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LieAlgebras[Codifferential] - calculate the codifferential of a multi-vector defined on a Lie algebra with coefficients in a representation

Calling Sequences

Codifferential(Z)

Parameters

Z     - a multi-vector defined on a Lie algebra, or on a Lie algebra with coefficients in a representation $V$

Description

 • Let  be a Lie algebra. The codifferential of monomial bi-vectors and tri-vectors on $\mathrm{𝔤}$ is defined by

and .

The formula for a general monomial multi-vector is

where the barred vectors are omitted from the wedge product. A general multi-vector of degree $p$ is a superposition of monomials of degree $p$. The definition of the codifferential is extended to all multi-vectors by linearity.

 • Let be a representation of $\mathrm{𝔤}$ on a vector space $V.$ For  and , write  For multi-vectors with coefficients in $V$, the above formulas for the codifferential are amended to

,

and, in general,

Again, these definitions are extended to all multi-vectors by linearity.

 • The command Codifferential computes the codifferential of a multi-vector $Z$. Note that if has degree $p$, then has degree
 • The co-differential satisfies It commutes with the Lie derivative Z and satisfies, for any vector $X$,

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

First initialize a 5-dimensional Lie algebra.

 > LD1 := LieAlgebraData([[x2, x3] = x1, [x2, x5] = x3, [x4, x5] = x4], [x1, x2, x3, x4, x5], alg);
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.1)
 > DGsetup(LD1);
 ${\mathrm{Lie algebra: alg}}$ (2.2)

Define a bi-vector and calculate its codifferential.

 alg > Z := evalDG(a*e2 &w e3 + b*e2 &w e5 + c*e2 &w e4);
 ${Z}{:=}{a}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{+}{c}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e4}}{+}{b}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}$ (2.3)
 alg > Codifferential(Z);
 ${a}{}{\mathrm{e1}}{+}{b}{}{\mathrm{e3}}$ (2.4)

Define a tri-vector and calculate its codifferential.

 alg > Z := evalDG(a*e2 &w e3 &w e4 + b*e3 &w e4 &w e5);
 ${Z}{:=}{a}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}{+}{b}{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}{}{\bigwedge }{}{\mathrm{e5}}$ (2.5)
 alg > W := Codifferential(Z);
 ${W}{:=}{a}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e4}}{-}{b}{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}$ (2.6)

Check that

 alg > Codifferential(W);
 ${0}{}{\mathrm{e1}}$ (2.7)

Example 2.

In this example we calculate the codifferentials for some multi-vectors defined on a Lie algebra with coefficients in a representation. For this example we shall use the Lie algebra $\mathrm{so}\left(4\right)$and its standard 4-dimensional representation. To create the computational environment we use the commands SimpleLieAlgebraData, StandardRepresentation and Representation.

 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (2.8)
 Alg1 > DGsetup(LD2);
 ${\mathrm{Lie algebra: so4}}$ (2.9)
 so4 > A := StandardRepresentation(so4);
 ${A}{:=}\left[\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]$ (2.10)

Create a 4-dimensional vector space to serve as the representation space.

 so4 > DGsetup([w1, w2, w3, w4], V);
 ${\mathrm{frame name: V}}$ (2.11)
 Alg1 > rho := Representation(so4, V, A);
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]\right]$ (2.12)

Initialize the Lie algebra $\mathrm{so4}$ with coefficients in the standard representation.



 V > DGsetup(so4, rho, so4V);
 ${\mathrm{Lie algebra with coefficients: so4V}}$ (2.13)

Calculate the codifferential of a bi-vector.

 V > Z := evalDG(w1*e1 &w e2);
 ${Z}{:=}{\mathrm{w1}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}$ (2.14)
 so4V > Codifferential(Z);
 ${-}{\mathrm{w3}}{}{\mathrm{e1}}{+}{\mathrm{w2}}{}{\mathrm{e2}}{+}{\mathrm{w1}}{}{\mathrm{e4}}$ (2.15)

Calculate the codifferential of a multi-vector of degree 4.

 so4V > Z := evalDG(w4*e1 &w e2 &w e5 &w e6);
 ${Z}{:=}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}{}{\bigwedge }{}{\mathrm{e6}}$ (2.16)
 so4V > W := Codifferential(Z);
 ${W}{:=}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e4}}{+}{\mathrm{w3}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}{-}{\mathrm{w2}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e6}}{-}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e5}}{-}{\mathrm{w4}}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e6}}{+}{\mathrm{w4}}{}{\mathrm{e4}}{}{\bigwedge }{}{\mathrm{e5}}{}{\bigwedge }{}{\mathrm{e6}}$ (2.17)

Check that 

 so4V > Codifferential(W);
 ${0}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}$ (2.18)