Free Lie Algebras Tests

Pensieve header: Tests for FreeLie.nb.

Global Definitions

SetDirectory["C:\\drorbn\\AcademicPensieve\\2013-05"];
<< FreeLie.m

Words and Lyndon Words

{LyndonQ[AW@"abba"], LyndonQ[AW@"ababb"]}

{AllWords[3, {"1", "2"}], AllLyndonWords[{3}, {"1", "2"}]}

Table[Length[AllLyndonWords[k, {"1", "2"}]], {k,10}]

Table[Length[AllLyndonWords[k, {"1", "2", "3"}]], {k,10}]

BracketForm[LW["12122"]]

TopBracketForm[LW["12122"]]

The Bracket for Lie Elements

b[LW["112"], LW["122"]]

Outer[b, AllLyndonWords[{3}, {"1", "2"}], AllLyndonWords[{3}, {"1", "2"}]] // MatrixForm

Union[Flatten[Outer[(b[#1,#2]+b[#2,#1])&,
AllLyndonWords[{6}, {"1", "2"}], AllLyndonWords[{6}, {"1", "2"}]
]]]

Outer[(b[#1,b[#2,#3]]+b[#2,b[#3,#1]]+b[#3,b[#1,#2]])&,
AllLyndonWords[{5}, {"1", "2"}], AllLyndonWords[{5}, {"1", "2"}], AllLyndonWords[{5}, {"1", "2"}]
] // Flatten // Union

LieSeries

Print /@ {ts1=⟨"1122"⟩ // MakeLieSeries, ts1[], ts1 /@ Range[6]};

α=RandomLieSeries[{"a", "b"}]

α@{5}

{ts2=⟨"122"⟩+⟨"11122"⟩ // MakeLieSeries, ts3=b[ts1,ts2], ts3[], ts3 /@ Range[10]}

{ts4=EulerE[ts3], ts4[], ts4 /@ Range[10]}

adPower, adSeries, and Ad

{xs=MakeLieSeries[LW["x"]], ys=MakeLieSeries[LW["y"]], ts5=adPower[0,xs][ys], ts5[], ts5 /@ Range[5]}

adPower[3,xs][ys]@{5}

{adSeries[E^(-ad), xs][ys]@{5}, adSeries[E^(-ad), ys][xs]@{5}}

Ad[xs][ys][5]

Ad[xs][ys][]

LieDerivation, DerivationPower, DerivationSeries

Print /@ {
ld1=LieDerivation[{⟨1⟩ -> b[⟨3⟩, ⟨1⟩]}],
ld1[],
(# -> ld1[#][{4}])& /@ AllLyndonWords[{3}, {"1", "2"}],
(⟨"112"⟩ // ld1 //ld1)[{5}]
};

⟨"112"⟩ // MakeLieSeries // DerivationExp[LieDerivation[{⟨1⟩ -> b[⟨3⟩, ⟨1⟩]}]] // ShowLieSeries[6]

⟨"122"⟩ // MakeLieSeries // DerivationExp[LieDerivation[{⟨1⟩ -> b[⟨3⟩, ⟨1⟩]}]] // ShowLieSeries[6]

LieMorphism

Print /@ {
lm0=LieMorphism[{LW["x"] -> LW["y"]}],
LW["x"] // lm0,
AW["x"] // lm0,
CW["x"] // lm0};

Print /@ {
lm1=LieMorphism[{LW["x"] -> Ad[LW["y"]][LW["x"]]}],
lm1[],
lm1[LW["y"]],
lm1[LW["x"]],
lm1[LW["x"]][4],
lm1[⟨"xxy"⟩],
lm1[⟨"xxy"⟩][8],
lm1[AW["x"]],
lm1[CW["x"]]
};

StableApply

BCH

{BCHBase, BCHBase[], BCHBase[8]}

{BCH[LW["y"], LW["z"]], BCH[LW["y"], LW["z"]][6]}

{
t1=BCH[LW["x"], BCH[LW["y"], LW["z"]]],
t2=BCH[BCH[LW["x"], LW["y"]], LW["z"]],
t1==t2,
Table[t1[d]==t2[d], {d,10 }]
} // Timing

AW, ASeries, ι, σ

ι[BCHBase[3]]

{as=ι[BCHBase], as[5]}

(#->σ[1, #])& /@ AllLyndonWords[{5}, {"1","2"}]

{σ["x", BCHBase][5], σ["y", BCHBase][5]}

CW, CWSeries, tr, div

tr[AW["yxxyxx"]]

t1=σ["y", BCHBase] //tr

t1[5]

{div["x", BCHBase][7], div["y", BCHBase][7]}

t1=MakeCWSeries[CW["xyxyyyy"]] // LieDerivation[{LW["x"] -> MakeLieSeries[b[LW["x"], LW["z"]]]}]

t1 /@ Range[10]

The Meta-Cocycle JA

Print /@ {y0=LW["y"], μ0=BCHBase,
JA[0,y0,μ0, s],
JA[1,y0,μ0, s],
JA[2,y0,μ0, s],
JA[y0, μ0]
};

$SeriesCompareDegree=$SeriesShowDegree=8;
JA[3,y0,μ0, s]≡JA[4,y0,μ0, s]

JA[y0, μ0][6]