adjoint.RdThe adjoint \(\mathrm{ad}_X\) of \(X\) is a map from a Lie group \(G\) to the endomorphism group of \(G\) defined by
$$\mathrm{ad}_X(Y)=\left[X,Y\right]$$
ad(x)Vignette adjoint gives more description
x <- rfalg()
y <- rfalg()
f <- ad(x)
f(y)
#> free algebra element algebraically equal to
#> - 7aaababba - 14aaabaccc - 35aaabb - 42aaabbb - 21aaabcaaa - 49aaabcbbb -
#> 28aaabccb - aabba - 2aaccc - 5ab - 6abb + abbaa + 7abbaaaab + 10abbab +
#> 6abbabcac + 4abbabccc - 3acaaa - 7acbbb - 4accb + 2accca + 14acccaaab + 20acccb
#> + 12acccbcac + 8acccbccc + 5ba + 35baaab - 10babba - 20baccc + 6bba + 42bbaaab
#> + 36bbbcac + 24bbbccc + 30bbcac + 20bbccc - 30bcaaa - 6bcacabba - 12bcacaccc -
#> 30bcacb - 36bcacbb - 18bcaccaaa - 42bcaccbbb - 24bcacccb - 70bcbbb - 40bccb -
#> 4bcccabba - 8bcccaccc - 20bcccb - 24bcccbb - 12bccccaaa - 28bccccbbb -
#> 16bcccccb + 3caaaa + 21caaaaaab + 30caaab + 18caaabcac + 12caaabccc + 7cbbba +
#> 49cbbbaaab + 70cbbbb + 42cbbbbcac + 28cbbbbccc + 4ccba + 28ccbaaab + 40ccbb +
#> 24ccbbcac + 16ccbbccc
f(f(y)) # [x, [x, y]]
#> free algebra element algebraically equal to
#> + 7aaababbaabba + 14aaababbaaccc + 35aaababbab + 42aaababbabb + 21aaababbacaaa
#> + 49aaababbacbbb + 28aaababbaccb + 14aaabacccabba + 28aaabacccaccc +
#> 70aaabacccb + 84aaabacccbb + 42aaabaccccaaa + 98aaabaccccbbb + 56aaabacccccb +
#> 35aaabbabba + 70aaabbaccc + 175aaabbb + 42aaabbbabba + 84aaabbbaccc +
#> 420aaabbbb + 252aaabbbbb + 126aaabbbcaaa + 294aaabbbcbbb + 168aaabbbccb +
#> 105aaabbcaaa + 245aaabbcbbb + 140aaabbccb + 21aaabcaaaabba + 42aaabcaaaaccc +
#> 105aaabcaaab + 126aaabcaaabb + 63aaabcaaacaaa + 147aaabcaaacbbb + 84aaabcaaaccb
#> + 49aaabcbbbabba + 98aaabcbbbaccc + 245aaabcbbbb + 294aaabcbbbbb +
#> 147aaabcbbbcaaa + 343aaabcbbbcbbb + 196aaabcbbbccb + 28aaabccbabba +
#> 56aaabccbaccc + 140aaabccbb + 168aaabccbbb + 84aaabccbcaaa + 196aaabccbcbbb +
#> 112aaabccbccb + aabbaabba + 2aabbaaccc + 5aabbab + 6aabbabb + 3aabbacaaa +
#> 7aabbacbbb + 4aabbaccb + 2aacccabba + 4aacccaccc + 10aacccb + 12aacccbb +
#> 6aaccccaaa + 14aaccccbbb + 8aacccccb + 5ababba + 10abaccc + 25abb -
#> 14abbaaaababba - 28abbaaaabaccc - 70abbaaaabb - 84abbaaaabbb - 42abbaaaabcaaa -
#> 98abbaaaabcbbb - 56abbaaaabccb - 2abbaaabba - 4abbaaaccc - 10abbaab - 12abbaabb
