MasseyProduct - Maple Help

LieAlgebras[MasseyProduct] - calculate the Massey product of a pair of forms

Calling Sequences

MasseyProduct(${\mathbf{α}},$${\mathbf{β}})$

Parameters

$\mathrm{α}$    - a $p$-form defined on a Lie algebra $\mathrm{𝔤}$ with coefficients in $\mathrm{𝔤}$

$\mathrm{β}$     - a $q$-form defined on a Lie algebra $\mathrm{𝔤}$ with coefficients in $\mathrm{𝔤}$

Description

 • The Massey product of a pair of forms  and is the 3-form  defined by

.

In general, if  and then the Massey product is the form defined by

+ cyclic permutations.

 • The Massey product plays an important role in the construction of the deformations of a Lie algebra.

Examples

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

Example 1.

First initialize a Lie algebra from a list of structure equations.

 > $\mathrm{StrEq}≔\left[\left[\mathrm{x2},\mathrm{x3}\right]=\mathrm{x1},\left[\mathrm{x2},\mathrm{x5}\right]=\mathrm{x3},\left[\mathrm{x4},\mathrm{x5}\right]=\mathrm{x4}\right]$
 ${\mathrm{StrEq}}{:=}\left[\left[{\mathrm{x2}}{,}{\mathrm{x3}}\right]{=}{\mathrm{x1}}{,}\left[{\mathrm{x2}}{,}{\mathrm{x5}}\right]{=}{\mathrm{x3}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x5}}\right]{=}{\mathrm{x4}}\right]$ (2.1)
 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\mathrm{StrEq},\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}\right],\mathrm{alg}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.2)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.3)

We define the adjoint representation and use this to construct the corresponding Lie algebra with coefficients.

 alg > $\mathrm{DGsetup}\left(\left[\mathrm{w1},\mathrm{w2},\mathrm{w3},\mathrm{w4},\mathrm{w5}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.4)
 alg > $\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{alg},V,\mathrm{Adjoint}\left(\mathrm{alg}\right)\right):$
 alg > $\mathrm{DGsetup}\left(\mathrm{alg},\mathrm{\rho },\mathrm{algV}\right)$
 ${\mathrm{Lie algebra with coefficients: algV}}$ (2.5)

Here is a pair of 2-forms on and their Massey product.

 alg > $\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{w1}\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ2}\right)$
 ${\mathrm{α}}{:=}{\mathrm{w1}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$ (2.6)
 algV > $\mathrm{\beta }≔\mathrm{evalDG}\left(\mathrm{w2}\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ4}\right)$
 ${\mathrm{β}}{:=}{\mathrm{w2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ4}}$ (2.7)
 algV > $\mathrm{MasseyProduct}\left(\mathrm{\alpha },\mathrm{\beta }\right)$
 ${\mathrm{w2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ4}}$ (2.8)

Here is a pair of 3-forms on and their Massey product.

 algV > $\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{w1}\left(\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ3}\right)$
 ${\mathrm{α}}{:=}{\mathrm{w1}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.9)
 algV > $\mathrm{\beta }≔\mathrm{evalDG}\left(\mathrm{w4}\left(\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ4}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ5}\right)$
 ${\mathrm{β}}{:=}{\mathrm{w4}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}$ (2.10)
 algV > $\mathrm{MasseyProduct}\left(\mathrm{\alpha },\mathrm{\beta }\right)$
 ${\mathrm{w4}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}$ (2.11)