canonical basis

The canonical basis for the irreducible representations of a quantized enveloping algebra of

type $ADE$ and also for the plus part of that algebra was introduced by Lusztig {{harvtxt|Lusztig|1990}} by

two methods: an algebraic one (using a braid group action and PBW bases) and a topological one

(using intersection cohomology). Specializing the parameter $q$ to $q=1$ yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was

not known earlier. Specializing the parameter $q$ to $q=0$ 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;{{harvtxt|Kashiwara|1990}} it is sometimes called the crystal basis.

The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara {{harvtxt|Kashiwara|1991}} (by an algebraic method) and by Lusztig {{harvtxt|Lusztig|1991}} (by a topological method).

There is a general concept underlying these bases:

Consider the ring of integral Laurent polynomials $\backslash mathcal\{Z\}:=\backslash mathbb\{Z\}\backslash left[v,v^\{-1\}\backslash right]$ with its two subrings $\backslash mathcal\{Z\}^\{\backslash pm\}:=\backslash mathbb\{Z\}\backslash left[v^\{\backslash pm\; 1\}\backslash right]$ and the automorphism $\backslash overline\{\backslash cdot\}$ defined by $\backslash overline\{v\}:=v^\{-1\}$.

A precanonical structure on a free $\backslash mathcal\{Z\}$-module $F$ consists of

• A standard basis $(t\_i)\_\{i\backslash in\; I\}$ of $F$,
• An interval finite partial order on $I$, that is, $(-\backslash infty,i]\; :=\; \backslash \{j\backslash in\; I\; \backslash mid\; j\backslash leq\; i\backslash \}$ is finite for all $i\backslash in\; I$,
• A dualization operation, that is, a bijection $F\backslash to\; F$ of order two that is $\backslash overline\{\backslash cdot\}$-semilinear and will be denoted by $\backslash overline\{\backslash cdot\}$ as well.

If a precanonical structure is given, then one can define the $\backslash mathcal\{Z\}^\{\backslash pm\}$ submodule $F^\{\backslash pm\}\; :=\; \backslash sum\; \backslash mathcal\{Z\}^\{\backslash pm\}\; t\_j$ of $F$.

A canonical basis of the precanonical structure is then a $\backslash mathcal\{Z\}$-basis $(c\_i)\_\{i\backslash in\; I\}$ of $F$ that satisfies:

• $\backslash overline\{c\_i\}=c\_i$ and
• $c\_i\; \backslash in\; \backslash sum\_\{j\backslash leq\; i\}\; \backslash mathcal\{Z\}^+\; t\_j\; \backslash text\{\; and\; \}\; c\_i\; \backslash equiv\; t\_i\; \backslash mod\; vF^+$

for all $i\backslash in\; I$.

One can show that there exists at most one canonical basis for each precanonical structure.{{harvtxt|Lusztig|1993|p=194}} A sufficient condition for existence is that the polynomials $r\_\{ij\}\backslash in\backslash mathcal\{Z\}$ defined by $\backslash overline\{t\_j\}=\backslash sum\_i\; r\_\{ij\}\; t\_i$ satisfy $r\_\{ii\}=1$ and $r\_\{ij\}\backslash neq\; 0\; \backslash implies\; i\backslash leq\; j$.

A canonical basis induces an isomorphism from $\backslash textstyle\; F^+\backslash cap\; \backslash overline\{F^+\}\; =\; \backslash sum\_i\; \backslash mathbb\{Z\}c\_i$ to $F^+/vF^+$.

Let $(W,S)$ be a Coxeter group. The corresponding Iwahori-Hecke algebra $H$ has the standard basis $(T\_w)\_\{w\backslash in\; W\}$, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by $\backslash overline\{T\_w\}:=T\_\{w^\{-1\}\}^\{-1\}$. This is a precanonical structure on $H$ that satisfies the sufficient condition above and the corresponding canonical basis of $H$ is the Kazhdan–Lusztig basis

: $C\_w\text{'}\; =\; \backslash sum\_\{y\backslash leq\; w\}\; P\_\{y,w\}(v^2)\; T\_w$

with $P\_\{y,w\}$ being the Kazhdan–Lusztig polynomials.

