 Codifferential - Maple Help

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

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

Example 1.

First initialize a 5-dimensional Lie algebra.

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

Define a bi-vector and calculate its codifferential.

 alg > ${Z}{≔}{\mathrm{evalDG}}{}\left({a}{}{\mathrm{e2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e3}}{+}{b}{}{\mathrm{e2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e5}}{+}{c}{}{\mathrm{e2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e4}}\right)$
 ${Z}{:=}{a}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{+}{c}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e4}}{+}{b}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}$ (2.3)
 alg > ${\mathrm{Codifferential}}{}\left({Z}\right)$
 ${a}{}{\mathrm{e1}}{+}{b}{}{\mathrm{e3}}$ (2.4)

Define a tri-vector and calculate its codifferential.

 alg > ${Z}{≔}{\mathrm{evalDG}}{}\left({a}{}\left({\mathrm{e2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e3}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e4}}{+}{b}{}\left({\mathrm{e3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e4}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e5}}\right)$
 ${Z}{:=}{a}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}{+}{b}{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}{}{\bigwedge }{}{\mathrm{e5}}$ (2.5)
 alg > ${W}{≔}{\mathrm{Codifferential}}{}\left({Z}\right)$
 ${W}{:=}{a}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e4}}{-}{b}{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e4}}$ (2.6)

Check that

 alg > ${\mathrm{Codifferential}}{}\left({W}\right)$
 ${0}{}{\mathrm{e1}}$ (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.

 > ${\mathrm{LD2}}{≔}{\mathrm{SimpleLieAlgebraData}}{}\left({"so\left(4\right)"}{,}{\mathrm{so4}}\right)$
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (2.8)
 Alg1 > ${\mathrm{DGsetup}}{}\left({\mathrm{LD2}}\right)$
 ${\mathrm{Lie algebra: so4}}$ (2.9)
 so4 > ${A}{≔}{\mathrm{StandardRepresentation}}{}\left({\mathrm{so4}}\right)$
 ${A}{:=}\left[\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]$ (2.10)

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

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

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



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

Calculate the codifferential of a bi-vector.

 V > ${Z}{≔}{\mathrm{evalDG}}{}\left({\mathrm{w1}}{}{\mathrm{e1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e2}}\right)$
 ${Z}{:=}{\mathrm{w1}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}$ (2.14)
 so4V > ${\mathrm{Codifferential}}{}\left({Z}\right)$
 ${-}{\mathrm{w3}}{}{\mathrm{e1}}{+}{\mathrm{w2}}{}{\mathrm{e2}}{+}{\mathrm{w1}}{}{\mathrm{e4}}$ (2.15)

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

 so4V > ${Z}{≔}{\mathrm{evalDG}}{}\left({\mathrm{w4}}{}\left(\left({\mathrm{e1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e5}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{e6}}\right)$
 ${Z}{:=}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}{}{\bigwedge }{}{\mathrm{e6}}$ (2.16)
 so4V > ${W}{≔}{\mathrm{Codifferential}}{}\left({Z}\right)$
 ${W}{:=}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e4}}{+}{\mathrm{w3}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e5}}{-}{\mathrm{w2}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e6}}{-}{\mathrm{w4}}{}{\mathrm{e1}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e5}}{-}{\mathrm{w4}}{}{\mathrm{e2}}{}{\bigwedge }{}{\mathrm{e3}}{}{\bigwedge }{}{\mathrm{e6}}{+}{\mathrm{w4}}{}{\mathrm{e4}}{}{\bigwedge }{}{\mathrm{e5}}{}{\bigwedge }{}{\mathrm{e6}}$ (2.17)

Check that 

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