Cohomology - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : Cohomology

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( $α$ , H, h)

CohomologyDecomposition( $α$ , H, R)

Parameters

h - a list of vectors in a Lie algebra $&gfr;$ defining a subalgebra $&hfr; \subset &gfr;$

C - a list of lists $C = [C_{p - 1}, C_{p}, C_{p + 1}, .. &period;, C_{q + 1}],$ where $C_{k}$ is a list of $k$ -forms

$α$ - a $&hfr;$ $-$ relative, closed $p -$ form on $&gfr;$

H - a list of closed $p$ -forms on $&gfr;$ defining the basis for the (relative) cohomology of $&gfr;$ in degree $p$

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

Description

Examples

Description

•

Let $&gfr;$ be a $n$ -dimensional (real) Lie algebra. Let ${&gfr;}^{&ast;}$ be the dual space of $&gfr;$ (the space of 1-forms on $&gfr;)$ . When initializing a Lie algebra with DGsetup, the default labelling is $\{e_{1}, e_{2}, .. &period;, e_{n}\}$ for the basis vectors and $\{θ^{1}, θ^{2}, .. &period;, θ^{n}\}$ for the 1-forms. Denote by $Λ^{p} ({&gfr;}^{&ast;})$ the $p -$ forms on $&gfr;$ : these are the alternating mult-linear maps $ω &colon; &gfr; \times &gfr; \cdot \cdot \cdot \times &gfr; \to &reals;$ . Let $ρ &colon; &gfr; \to gl (V)$ be a representation of $&gfr; &period;$ If $\{x_{1}, x_{2}, .. &period; x_{m}\}$ is a basis for $V,$ let $ρ (e_{i}) (x_{α}) = B_{α i}^{β}$ $x_{β}$ and denote by $Λ^{p} (𝔤^{&ast;}, V)$ the $p -$ forms on $&gfr;$ with coefficients in $V$ . These are the alternating mult-linear maps $ω &colon; &gfr; \times &gfr; \cdot \cdot \cdot \times &gfr; \to V$ . Any form $ω \in Λ^{p} (g^{&ast;}, V)$ can be written as

$ω = A_{i_{1} i_{2} \cdot \cdot \cdot i_{p}}^{α} x_{α} θ^{i_{1}} \land θ^{i_{2}} \land \cdot \cdot \cdot \land θ^{i_{p}} &period;$

The exterior derivative $d &colon; Λ^{p} (g^{&ast;}, V) \to Λ^{p + 1} (g^{&ast;}, V)$ is defined by the rules $d (θ^{i}) = - \frac{1}{2} C_{jk}^{i} θ^{j} \land θ^{k}$ and $d (x_{α}) = B_{α i}^{β} x_{β} θ^{i}$ . If $&hfr; \subset &gfr;$ is a subalgebra of $&gfr;$ , then the space of $&hfr; -$ relative $p -$ forms on $&gfr;$ with coefficients in $V$ is

$Λ^{p} ({&gfr;}^{&ast;}, &hfr;, V) = {ω \in Λ^{p} (g^{&ast;}, V) | ι_{y} (ω) = 0$ and $ι_{y} (dω) = 0$ for all $y \in &hfr;$ $}$ .

A $p$ -form $ω \in$ $Λ^{p} (𝔤^{&ast;}, 𝔥, V)$ is closed if $d (ω) = 0$ and exact if there a $(p - 1)$ -form $η \in Λ^{p - 1} (𝔤^{&ast;}, hfr, V)$ such that $ω = d (η)$ . The $&hfr; -$ relative , $p$ -dimensional Lie algebra cohomology of $&gfr;$ with coefficients in the representation $V$ is the space of closed $p$ -forms module the exact $p$ -forms, that is,

$H^{p} (𝔤, 𝔥, V) = \frac{{ω \in Λ^{p} (𝔤^{&ast;}, 𝔥, V) | d (ω) = 0}}{{ω \in Λ^{p} (𝔤^{&ast;}, 𝔥, V) | ω = d (η)}}$ .

The cohomology $H^{p} (&gfr;)$ of $&gfr;$ , the relative Lie algebra cohomology $H^{p} (&gfr;, &hfr;),$ and the cohomology $H^{p} (&gfr;, V)$ of $&gfr;$ 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 $C = [C_{1}, C_{2}, C_{3} .. &period;]$ of all relative chains $Λ^{p} (𝔤^{&ast;}, &hfr;, V)$ .

•

The command Cohomology(C) computes the cohomology of the sequence of forms $C = [C_{p - 1}, C_{p}, C_{p + 1}, .. &period;, C_{q + 1}]$ . This requires that $d (C_{i}) \subset C_{i + 1}$ for all $i = p - 1, p, p + 1, .. &period;, q$ . If $C_{p - 1}$ is a list of $p - 1$ forms on $&gfr;,$ then Cohomology(C) returns a list $H = [H_{p}, H_{p + 1}, .. &period;, H_{q}],$ where $H_{k}$ is a basis for the cohomology in $C_{k}$ .