If we are given an n × n matrix $A$ and wish to find a matrix $J$ in Jordan normal form, similar to $A$, 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 $D$ 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 $A$ possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If $\backslash lambda$ is an eigenvalue of $A$ of algebraic multiplicity $\backslash mu$, then $A$ will have $\backslash mu$ linearly independent generalized eigenvectors corresponding to $\backslash lambda$.

For any given n × n matrix $A$, 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 $A$ 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 $\backslash mathbf\; x\_\{m-1\},\; \backslash mathbf\; x\_\{m-2\},\; \backslash ldots\; ,\; \backslash mathbf\; x\_1$ that are in the Jordan chain generated by $\backslash mathbf\; x\_m$ are also in the canonical basis.{{harvtxt|Bronson|1970|pp=196,197}}

Let $\backslash lambda\_i$ be an eigenvalue of $A$ of algebraic multiplicity $\backslash mu\_i$. First, find the ranks (matrix ranks) of the matrices $(A\; -\; \backslash lambda\_i\; I),\; (A\; -\; \backslash lambda\_i\; I)^2,\; \backslash ldots\; ,\; (A\; -\; \backslash lambda\_i\; I)^\{m\_i\}$. The integer $m\_i$ is determined to be the first integer for which $(A\; -\; \backslash lambda\_i\; I)^\{m\_i\}$ has rank $n\; -\; \backslash mu\_i$ (n being the number of rows or columns of $A$, that is, $A$ is n × n).

Now define

:$\backslash rho\_k\; =\; \backslash operatorname\{rank\}(A\; -\; \backslash lambda\_i\; I)^\{k-1\}\; -\; \backslash operatorname\{rank\}(A\; -\; \backslash lambda\_i\; I)^k\; \backslash qquad\; (k\; =\; 1,\; 2,\; \backslash ldots\; ,\; m\_i).$

The variable $\backslash rho\_k$ designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue $\backslash lambda\_i$ that will appear in a canonical basis for $A$. Note that

:$\backslash operatorname\{rank\}(A\; -\; \backslash lambda\_i\; I)^0\; =\; \backslash operatorname\{rank\}(I)\; =\; n\; .$

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).{{harvtxt|Bronson|1970|pp=197,198}}

This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.{{harvtxt|Nering|1970|pp=122,123}}

The matrix

:$A\; =\; \backslash begin\{pmatrix\}$

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}

has eigenvalues $\backslash lambda\_1\; =\; 4$ and $\backslash lambda\_2\; =\; 5$ with algebraic multiplicities $\backslash mu\_1\; =\; 4$ and $\backslash mu\_2\; =\; 2$, but geometric multiplicities $\backslash gamma\_1\; =\; 1$ and $\backslash gamma\_2\; =\; 1$.

For $\backslash lambda\_1\; =\; 4,$ we have $n\; -\; \backslash mu\_1\; =\; 6\; -\; 4\; =\; 2,$

:$(A\; -\; 4I)$ has rank 5,

:$(A\; -\; 4I)^2$ has rank 4,

:$(A\; -\; 4I)^3$ has rank 3,

:$(A\; -\; 4I)^4$ has rank 2.

Therefore $m\_1\; =\; 4.$

:$\backslash rho\_4\; =\; \backslash operatorname\{rank\}(A\; -\; 4I)^3\; -\; \backslash operatorname\{rank\}(A\; -\; 4I)^4\; =\; 3\; -\; 2\; =\; 1,$

:$\backslash rho\_3\; =\; \backslash operatorname\{rank\}(A\; -\; 4I)^2\; -\; \backslash operatorname\{rank\}(A\; -\; 4I)^3\; =\; 4\; -\; 3\; =\; 1,$

:$\backslash rho\_2\; =\; \backslash operatorname\{rank\}(A\; -\; 4I)^1\; -\; \backslash operatorname\{rank\}(A\; -\; 4I)^2\; =\; 5\; -\; 4\; =\; 1,$

:$\backslash rho\_1\; =\; \backslash operatorname\{rank\}(A\; -\; 4I)^0\; -\; \backslash operatorname\{rank\}(A\; -\; 4I)^1\; =\; 6\; -\; 5\; =\; 1.$

Thus, a canonical basis for $A$ will have, corresponding to $\backslash lambda\_1\; =\; 4,$ one generalized eigenvector each of ranks 4, 3, 2 and 1.

