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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

Examples

Description

• 

Let 𝔤  be a Lie algebra. The codifferential  of monomial bi-vectors and tri-vectors on 𝔤 is defined by

 x1  x2 = x1, x2   and   x1 x2  x3 = x1, x2 x3  x1 ,x2 x3  + x2, x3 x1  .

The formula for a general monomial multi-vector is

 x1  x2  xp &equals;i<j1i &plus;j &plus;1xi &comma; xj x1    xi   xj  xp

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 &rho;&colon; &gfr; glV be a representation of &gfr; on a vector space V&period; For x  &gfr; and w  V, write &rho;xw &equals; x w&period; For multi-vectors with coefficients in V, the above formulas for the codifferential are amended to

 

w x1  x2 &equals; x1wx2  x2 wx1 &plus; wx1&comma; x2,

 

w x1 x2  x3 &equals;x1&grave;wx2x3  x2 wx1x3  &plus; x3wx1x2 &plus; x1&comma; x2 x3  x1 &comma;x2 x3  &plus; x2&comma; x3 x1  and, in general,

w x1  x2  xp &equals;i&equals;1p1i &plus;1xi w x1    xi  xp  &plus; i<j1i &plus;j &plus;1xi &comma; xj x1    xi   xj  xp&period;

 

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 Z has degree p, then Z has degree  p1.

• 

The co-differential satisfies 2 &equals;0  It commutes with the Lie derivative Z and satisfies, for any vector X, &Zscr;XZ &equals; X Z &plus;  X Z &period;

Examples

withDifferentialGeometry&colon;withLieAlgebras&colon;

 

Example 1.

 

First initialize a 5-dimensional Lie algebra.

LD1LieAlgebraDatax2&comma;x3&equals;x1&comma;x2&comma;x5&equals;x3&comma;x4&comma;x5&equals;x4&comma;x1&comma;x2&comma;x3&comma;x4&comma;x5&comma;alg

LD1:=e2&comma;e3&equals;e1&comma;e2&comma;e5&equals;e3&comma;e4&comma;e5&equals;e4

(2.1)

DGsetupLD1

Lie algebra: alg

(2.2)

 

Define a bi-vector and calculate its codifferential.

alg > 

ZevalDGae2 &w e3&plus;be2 &w e5&plus;ce2 &w e4

Z:=ae2e3&plus;ce2e4&plus;be2e5

(2.3)
alg > 

CodifferentialZ

ae1&plus;be3

(2.4)

 

Define a tri-vector and calculate its codifferential.

alg > 

ZevalDGae2 &w e3 &w e4&plus;be3 &w e4 &w e5

Z:=ae2e3e4&plus;be3e4e5

(2.5)
alg > 

WCodifferentialZ

W:=ae1e4be3e4

(2.6)

 

Check that 2Z &equals;0. 

alg > 

CodifferentialW

0e1

(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 so4and its standard 4-dimensional representation. To create the computational environment we use the commands SimpleLieAlgebraData, StandardRepresentation and Representation.

 

LD2SimpleLieAlgebraDataso(4)&comma;so4

LD2:=e1&comma;e2&equals;e4&comma;e1&comma;e3&equals;e5&comma;e1&comma;e4&equals;e2&comma;e1&comma;e5&equals;e3&comma;e2&comma;e3&equals;e6&comma;e2&comma;e4&equals;e1&comma;e2&comma;e6&equals;e3&comma;e3&comma;e5&equals;e1&comma;e3&comma;e6&equals;e2&comma;e4&comma;e5&equals;e6&comma;e4&comma;e6&equals;e5&comma;e5&comma;e6&equals;e4

(2.8)
Alg1 > 

DGsetupLD2

Lie algebra: so4

(2.9)
so4 > 

AStandardRepresentationso4

A:=0100100000000000&comma;0010000010000000&comma;0001000000001000&comma;0000001001000000&comma;0000000100000100&comma;0000000000010010

(2.10)

 

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

so4 > 

DGsetupw1&comma;w2&comma;w3&comma;w4&comma;V

frame name: V

(2.11)
Alg1 > 

&rho;Representationso4&comma;V&comma;A

&rho;:=e1&comma;0100100000000000&comma;e2&comma;0010000010000000&comma;e3&comma;0001000000001000&comma;e4&comma;0000001001000000&comma;e5&comma;0000000100000100&comma;e6&comma;0000000000010010

(2.12)

 

Initialize the Lie algebra so4 with coefficients in the standard representation.

V > 

DGsetupso4&comma;&rho;&comma;so4V

Lie algebra with coefficients: so4V

(2.13)

 

Calculate the codifferential of a bi-vector.

V > 

ZevalDGw1e1 &w e2

Z:=w1e1e2

(2.14)
so4V > 

CodifferentialZ

w3e1&plus;w2e2&plus;w1e4

(2.15)

 

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

so4V > 

ZevalDGw4e1 &w e2 &w e5 &w e6

Z:=w4e1e2e5e6

(2.16)
so4V > 

WCodifferentialZ

W:=w4e1e2e4&plus;w3e1e2e5w2e1e2e6w4e1e3e5w4e2e3e6&plus;w4e4e5e6

(2.17)

 

Check that 2Z &equals;0.  

so4V > 

CodifferentialW

0e1e2

(2.18)

See Also

DifferentialGeometry

LieAlgebras

Adjoint

ExteriorDerivative

Representation

SimpleLieAlgebraData

StandardRepresentation

 


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