#> + abbaabbaa + 7abbaabbaaaab + 10abbaabbab + 6abbaabbabcac + 4abbaabbabccc -
#> 6abbaacaaa - 14abbaacbbb - 8abbaaccb + 2abbaaccca + 14abbaacccaaab +
#> 20abbaacccb + 12abbaacccbcac + 8abbaacccbccc + 5abbaba + 35abbabaaab -
#> 20abbababba - 40abbabaccc - 50abbabb + 12abbabba + 42abbabbaaab - 60abbabbb +
#> 36abbabbbcac + 24abbabbbccc + 30abbabbcac + 20abbabbccc - 60abbabcaaa -
#> 12abbabcacabba - 24abbabcacaccc - 60abbabcacb - 72abbabcacbb - 36abbabcaccaaa -
#> 84abbabcaccbbb - 48abbabcacccb - 140abbabcbbb - 80abbabccb - 8abbabcccabba -
#> 16abbabcccaccc - 40abbabcccb - 48abbabcccbb - 24abbabccccaaa - 56abbabccccbbb -
#> 32abbabcccccb + 3abbacaaaa + 21abbacaaaaaab + 30abbacaaab + 18abbacaaabcac +
#> 12abbacaaabccc + 7abbacbbba + 49abbacbbbaaab + 70abbacbbbb + 42abbacbbbbcac +
#> 28abbacbbbbccc + 4abbaccba + 28abbaccbaaab + 40abbaccbb + 24abbaccbbcac +
#> 16abbaccbbccc + 12abbaccc + 60abbb + 36abbbb + 18abbcaaa + 42abbcbbb + 24abbccb
#> + 15abcaaa + 35abcbbb + 20abccb + 3acaaaabba + 6acaaaaccc + 15acaaab +
#> 18acaaabb + 9acaaacaaa + 21acaaacbbb + 12acaaaccb + 7acbbbabba + 14acbbbaccc +
#> 35acbbbb + 42acbbbbb + 21acbbbcaaa + 49acbbbcbbb + 28acbbbccb + 4accbabba +
#> 8accbaccc + 20accbb + 24accbbb + 12accbcaaa + 28accbcbbb + 16accbccb -
#> 28acccaaababba - 56acccaaabaccc - 140acccaaabb - 168acccaaabbb - 84acccaaabcaaa
#> - 196acccaaabcbbb - 112acccaaabccb - 4acccaabba - 8acccaaccc - 20acccab -
#> 24acccabb + 2acccabbaa + 14acccabbaaaab + 20acccabbab + 12acccabbabcac +
#> 8acccabbabccc - 12acccacaaa - 28acccacbbb - 16acccaccb + 4acccaccca +
#> 28acccacccaaab + 40acccacccb + 24acccacccbcac + 16acccacccbccc + 10acccba +
#> 70acccbaaab - 40acccbabba - 80acccbaccc - 100acccbb + 12acccbba + 84acccbbaaab
#> - 120acccbbb + 72acccbbbcac + 48acccbbbccc + 60acccbbcac + 40acccbbccc -
#> 120acccbcaaa - 24acccbcacabba - 48acccbcacaccc - 120acccbcacb - 144acccbcacbb -
#> 72acccbcaccaaa - 168acccbcaccbbb - 96acccbcacccb - 280acccbcbbb - 160acccbccb -
#> 16acccbcccabba - 32acccbcccaccc - 80acccbcccb - 96acccbcccbb - 48acccbccccaaa -
#> 112acccbccccbbb - 64acccbcccccb + 6accccaaaa + 42accccaaaaaab + 60accccaaab +
#> 36accccaaabcac + 24accccaaabccc + 14accccbbba + 98accccbbbaaab + 140accccbbbb +
#> 84accccbbbbcac + 56accccbbbbccc + 8acccccba + 56acccccbaaab + 80acccccbb +
#> 48acccccbbcac + 32acccccbbccc - 70baaababba - 140baaabaccc - 350baaabb -
