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Details for SimpleLieAlgebraData - definitions of the classical matrix algebras

Description

The following two tables describe the Lie algebras which can be initialized with the command SimpleLieAlgebraData .

The Classical Simple Real Matrix Algebras

 Name Dim Type Rank Matrices Conditions Examples sl(n) ${n}^{2}-1$ A $\left[A\right]$ Example 1. $\mathrm{su}\left(n\right)$ A $n-1$ Example 2 $\mathrm{su}\left(p,q\right)$ version 1 A $n-1$ ${Z}_{1}$,  arbitrary, ${Z}_{2}+{Z}_{2}^{†}=0,$ skew-symmetric symmetric Example 3 version 2 A $n-1$ $,$ arbitrary Example 3 ${\mathrm{su}}^{*}\left(n\right)$ A $n-1$ Example 4 $\mathrm{so}\left(p,q\right)$ version 1 B $m$ $\left[\begin{array}{rrr}A& B& C\\ \mathrm{D}& -{A}^{t}& E\\ -{E}^{}& -{C}^{t}& F\end{array}\right]$ arbitrary , Example 5 $\mathrm{so}\left(p,q\right)$ version 2 B $m$ $\left[\begin{array}{rr}A& B\\ -{B}^{t}& C\end{array}\right]$ A arbitrary Example 5 C $m$ $\left[\begin{array}{rr}A& B\\ C& -{A}^{t}\end{array}\right]$ Example 6 C $m$ arbitrary  = 0, , , , Example 7 $\mathrm{sp}\left(n\right)$ $n\left(n+1\right)$ C $m$ Example 7  version 1 D $m$ $\left[\begin{array}{rrr}A& B& C\\ \mathrm{D}& -{A}^{t}& E\\ -{E}^{}& -{C}^{t}& F\end{array}\right]$ arbitrary , Example 8 version 2 D $m$ $\left[\begin{array}{rr}A& B\\ -{B}^{t}& C\end{array}\right]$ A arbitrary Example 9 ${\mathrm{so}}^{*}\left(n\right)$ D $m$ , Example 10

The following algebras can also be initialized with the command SimpleLieAlgebraData .

Other Classical Real Matrix Algebras

 Name Dim Matrices Conditions Examples $\mathrm{gl}\left(n,\mathrm{ℝ}\right)$ $\left[\begin{array}{r}A\end{array}\right]$ Z, Example 11 Z, Example 11 $\mathrm{sl}\left(n,\mathrm{ℂ}\right)$ Z, Z, trace-free Example 12 ${n}^{2}$ arbitrary Example 13 $\mathrm{so}\left(n,\mathrm{ℂ}\right)$ Z, Z, skew-symmetric Example 14  Example 15 $\mathrm{sol}\left(n\right)$ $\frac{1}{2}n\left(n+1\right)$ $\left[\begin{array}{r}A\end{array}\right]$ upper triangular Example 16 $\mathrm{nil}\left(n\right)$ $\frac{1}{2}n\left(n-1\right)$ $\left[\begin{array}{r}A\end{array}\right]$ strictly upper triangular Example 17

Examples

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

Example 1. $\mathbit{sl}\left(\mathbit{n}\right)$

 > $\mathrm{LD1}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(3\right)",\mathrm{sl3}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD1}\right)$

Example 2. $\mathbit{su}\left(\mathbit{n}\right)$

 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("su\left(3\right)",\mathrm{su3}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD2}\right)$

Example 3.

 > $\mathrm{LD3I}≔\mathrm{SimpleLieAlgebraData}\left("su\left(3,1\right)",\mathrm{su31I}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD3I}\right)$
 > $\mathrm{LD3II}≔\mathrm{SimpleLieAlgebraData}\left("su\left(3,1\right)",\mathrm{su31II},\mathrm{version}=2\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD3II}\right)$

Example 4. ${\mathbit{su}}^{\mathbf{*}}\left(\mathbit{n}\right)$

 > $\mathrm{LD4}≔\mathrm{SimpleLieAlgebraData}\left("su*\left(4\right)",\mathrm{sus4}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD4}\right)$

Example 5.

 > $\mathrm{LD5I}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3,2\right)",\mathrm{su32I}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD5I}\right)$
 > $\mathrm{LD5II}≔\mathrm{SimpleLieAlgebraData}\left("su\left(3,2\right)",\mathrm{su32II},\mathrm{version}=2\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD3II}\right)$

Example 6.

 > $\mathrm{LD6}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, R\right)",\mathrm{sp4R}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD6}\right)$

Example 7.

 > $\mathrm{LD7}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(2, 2\right)",\mathrm{sp22}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD7}\right)$

Example 8. $\mathbit{sp}\left(\mathbit{n}\right)$

 > $\mathrm{LD8}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4\right)",\mathrm{sp4}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD8}\right)$

Example 9. $\mathbit{so}\left(\mathbit{p}\mathbf{,}\mathbit{q}\right)$

 > $\mathrm{LD9I}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3,1\right)",\mathrm{so31}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD9I}\right)$
 > $\mathrm{LD9II}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3,1\right)",\mathrm{so31},\mathrm{version}=2\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD9II}\right)$

Example 10. ${\mathbit{so}}^{\mathbf{*}}\left(\mathbit{n}\right)$

 > $\mathrm{LD10}≔\mathrm{SimpleLieAlgebraData}\left("so*\left(4\right)",\mathrm{sos}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD10}\right)$

Example 11.

 > $\mathrm{LD11R}≔\mathrm{SimpleLieAlgebraData}\left("gl\left(2, R\right)",\mathrm{gl2R}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD11R}\right)$
 > $\mathrm{LD11C}≔\mathrm{SimpleLieAlgebraData}\left("gl\left(2, C\right)",\mathrm{gl2C}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD11C}\right)$

Example 12.

 > $\mathrm{LD12}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(2, C\right)",\mathrm{sl2C}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD12}\right)$

Example 13.

 > $\mathrm{LD13}≔\mathrm{SimpleLieAlgebraData}\left("u\left(2,1\right)",\mathrm{u21}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD13}\right)$

Example 14.

 > $\mathrm{LD14}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3, C\right)",\mathrm{so3C}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD14}\right)$

Example 15.

 > $\mathrm{LD15}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, C\right)",\mathrm{sp4C}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD15}\right)$

Example 16. $\mathbit{sol}\left(\mathbit{n}\right)$

 > $\mathrm{LD16}≔\mathrm{SimpleLieAlgebraData}\left("sol\left(4\right)",\mathrm{sol4}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD16}\right)$

Example 17. $\mathbit{nil}\left(\mathbit{n}\right)$

 > $\mathrm{LD17}≔\mathrm{SimpleLieAlgebraData}\left("nil\left(4\right)",\mathrm{nil4}\right):$
 > $\mathrm{StandardRepresentation}\left(\mathrm{LD17}\right)$