[Verse 1]
When groups get curved and complicated
Smooth manifolds with operations
Take the tangent at identity
Find the algebra that sets you free
Matrix groups with smooth transitions
Lead us to these compositions
[Chorus]
From Lie groups to algebras we go
Take the tangent space to know
Bracket XY minus YX
Commutators show what's next
Linear space with bracket law
Simpler truth is what we saw
[Verse 2]
GL of n has all matrices
n squared dimensions, no restrictions
SL needs trace to equal zero
n squared minus one's our hero
Special linear, determinant one
Algebra follows what we've done
[Chorus]
From Lie groups to algebras we go
Take the tangent space to know
Bracket XY minus YX
Commutators show what's next
Linear space with bracket law
Simpler truth is what we saw
[Verse 3]
SO rotations, orthogonal
Skew symmetric, antisocial
X plus X transpose equals zero
n times n minus one divided by two
SU unitary and special
Traceless, anti-Hermitian spectacle
[Bridge]
The miracle we celebrate tonight
Connected groups, simply connected sight
The algebra determines all
Vector spaces answer the call
Study brackets, understand the group
Curved geometry in linear loop
[Chorus]
From Lie groups to algebras we go
Take the tangent space to know
Bracket XY minus YX
Commutators show what's next
Linear space with bracket law
Simpler truth is what we saw
[Outro]
When the group seems hard to see
Find its infinitesimal key
Tangent space at identity
Holds the group's geometry