#> 420baaabbb - 210baaabcaaa - 490baaabcbbb - 280baaabccb - 10baabba - 20baaccc -
#> 50bab - 60babb + 5babbaa + 35babbaaaab + 10babbaabba + 20babbaaccc + 100babbab
#> + 60babbabb + 30babbabcac + 20babbabccc + 30babbacaaa + 70babbacbbb +
#> 40babbaccb - 30bacaaa - 70bacbbb - 40baccb + 10baccca + 70bacccaaab +
#> 20bacccabba + 40bacccaccc + 200bacccb + 120bacccbb + 60bacccbcac + 40bacccbccc
#> + 60baccccaaa + 140baccccbbb + 80bacccccb + 25bba + 175bbaaab - 84bbaaababba -
#> 168bbaaabaccc - 420bbaaabb - 504bbaaabbb - 252bbaaabcaaa - 588bbaaabcbbb -
#> 336bbaaabccb - 12bbaabba - 24bbaaccc - 60bbab - 72bbabb - 50bbabba + 6bbabbaa +
#> 42bbabbaaaab + 60bbabbab + 36bbabbabcac + 24bbabbabccc - 36bbacaaa - 84bbacbbb
#> - 48bbaccb - 100bbaccc + 12bbaccca + 84bbacccaaab + 120bbacccb + 72bbacccbcac +
#> 48bbacccbccc + 60bbba + 420bbbaaab - 60bbbabba - 120bbbaccc + 36bbbba +
#> 252bbbbaaab + 216bbbbbcac + 144bbbbbccc + 360bbbbcac + 240bbbbccc - 180bbbcaaa
#> + 150bbbcac - 72bbbcacabba - 144bbbcacaccc - 360bbbcacb - 432bbbcacbb -
#> 216bbbcaccaaa - 504bbbcaccbbb - 288bbbcacccb - 420bbbcbbb - 240bbbccb +
#> 100bbbccc - 48bbbcccabba - 96bbbcccaccc - 240bbbcccb - 288bbbcccbb -
#> 144bbbccccaaa - 336bbbccccbbb - 192bbbcccccb - 150bbcaaa + 18bbcaaaa +
#> 126bbcaaaaaab + 180bbcaaab + 108bbcaaabcac + 72bbcaaabccc - 60bbcacabba -
#> 120bbcacaccc - 300bbcacb - 360bbcacbb - 180bbcaccaaa - 420bbcaccbbb -
#> 240bbcacccb - 350bbcbbb + 42bbcbbba + 294bbcbbbaaab + 420bbcbbbb +
#> 252bbcbbbbcac + 168bbcbbbbccc - 200bbccb + 24bbccba + 168bbccbaaab + 240bbccbb
#> + 144bbccbbcac + 96bbccbbccc - 40bbcccabba - 80bbcccaccc - 200bbcccb -
#> 240bbcccbb - 120bbccccaaa - 280bbccccbbb - 160bbcccccb + 15bcaaaa +
#> 105bcaaaaaab + 30bcaaaabba + 60bcaaaaccc + 300bcaaab + 180bcaaabb + 90bcaaabcac
#> + 60bcaaabccc + 90bcaaacaaa + 210bcaaacbbb + 120bcaaaccb + 6bcacabbaabba +
#> 12bcacabbaaccc + 30bcacabbab + 36bcacabbabb + 18bcacabbacaaa + 42bcacabbacbbb +
#> 24bcacabbaccb + 12bcacacccabba + 24bcacacccaccc + 60bcacacccb + 72bcacacccbb +
#> 36bcacaccccaaa + 84bcacaccccbbb + 48bcacacccccb + 30bcacbabba + 60bcacbaccc +
#> 150bcacbb + 36bcacbbabba + 72bcacbbaccc + 360bcacbbb + 216bcacbbbb +
#> 108bcacbbcaaa + 252bcacbbcbbb + 144bcacbbccb + 90bcacbcaaa + 210bcacbcbbb +
