0&0&0\\0&0&4&1&0&0\\0&0&0&4&0&0\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where the matrix
is a generalized modal matrix for
and
.[10]
See also
Notes
- ^ Bronson (1970, p. 196)
- ^ Lusztig (1990)
- ^ Kashiwara (1990)
- ^ Kashiwara (1991)
- ^ Lusztig (1991)
- ^ Lusztig (1993, p. 194)
- ^ Bronson (1970, pp. 196, 197)
- ^ Bronson (1970, pp. 197, 198)
- ^ Nering (1970, pp. 122, 123)
- ^ Bronson (1970, p. 203)
References
- Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
- Deng, Bangming; Ju, Jie; Parshall, Brian; Wang, Jianpan (2008), Finite Dimensional Algebras and Quantum Groups, Mathematical surveys and monographs, vol. 150, Providence, R.I.: American Mathematical Society, ISBN 9780821875315
- Kashiwara, Masaki (1990), "Crystalizing the q-analogue of universal enveloping algebras", Communications in Mathematical Physics, 133 (2): 249–260, Bibcode:1990CMaPh.133..249K, doi:10.1007/bf02097367, ISSN 0010-3616, MR 1090425, S2CID 121695684
- Kashiwara, Masaki (1991), "On crystal bases of the q-analogue of universal enveloping algebras", Duke Mathematical Journal, 63 (2): 465–516, doi:10.1215/S0012-7094-91-06321-0, ISSN 0012-7094, MR 1115118
- Lusztig, George (1990), "Canonical bases arising from quantized enveloping algebras", Journal of the American Mathematical Society, 3 (2): 447–498, doi:10.2307/1990961, ISSN 0894-0347, JSTOR 1990961, MR 1035415
- Lusztig, George (1991), "Quivers, perverse sheaves and quantized enveloping algebras", Journal of the American Mathematical Society, 4 (2): 365–421, doi:10.2307/2939279, ISSN 0894-0347, JSTOR 2939279, MR 1088333
- Lusztig, George (1993), Introduction to quantum groups, Boston, MA: Birkhauser Boston, ISBN 0-8176-3712-5, MR 1227098
- Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646