•

The command CohomologyDecomposition(alpha, H, h) returns a pair of forms $β, δ$ such that $α = β + d (δ),$ where $β$ is a linear combination of the cohomology representatives given by $H$ and where $δ$ is a $&hfr; -$ relative form The form $β$ is uniquely determined, the form $δ$ is not. In particular, if the closed form $α$ is exact, then $β = 0$ .

Examples

>	$with (DifferentialGeometry) &colon;$ $with (LieAlgebras) &colon;$

Example 1.

First we initialize a Lie algebra.

>	$L1 ≔_DG ([[LieAlgebra, Alg1, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]])$

$L1 ≔ [[e2, e3] = e1, [e2, e5] = e3, [e4, e5] = e4]$

(2.1)

>	$DGsetup (L1)$

$Lie algebra: Alg1$

(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 >

$C ≔ RelativeChains ([])$

$C ≔ [[], [θ1, θ2, θ3, θ4, θ5], [θ1 ⋀ θ2, θ1 ⋀ θ3, θ1 ⋀ θ4, θ1 ⋀ θ5, θ2 ⋀ θ3, θ2 ⋀ θ4, θ2 ⋀ θ5, θ3 ⋀ θ4, θ3 ⋀ θ5, θ4 ⋀ θ5], [θ1 ⋀ θ2 ⋀ θ3, θ1 ⋀ θ2 ⋀ θ4, θ1 ⋀ θ2 ⋀ θ5, θ1 ⋀ θ3 ⋀ θ4, θ1 ⋀ θ3 ⋀ θ5, θ1 ⋀ θ4 ⋀ θ5, θ2 ⋀ θ3 ⋀ θ4, θ2 ⋀ θ3 ⋀ θ5, θ2 ⋀ θ4 ⋀ θ5, θ3 ⋀ θ4 ⋀ θ5], [θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4, θ1 ⋀ θ2 ⋀ θ3 ⋀ θ5, θ1 ⋀ θ2 ⋀ θ4 ⋀ θ5, θ1 ⋀ θ3 ⋀ θ4 ⋀ θ5, θ2 ⋀ θ3 ⋀ θ4 ⋀ θ5], [θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4 ⋀ θ5], []]$

(2.3)

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

Alg1 >

$H ≔ Cohomology (C)$

$H ≔ [[θ5, θ2], [θ1 ⋀ θ2, θ3 ⋀ θ5], [θ1 ⋀ θ2 ⋀ θ3, θ1 ⋀ θ3 ⋀ θ5], [θ1 ⋀ θ2 ⋀ θ3 ⋀ θ5], []]$

(2.4)

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

Alg1 >

$map (nops, H)$

$[2, 2, 2, 1, 0]$

(2.5)

Example 2.

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

Alg1 >

$Cohomology (C [3 .. 5])$

$[[θ1 ⋀ θ2 ⋀ θ3, θ1 ⋀ θ3 ⋀ θ5]]$

(2.6)

Example 3.

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

Alg1 >

$α ≔ evalDG (θ4 &w θ5 - θ3 &w θ5 + 3 θ2 &wedge θ5 + 2 θ1 &w θ2)$

$α ≔ 2 θ1 ⋀ θ2 + 3 θ2 ⋀ θ5 - θ3 ⋀ θ5 + θ4 ⋀ θ5$

(2.7)

Alg1 >

$ExteriorDerivative (α)$

$0 θ1 ⋀ θ2 ⋀ θ3$

(2.8)

Alg1 >

$β, δ ≔ CohomologyDecomposition (α, H [2])$

$β, δ ≔ 2 θ1 ⋀ θ2 - θ3 ⋀ θ5, - 3 θ3 - θ4$

(2.9)

Alg1 >

$α &minus (β &plus ExteriorDerivative (δ))$

$0 θ1 ⋀ θ2$

(2.10)

Example 4.

First we initialize a Lie algebra.

Alg1 >

$L2 ≔_DG ([[LieAlgebra, Alg2, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]])$

$L2 ≔ [[e2, e3] = e1, [e2, e5] = e3, [e4, e5] = e4]$

(2.11)

Alg1 >

$DGsetup (L2)$

$Lie algebra: Alg2$

(2.12)

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

Alg2 >

$S ≔ [e1, e2]$

(2.13)

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

Alg2 >

$C ≔ RelativeChains (S)$

$C ≔ [[], [θ4, θ5], [- θ3 ⋀ θ5, - θ4 ⋀ θ5], [- θ3 ⋀ θ4 ⋀ θ5], []]$

(2.14)

Alg2 >

$H ≔ Cohomology (C)$

$H ≔ [[θ5], [- θ3 ⋀ θ5], [- θ3 ⋀ θ4 ⋀ θ5]]$

(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 >

$L3 ≔ Library :- Retrieve (Winternitz, 1, [4, 7], Alg3)$

$L3 ≔ [[e1, e4] = 2 e1, [e2, e3] = e1, [e2, e4] = e2, [e3, e4] = e2 + e3]$

(2.16)

Rep1 >

$DGsetup (L3)$

$Lie algebra: Alg3$

(2.17)

Define the representation space $V &period;$

Alg3 >

$DGsetup ([x1, x2, x3, x4], V)$

$frame name: V$

(2.18)

Define the adjoint representation.