#> 120bcacbccb + 18bcaccaaaabba + 36bcaccaaaaccc + 90bcaccaaab + 108bcaccaaabb +
#> 54bcaccaaacaaa + 126bcaccaaacbbb + 72bcaccaaaccb + 42bcaccbbbabba +
#> 84bcaccbbbaccc + 210bcaccbbbb + 252bcaccbbbbb + 126bcaccbbbcaaa +
#> 294bcaccbbbcbbb + 168bcaccbbbccb + 24bcacccbabba + 48bcacccbaccc + 120bcacccbb
#> + 144bcacccbbb + 72bcacccbcaaa + 168bcacccbcbbb + 96bcacccbccb + 35bcbbba +
#> 245bcbbbaaab + 70bcbbbabba + 140bcbbbaccc + 700bcbbbb + 420bcbbbbb +
#> 210bcbbbbcac + 140bcbbbbccc + 210bcbbbcaaa + 490bcbbbcbbb + 280bcbbbccb +
#> 20bccba + 140bccbaaab + 40bccbabba + 80bccbaccc + 400bccbb + 240bccbbb +
#> 120bccbbcac + 80bccbbccc + 120bccbcaaa + 280bccbcbbb + 160bccbccb +
#> 4bcccabbaabba + 8bcccabbaaccc + 20bcccabbab + 24bcccabbabb + 12bcccabbacaaa +
#> 28bcccabbacbbb + 16bcccabbaccb + 8bcccacccabba + 16bcccacccaccc + 40bcccacccb +
#> 48bcccacccbb + 24bcccaccccaaa + 56bcccaccccbbb + 32bcccacccccb + 20bcccbabba +
#> 40bcccbaccc + 100bcccbb + 24bcccbbabba + 48bcccbbaccc + 240bcccbbb +
#> 144bcccbbbb + 72bcccbbcaaa + 168bcccbbcbbb + 96bcccbbccb + 60bcccbcaaa +
#> 140bcccbcbbb + 80bcccbccb + 12bccccaaaabba + 24bccccaaaaccc + 60bccccaaab +
#> 72bccccaaabb + 36bccccaaacaaa + 84bccccaaacbbb + 48bccccaaaccb + 28bccccbbbabba
#> + 56bccccbbbaccc + 140bccccbbbb + 168bccccbbbbb + 84bccccbbbcaaa +
#> 196bccccbbbcbbb + 112bccccbbbccb + 16bcccccbabba + 32bcccccbaccc + 80bcccccbb +
#> 96bcccccbbb + 48bcccccbcaaa + 112bcccccbcbbb + 64bcccccbccb - 42caaaaaababba -
#> 84caaaaaabaccc - 210caaaaaabb - 252caaaaaabbb - 126caaaaaabcaaa -
#> 294caaaaaabcbbb - 168caaaaaabccb - 6caaaaabba - 12caaaaaccc - 30caaaab -
#> 36caaaabb + 3caaaabbaa + 21caaaabbaaaab + 30caaaabbab + 18caaaabbabcac +
#> 12caaaabbabccc - 18caaaacaaa - 42caaaacbbb - 24caaaaccb + 6caaaaccca +
#> 42caaaacccaaab + 60caaaacccb + 36caaaacccbcac + 24caaaacccbccc + 15caaaba +
#> 105caaabaaab - 60caaababba - 120caaabaccc - 150caaabb + 18caaabba +
#> 126caaabbaaab - 180caaabbb + 108caaabbbcac + 72caaabbbccc + 90caaabbcac +
#> 60caaabbccc - 180caaabcaaa - 36caaabcacabba - 72caaabcacaccc - 180caaabcacb -
#> 216caaabcacbb - 108caaabcaccaaa - 252caaabcaccbbb - 144caaabcacccb -
#> 420caaabcbbb - 240caaabccb - 24caaabcccabba - 48caaabcccaccc - 120caaabcccb -
#> 144caaabcccbb - 72caaabccccaaa - 168caaabccccbbb - 96caaabcccccb + 9caaacaaaa +
#> 63caaacaaaaaab + 90caaacaaab + 54caaacaaabcac + 36caaacaaabccc + 21caaacbbba +