For $\backslash lambda\_2\; =\; 5,$ we have $n\; -\; \backslash mu\_2\; =\; 6\; -\; 2\; =\; 4,$

:$(A\; -\; 5I)$ has rank 5,

:$(A\; -\; 5I)^2$ has rank 4.

Therefore $m\_2\; =\; 2.$

:$\backslash rho\_2\; =\; \backslash operatorname\{rank\}(A\; -\; 5I)^1\; -\; \backslash operatorname\{rank\}(A\; -\; 5I)^2\; =\; 5\; -\; 4\; =\; 1,$

:$\backslash rho\_1\; =\; \backslash operatorname\{rank\}(A\; -\; 5I)^0\; -\; \backslash operatorname\{rank\}(A\; -\; 5I)^1\; =\; 6\; -\; 5\; =\; 1.$

Thus, a canonical basis for $A$ will have, corresponding to $\backslash lambda\_2\; =\; 5,$ one generalized eigenvector each of ranks 2 and 1.

A canonical basis for $A$ is

:$$

\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\}.

$\backslash mathbf\; x\_1$ is the ordinary eigenvector associated with $\backslash lambda\_1$.

$\backslash mathbf\; x\_2,\; \backslash mathbf\; x\_3$ and $\backslash mathbf\; x\_4$ are generalized eigenvectors associated with $\backslash lambda\_1$.

$\backslash mathbf\; y\_1$ is the ordinary eigenvector associated with $\backslash lambda\_2$.

$\backslash mathbf\; y\_2$ is a generalized eigenvector associated with $\backslash lambda\_2$.

A matrix $J$ in Jordan normal form, similar to $A$ is obtained as follows:

:$$

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},

:$J\; =\; \backslash begin\{pmatrix\}$

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},

where the matrix $M$ is a generalized modal matrix for $A$ and $AM\; =\; MJ$.{{harvtxt|Bronson|1970|p=203}}

• Canonical form
• Change of basis
• Normal basis
• Normal form (disambiguation)
• Polynomial basis

• {{ citation | first1 = Richard | last1 = Bronson | year = 1970 | lccn = 70097490 | title = Matrix Methods: An Introduction | publisher = Academic Press | location = New York }}
• {{Citation |last1 = Deng |first1 = Bangming |last2 = Ju |first2 = Jie |last3 = Parshall |first3 = Brian |last4 = Wang |first4 = Jianpan |title = Finite Dimensional Algebras and Quantum Groups |url = https://books.google.com/books?id=k9aUvxN2w2MC |publisher = American Mathematical Society |location = Providence, R.I. |series = Mathematical surveys and monographs |isbn = 9780821875315 |year = 2008 |volume = 150 }}
• {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | title=Crystalizing the q-analogue of universal enveloping algebras | url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-133/issue-2/Crystalizing-the-q-analogue-of-universal-enveloping-algebras/cmp/1104201397.full | mr=1090425 | year=1990 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=133 | issue=2 | pages=249–260 | doi=10.1007/bf02097367| bibcode=1990CMaPh.133..249K | s2cid=121695684 | url-access=subscription }}
• {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | title=On crystal bases of the q-analogue of universal enveloping algebras | url=https://doi.org/10.1215/S0012-7094-91-06321-0 | mr=1115118 | year=1991 | journal=Duke Mathematical Journal | issn=0012-7094 | volume=63 | issue=2 | pages=465–516 | doi=10.1215/S0012-7094-91-06321-0 | url-access=subscription }}
• {{Citation | last1=Lusztig | first1=George | author1-link=George Lusztig | title=Canonical bases arising from quantized enveloping algebras | doi=10.2307/1990961 | mr=1035415 | year=1990 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=3 | issue=2 | pages=447–498| jstor=1990961 | doi-access=free }}
• {{Citation | last1=Lusztig | first1=George | author1-link=George Lusztig | title=Quivers, perverse sheaves and quantized enveloping algebras | doi=10.2307/2939279 | mr=1088333 | year=1991 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=4 | issue=2 | pages=365–421| jstor=2939279 | doi-access=free }}
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• {{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = Wiley | location = New York | lccn = 76091646 }}

Category:Linear algebra

Category:Abstract algebra

Category:Lie algebras

Category:Representation theory

Category:Quantum groups