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LieAlgebras[LieAlgebraWithCoefficientsData] - calculate the structure equations for a Lie algebra with coefficients in a representation

Calling Sequences

LieAlgebraWithCoefficientsData(rho$,$algC)

LieAlgebraWithCoefficientsData(alg$,$VM, algC$)$

LieAlgebraWithCoefficientsData(algC)

Parameters

$\mathrm{ρ}$       - a representation of a Lie algebra

algC    - name or a string, the name to be assigned to the Lie algebra with coefficients

V       - name of the representation space used to define the Lie algebra with coefficients

M       - a list of square matrices which form a Lie algebra

Description

 • Let $V$ be a linear space with basis ; let be a Lie algebra with basis  and dual basis ${\mathrm{θ}}^{1}$,; and let  be a representation of $\mathrm{𝔤}$. The representation defines the multiplication ${e}_{i}\cdot {x}^{a}$ = $\mathrm{ρ}\left({e}_{i}\right)\left({x}^{a}\right).$ Let be the vector space of $p$-forms with coefficients in the representation space $V$. A form if for all vectors ,  For example, the general 1-formand 2-form with coefficients in $V$ can be written as sums

and  ,

where the coefficients and are constants. The spaces play an important role in a number of constructions in Lie theory (See, for example, Cohomology, Deformation, MasseyProduct, KostantLaplacian). To work with forms defined on Lie algebras with coefficients in a representation, one first uses the commands LieAlgebraWithCoefficientsData and DGsetup -- in much the same way that one uses LieAlgebraData and AlgebraData to calculate the structure equations for a Lie algebra or a general non-commutative algebra.

 • The output of the LieAlgebraWithCoefficientsData is a data structure which can be passed to the command DGsetup.The structure equations are displayed.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We use the 6 dimensional Lie algebra and its standard representation by skew-symmetric matrices to illustrate the 3 calling sequences for LieAlgebraWithCoefficientsData. First, use the command SimpleLieAlgebraData to retrieve the structure equations for $\mathrm{so}\left(4\right)$.

 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (1)
 > DGsetup(LD);
 ${\mathrm{Lie algebra: so4}}$ (2)

Use the command StandardRepresentation to retrieve the matrices for the standard representation.

 so4 > M := StandardRepresentation(so4);
 ${M}{:=}\left[\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]$ (3)

Define a 4-dimensional representation space and the representation $\mathrm{ρ}.$

 so4 > DGsetup([x1, x2,x3,x4], V);
 ${\mathrm{frame name: V}}$ (4)
 V > rho := Representation(so4, V, M);
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]\right]$ (5)

Use the first calling sequence to calculate the structure equations for $\mathrm{so}\left(4\right)$with coefficients in the representation $\mathrm{ρ}$.

 so4 > LC1 := LieAlgebraWithCoefficientsData(rho, algC1);
 ${\mathrm{LC1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{x1}}{=}{\mathrm{x2}}{,}{\mathrm{e1}}{.}{\mathrm{x2}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e2}}{.}{\mathrm{x1}}{=}{\mathrm{x3}}{,}{\mathrm{e2}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e3}}{.}{\mathrm{x1}}{=}{\mathrm{x4}}{,}{\mathrm{e3}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e4}}{.}{\mathrm{x2}}{=}{\mathrm{x3}}{,}{\mathrm{e4}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e5}}{.}{\mathrm{x2}}{=}{\mathrm{x4}}{,}{\mathrm{e5}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e6}}{.}{\mathrm{x3}}{=}{\mathrm{x4}}{,}{\mathrm{e6}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x3}}\right]$ (6)

Initialize.

 so4 > DGsetup(LC1);
 ${\mathrm{Lie algebra with coefficients: algC1}}$ (7)

Here is a sample calculation using a 2-form form on $\mathrm{so4}$ with coefficients in $V$.

 algC > alpha := evalDG(x3*theta1 &w theta2);
 ${\mathrm{α}}{:=}{\mathrm{x3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$ (8)
 algC > ExteriorDerivative(alpha);
 ${-}{\mathrm{x2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ4}}{+}{\mathrm{x4}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{x3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{\mathrm{x3}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}$ (9)

The second calling sequence simply allows one to calculate the structure equations (6) directly from the matrices (3) without having to first define the representation $\mathrm{ρ}$.

 so4 > LieAlgebraWithCoefficientsData(so4, V, M, algC2);
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{x1}}{=}{\mathrm{x2}}{,}{\mathrm{e1}}{.}{\mathrm{x2}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e2}}{.}{\mathrm{x1}}{=}{\mathrm{x3}}{,}{\mathrm{e2}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e3}}{.}{\mathrm{x1}}{=}{\mathrm{x4}}{,}{\mathrm{e3}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e4}}{.}{\mathrm{x2}}{=}{\mathrm{x3}}{,}{\mathrm{e4}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e5}}{.}{\mathrm{x2}}{=}{\mathrm{x4}}{,}{\mathrm{e5}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e6}}{.}{\mathrm{x3}}{=}{\mathrm{x4}}{,}{\mathrm{e6}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x3}}\right]$ (10)

The third calling sequence retrieves the structure equations of a previously defined Lie algebra with coefficients in a representation.

 algC > LieAlgebraWithCoefficientsData(algC1);
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{x1}}{=}{\mathrm{x2}}{,}{\mathrm{e1}}{.}{\mathrm{x2}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e2}}{.}{\mathrm{x1}}{=}{\mathrm{x3}}{,}{\mathrm{e2}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e3}}{.}{\mathrm{x1}}{=}{\mathrm{x4}}{,}{\mathrm{e3}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x1}}{,}{\mathrm{e4}}{.}{\mathrm{x2}}{=}{\mathrm{x3}}{,}{\mathrm{e4}}{.}{\mathrm{x3}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e5}}{.}{\mathrm{x2}}{=}{\mathrm{x4}}{,}{\mathrm{e5}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x2}}{,}{\mathrm{e6}}{.}{\mathrm{x3}}{=}{\mathrm{x4}}{,}{\mathrm{e6}}{.}{\mathrm{x4}}{=}{-}{\mathrm{x3}}\right]$ (11)