#> 147caaacbbbaaab + 210caaacbbbb + 126caaacbbbbcac + 84caaacbbbbccc + 12caaaccba
#> + 84caaaccbaaab + 120caaaccbb + 72caaaccbbcac + 48caaaccbbccc - 98cbbbaaababba
#> - 196cbbbaaabaccc - 490cbbbaaabb - 588cbbbaaabbb - 294cbbbaaabcaaa -
#> 686cbbbaaabcbbb - 392cbbbaaabccb - 14cbbbaabba - 28cbbbaaccc - 70cbbbab -
#> 84cbbbabb + 7cbbbabbaa + 49cbbbabbaaaab + 70cbbbabbab + 42cbbbabbabcac +
#> 28cbbbabbabccc - 42cbbbacaaa - 98cbbbacbbb - 56cbbbaccb + 14cbbbaccca +
#> 98cbbbacccaaab + 140cbbbacccb + 84cbbbacccbcac + 56cbbbacccbccc + 35cbbbba +
#> 245cbbbbaaab - 140cbbbbabba - 280cbbbbaccc - 350cbbbbb + 42cbbbbba +
#> 294cbbbbbaaab - 420cbbbbbb + 252cbbbbbbcac + 168cbbbbbbccc + 210cbbbbbcac +
#> 140cbbbbbccc - 420cbbbbcaaa - 84cbbbbcacabba - 168cbbbbcacaccc - 420cbbbbcacb -
#> 504cbbbbcacbb - 252cbbbbcaccaaa - 588cbbbbcaccbbb - 336cbbbbcacccb -
#> 980cbbbbcbbb - 560cbbbbccb - 56cbbbbcccabba - 112cbbbbcccaccc - 280cbbbbcccb -
#> 336cbbbbcccbb - 168cbbbbccccaaa - 392cbbbbccccbbb - 224cbbbbcccccb +
#> 21cbbbcaaaa + 147cbbbcaaaaaab + 210cbbbcaaab + 126cbbbcaaabcac + 84cbbbcaaabccc
#> + 49cbbbcbbba + 343cbbbcbbbaaab + 490cbbbcbbbb + 294cbbbcbbbbcac +
#> 196cbbbcbbbbccc + 28cbbbccba + 196cbbbccbaaab + 280cbbbccbb + 168cbbbccbbcac +
#> 112cbbbccbbccc - 56ccbaaababba - 112ccbaaabaccc - 280ccbaaabb - 336ccbaaabbb -
#> 168ccbaaabcaaa - 392ccbaaabcbbb - 224ccbaaabccb - 8ccbaabba - 16ccbaaccc -
#> 40ccbab - 48ccbabb + 4ccbabbaa + 28ccbabbaaaab + 40ccbabbab + 24ccbabbabcac +
#> 16ccbabbabccc - 24ccbacaaa - 56ccbacbbb - 32ccbaccb + 8ccbaccca + 56ccbacccaaab
#> + 80ccbacccb + 48ccbacccbcac + 32ccbacccbccc + 20ccbba + 140ccbbaaab -
#> 80ccbbabba - 160ccbbaccc - 200ccbbb + 24ccbbba + 168ccbbbaaab - 240ccbbbb +
#> 144ccbbbbcac + 96ccbbbbccc + 120ccbbbcac + 80ccbbbccc - 240ccbbcaaa -
#> 48ccbbcacabba - 96ccbbcacaccc - 240ccbbcacb - 288ccbbcacbb - 144ccbbcaccaaa -
#> 336ccbbcaccbbb - 192ccbbcacccb - 560ccbbcbbb - 320ccbbccb - 32ccbbcccabba -
#> 64ccbbcccaccc - 160ccbbcccb - 192ccbbcccbb - 96ccbbccccaaa - 224ccbbccccbbb -
#> 128ccbbcccccb + 12ccbcaaaa + 84ccbcaaaaaab + 120ccbcaaab + 72ccbcaaabcac +
#> 48ccbcaaabccc + 28ccbcbbba + 196ccbcbbbaaab + 280ccbcbbbb + 168ccbcbbbbcac +
#> 112ccbcbbbbccc + 16ccbccba + 112ccbccbaaab + 160ccbccbb + 96ccbccbbcac +
#> 64ccbccbbccc