compute relative Lie algebra cohomology with coefficients in a representation - Maple Programming Help

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LieAlgebras[Cohomology] -  compute  relative Lie algebra cohomology with coefficients in a representation

LieAlgebras[RelativeChains] - find the vector space of forms on a Lie algebra relative to a given subalgebra

LieAlgebras[ CohomologyDecomposition] -  decompose a closed form into the sum of an exact form and a form defining a cohomology class

Calling Sequences

RelativeChains(h)

Cohomology(C)

CohomologyDecomposition(, H, h)

CohomologyDecomposition(,  H, R)

Parameters

h         - a list of vectors in a Lie algebra defining a subalgebra

C         - a list of lists where is a list of $k$-forms

$\mathrm{α}$         - a $\mathrm{𝔥}$ $-$relative, closed $p-$form on $\mathrm{𝔤}$

H         - a list of closed $p$-forms on $\mathrm{𝔤}$ defining the basis for the (relative) cohomology of in degree $p$

R                 - a list of ($p$-1)-forms on defining the basis for the relative chains of in degree $p-1$

Description

 • Let be a $n$-dimensional (real) Lie algebra. Let be the dual space of (the space of 1-forms on $\mathrm{𝔤}\mathit{)}$. When initializing a Lie algebra with DGsetup, the default labelling is  for the basis vectors and for the 1-forms. Denote by ${\mathrm{Λ}}^{p}\left({\mathrm{𝔤}}^{*}\right)$ the $p-$forms on : these are the alternating mult-linear maps . Let be a representation ofIf is a basis for let ${x}_{\mathrm{β}}$ and denote by the $p-$forms on  with coefficients in $V$. These are the alternating mult-linear maps . Any form can be written as

The exterior derivative  is defined by the rules  and . If  is a subalgebra of $\mathrm{𝔤}$, then the space of $\mathrm{𝔥}\mathit{-}$relative $p-$forms on with coefficients in $V$ is

and  for all $}$ .

A $p$-form is closed if and exact if there a -form such that . The $\mathrm{𝔥}\mathit{-}$relative ,$p$-dimensional Lie algebra cohomology of $\mathrm{𝔤}$ with coefficients in the representation $V$ is the space of closed $p$ -forms module the exact $p$-forms, that is,

.

The cohomology ${H}^{p}\left(\mathrm{𝔤}\right)$ of $\mathrm{𝔤}$, the relative Lie algebra cohomology and the cohomology of with coefficients in a represention all play an important role in Lie theory, in the differential geometry and topology of homogeneous spaces and in the Cartan equivalence method. The text by D. B. Fuks (Chapter 1) and the papers by Hochschild and Koszul contain the basic material on Lie Algebra cohomology. Also, see the help pages Deformation, Extensions, KostantCodifferential.

 • The LieAlgebra package currently contains 3 commands: RelativeChains, Cohomology, and CohomologyDecomposition for finding Lie algebra cohomology.
 • The command RelativeChains(h) returns a list of all relative chains .
 • The command Cohomology(C) computes the cohomology of the sequence of forms . This requires that for all . If  is a list of forms onthen Cohomology(C) returns a list where is a basis for the cohomology in ${C}_{k}$.
 • The command CohomologyDecomposition(alpha, H, h) returns a pair of forms such that where is a linear combination of the cohomology representatives given by $H$ and where is a $\mathrm{𝔥}\mathit{-}$relative form The form $\mathrm{β}$ is uniquely determined, the form is not. In particular, if the closed form is exact, then .

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

First we initialize a Lie algebra.

 > L1 := _DG([["LieAlgebra", Alg1, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.1)
 > DGsetup(L1);
 ${\mathrm{Lie algebra: Alg1}}$ (2.2)

For this example we take h to be the trivial subspace. In this case the procedure RelativeChains simply returns a list of bases for the 1-forms on g, the 2-forms on g, the 3-forms on g, and so on.

 Alg1 > C := RelativeChains([]);
 $\left[\left[{}\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{2}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{2}{,}{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{2}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{2}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{1}{,}{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{2}{,}{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{5}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{}\right]\right]$ (2.3)

We pass the output of the RelativeChains program to the Cohomology program.

 Alg1 > H := Cohomology(C);
 $\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{}\right]\right]$ (2.4)

To read off the dimensions of the cohomology of g, use the nops and map command.

 Alg1 > map(nops, H);
 $\left[{2}{,}{2}{,}{2}{,}{1}{,}{0}\right]$ (2.5)

Example 2.

We continue with Example 1. To find the cohomology of just in degree 3, pass the Cohomology program to just the chains of degree 2 and 3 and 4.

 Alg1 > Cohomology(C[3..5]);
 $\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]\right]$ (2.6)

Example 3.

We continue with Example 1. Show that the 2-form  is closed and express $\mathrm{β}$ as a linear combination of the cohomology classes in ${H}^{2}$ and the exterior derivative of a 1-form.

 Alg1 > alpha := evalDG(theta4 &w theta5 - theta3 &w theta5 + 3*theta2 &wedge theta5 + 2*theta1 &w theta2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}\right]{,}\left[\left[{2}{,}{5}\right]{,}{3}\right]{,}\left[\left[{3}{,}{5}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)$ (2.7)
 Alg1 > ExteriorDerivative(alpha);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{0}\right]\right]\right]\right)$ (2.8)
 Alg1 > beta, delta := CohomologyDecomposition(alpha, H[2]);
 ${\mathrm{\beta }}{,}{\mathrm{\delta }}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}\right]{,}\left[\left[{3}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}{3}\right]{,}\left[\left[{4}\right]{,}{-}{1}\right]\right]\right]\right)$ (2.9)
 Alg1 > alpha &minus (beta &plus ExteriorDerivative(delta));
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right)$ (2.10)

Example 4.

First we initialize a Lie algebra.

 Alg1 > L2 := _DG([["LieAlgebra", Alg2, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]]);
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.11)
 Alg1 > DGsetup(L2);
 ${\mathrm{Lie algebra: Alg2}}$ (2.12)

Define a 2 dimensional subspace $h$ to be the vectors spanned by $S$..

 Alg2 > S := [e1, e2];
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Alg2}}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Alg2}}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.13)

Compute the relative chains with respect to the subspace $h$.

 Alg2 > C := RelativeChains(S);
 $\left[\left[{}\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{2}\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{3}\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right)\right]{,}\left[{}\right]\right]$ (2.14)
 Alg2 > H := Cohomology(C);
 $\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg2}}{,}{3}\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{-}{1}\right]\right]\right]\right)\right]\right]$ (2.15)

Example 5.

In this example we compute the cohomology of a 4-dimensional Lie algebra with coefficients in the adjoint representation. First define and initialize the Lie algebra.

 Rep1 > L3:= Library:-Retrieve("Winternitz", 1, [4,7], Alg3);
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.16)
 Rep1 > DGsetup(L3);
 ${\mathrm{Lie algebra: Alg3}}$ (2.17)

Define the representation space $V.$

 Alg3 > DGsetup([x1, x2, x3, x4], V);
 ${\mathrm{frame name: V}}$ (2.18)

 ${\mathrm{_DG}}{}\left(\left[\left[{"Representation"}{,}\left[\left[{\mathrm{Alg3}}{,}\right]\right]\right]\right]\right)$