V >	$ρ ≔ Representation (Alg3, V, Adjoint (Alg3))$

$ρ ≔ [[e1, [\begin{array}{c} 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]], [e2, [\begin{array}{c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]], [e3, [\begin{array}{c} 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]], [e4, [\begin{array}{c} −2 & 0 & 0 & 0 \\ 0 & −1 & −1 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]]]$

(2.19)

Alg3 >

$DGsetup (Alg3, ρ, Rep1)$

$Lie algebra with coefficients: Rep1$

(2.20)

Note that the chains are now linear functions of the coordinates on the representation space.

Rep1 >

$C ≔ RelativeChains ([])$

$C ≔ [[x1, x2, x3, x4], [x1 θ1, x1 θ2, x1 θ3, x1 θ4, x2 θ1, x2 θ2, x2 θ3, x2 θ4, x3 θ1, x3 θ2, x3 θ3, x3 θ4, x4 θ1, x4 θ2, x4 θ3, x4 θ4], [x1 θ1 ⋀ θ2, x1 θ1 ⋀ θ3, x1 θ1 ⋀ θ4, x1 θ2 ⋀ θ3, x1 θ2 ⋀ θ4, x1 θ3 ⋀ θ4, x2 θ1 ⋀ θ2, x2 θ1 ⋀ θ3, x2 θ1 ⋀ θ4, x2 θ2 ⋀ θ3, x2 θ2 ⋀ θ4, x2 θ3 ⋀ θ4, x3 θ1 ⋀ θ2, x3 θ1 ⋀ θ3, x3 θ1 ⋀ θ4, x3 θ2 ⋀ θ3, x3 θ2 ⋀ θ4, x3 θ3 ⋀ θ4, x4 θ1 ⋀ θ2, x4 θ1 ⋀ θ3, x4 θ1 ⋀ θ4, x4 θ2 ⋀ θ3, x4 θ2 ⋀ θ4, x4 θ3 ⋀ θ4], [x1 θ1 ⋀ θ2 ⋀ θ3, x1 θ1 ⋀ θ2 ⋀ θ4, x1 θ1 ⋀ θ3 ⋀ θ4, x1 θ2 ⋀ θ3 ⋀ θ4, x2 θ1 ⋀ θ2 ⋀ θ3, x2 θ1 ⋀ θ2 ⋀ θ4, x2 θ1 ⋀ θ3 ⋀ θ4, x2 θ2 ⋀ θ3 ⋀ θ4, x3 θ1 ⋀ θ2 ⋀ θ3, x3 θ1 ⋀ θ2 ⋀ θ4, x3 θ1 ⋀ θ3 ⋀ θ4, x3 θ2 ⋀ θ3 ⋀ θ4, x4 θ1 ⋀ θ2 ⋀ θ3, x4 θ1 ⋀ θ2 ⋀ θ4, x4 θ1 ⋀ θ3 ⋀ θ4, x4 θ2 ⋀ θ3 ⋀ θ4], [x1 θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4, x2 θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4, x3 θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4, x4 θ1 ⋀ θ2 ⋀ θ3 ⋀ θ4], []]$

(2.21)

Rep1 >

$Cohomology (C)$

$[[- \frac{x1 θ1}{3} - \frac{x2 θ2}{6} + (- \frac{x3}{6} + x2) θ3], [x3 θ2 ⋀ θ4], [], []]$

(2.22)

Example 6.

Finally, we compute the Lie algebra cohomology of Alg3 with coefficients in the adjoint representation, relative to the subalgebra spanned bu $\{e1\} &period;$

Rep1 >

$C ≔ RelativeChains ([e1])$

$C ≔ [[x3, x2, x1], [x3 θ2, x3 θ3, x3 θ4, x2 θ2, x2 θ3, x2 θ4, x1 θ2, x1 θ3, x1 θ4], [- x3 θ3 ⋀ θ4, - x2 θ2 ⋀ θ3, - x2 θ2 ⋀ θ4, - x2 θ3 ⋀ θ4, - x1 θ2 ⋀ θ3, - x1 θ2 ⋀ θ4, - x1 θ3 ⋀ θ4, - x3 θ2 ⋀ θ3, - x3 θ2 ⋀ θ4], [- x3 θ2 ⋀ θ3 ⋀ θ4, - x2 θ2 ⋀ θ3 ⋀ θ4, - x1 θ2 ⋀ θ3 ⋀ θ4], []]$

(2.23)

Rep1 >

$Cohomology (C)$

$[[x2 θ3], [- x3 θ2 ⋀ θ4], []]$

(2.24)

Rep1 >

Maple

Add-ons Maple

Student Success Platform

Améliorer les taux de retention

MapleSim

Add-ons MapleSim

Ingénierie systèmes

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim

Maple

Logiciel de mathématiques puissant facile à utiliser

Add-ons Maple

Student Success Platform

Améliorer les taux de retention

MapleSim

Modélisation avancée au niveau système

Add-ons MapleSim

Ingénierie systèmes

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim