CoRoot - Maple Help

LieAlgebras[CoRoot] - find the coroot of a root vector for a semi-simple Lie algebra

Calling Sequences

CoRoot(${\mathbf{α}}$, CSA, option)

Parameters

$\mathrm{α}$        - a vector, defining a root vector for a semi-simple Lie algebra

CSA      - a list of $r$ vectors in a Lie algebra, defining a Cartan subalgebra

option   - an  non-singular matrix, defining the restriction of the Killing form to the Cartan subalgebra

Description

 • Let $\mathrm{𝔤}$ be a semi-simple Lie algebra, a Cartan subalgebra, and the associated set of roots. Let be the Killing form. If then the coroot of $\mathrm{α}$ is the unique vector such that. Let  be a basis for and with inverse. Then , where .
 • The calling sequence CoRoot(${\mathbf{α}}$, CSA) returns the vector ${T}_{\mathrm{α}}$.
 • In a situation involving repeated calls to CoRoot, efficiency can be dramatically improved by using the optional 3rd argument to specify the restriction of the Killing form.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

We use the command SimpleLieAlgebraData to retrieve the structure equations for the rank 3 Lie algebra we initialize this algebra, and we calculate the coroots of several root vectors.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(4\right)",\mathrm{sl4}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e15}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl4}}$ (2.2)

We obtain the Cartan subalgebra and the positive roots using SimpleLieAlgebraProperties

 M > $P≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$
 sl4 > $\mathrm{CSA}≔P\left["CartanSubalgebra"\right]$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.3)
 sl4 > $\mathrm{\Delta }≔P\left["PositiveRoots"\right]$

Calculate the coroot for the first root ${\mathrm{Δ}}_{1}$.

 sl4 > $\mathrm{\alpha }≔\mathrm{\Delta }\left[1\right]$
 sl4 > $\mathrm{CoRoot}\left(\mathrm{\alpha },\mathrm{CSA}\right)$
 $\frac{{1}}{{8}}{}{\mathrm{e1}}{-}\frac{{1}}{{8}}{}{\mathrm{e2}}$ (2.4)

Calculate the coroot for the last root ${\mathrm{Δ}}_{6}$.

 sl4 > $\mathrm{\beta }≔\mathrm{\Delta }\left[-1\right]$
 sl4 > $\mathrm{CoRoot}\left(\mathrm{\beta },\mathrm{CSA}\right)$
 $\frac{{1}}{{8}}{}{\mathrm{e1}}$ (2.5)

Example 2.

We repeat the calculation the first coroot from Example 1 using the optional calling sequence. The restriction of the Killing form to the Cartan subalgebra is needed.

 sl4 > $B≔\mathrm{Killing}\left(\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]\right)$
 sl4 > $\mathrm{CoRoot}\left(\mathrm{\alpha },\mathrm{CSA},B\right)$
 $\frac{{1}}{{8}}{}{\mathrm{e1}}{-}\frac{{1}}{{8}}{}{\mathrm{e2}}$ (2.6)

Here is the same computation in components.

 sl4 > $T≔{B}^{-1}·\mathrm{\alpha }$