Pensieve header: Testing the AwCalculus package.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO4"];
<< FreeLie.m
<< AwCalculus.m
$SeriesShowDegree=4

{λ1,λ2, λ3}=Table[⟨1->RandomLieSeries[{1,2}], 2 -> RandomLieSeries[{1,2}]⟩, {3}]

{ω1,ω2, ω3}=Table[RandomCWSeries[{1,2}], {3}]

(Subscript[ζ, a])=El[
⟨1->RandomLieSeries[{1,2}], 2 -> RandomLieSeries[{1,2}]⟩,
RandomCWSeries[{1,2}]
]

(Subscript[ζ, b])=El[
⟨3->RandomLieSeries[{3,4}], 4 -> RandomLieSeries[{3,4}]⟩,
RandomCWSeries[{3,4}]
]

(Subscript[ζ, c])=El[
⟨1->RandomLieSeries[{1,2}], 2 -> RandomLieSeries[{1,2}]⟩,
RandomCWSeries[{1,2}]
]

Testing Jacobi for tb

tb[λ1, λ1]

j1=tb[tb[λ1,λ2],λ3]

j2=tb[tb[λ2,λ3],λ1]

j3=tb[tb[λ3,λ1],λ2]

j1+j2+j3

Testing BCH

BCH[λ1, λ2, ⟨1,2⟩, tb]

la=BCH[BCH[λ1, λ2, ⟨1,2⟩, tb], λ3, ⟨1,2⟩, tb]

BCH[λ2, λ3, ⟨1,2⟩, tb]

ra=BCH[λ1, BCH[λ2, λ3, ⟨1,2⟩, tb], ⟨1,2⟩, tb]

la-ra

Testing Associativity for (Subscript[E, l])**(Subscript[E, l])

{ζ1=El[λ1, ω1], ζ2=El[λ2, ω2], ζ3=El[λ3, ω3]}

(E^(Subscript[D, λ1]))[ω2]

t1=ζ1**ζ2

la=t1**ζ3

t2=ζ2**ζ3

ra=ζ1**t2

Testing Associativity for (Subscript[E, s])**(Subscript[E, s])

{ζ1=Es[λ1, ω1], ζ2=Es[λ2, ω2], ζ3=Es[λ3, ω3]}

t1=ζ1**ζ2

la=t1**ζ3

t2=ζ2**ζ3

ζ1**t2

Testing the (Subscript[E, l]) ↔ (Subscript[E, s]) Conversions

ζ1

ζ1 // Λ

ζ1 // Λ // Γ

ζ1≡(ζ1 // Λ // Γ)

{λ1, λ1 // Λ, t1=λ1 // Λ // Γ, (λ1-t1)@{6}}

{λ2, λ2 // Γ, t2=λ2 // Γ // Λ, (λ2-t2)@{6}}

{t1=λ2 // CC[-λ1], t2=λ2 //(E^(Subscript[D, λ1])), t1-t2}

$SeriesCompareDegree=6;
{t1=λ2 // CC[Γ[-λ1]], t2=λ2 //(E^(Subscript[D, λ1])), t1-t2, t1≡t2}

$SeriesCompareDegree=8;
{t1=λ2 // CC[λ1], t2=λ2 //(E^(Subscript[D, -Λ[λ1]])), t1-t2, t1≡t2}

Testing the Cocycle Property of j

$SeriesShowDegree=6

div[λ1]

j[λ1]

ζ1=El[λ1, ω1]; ζ2=El[λ2, ω2]; λ12=First[ζ1**ζ2]

j[λ12]

t1=j[λ1]

t2=j[λ2] // (E^(Subscript[D, λ1]))

t1+t2

j[λ12]-t1-t2

Testing dA

$SeriesShowDegree=5;

(Subscript[ζ, a])

(Subscript[ζ, a]) // dA

(Subscript[ζ, a]) // dA // Γ

(Subscript[ζ, a]) // Γ

(Subscript[ζ, a]) // Γ// dA

Testing dS

$SeriesShowDegree=5;

(Subscript[ζ, a])

(Subscript[ζ, a]) // dS // Γ

(Subscript[ζ, a]) // Γ// dS