Edited by Toby Bartels

**R**means the real numbers.**C**means the complex numbers.**H**means the quaternions.- G & g mean "general", which doesn't mean anything.
- L & l mean "linear" -- which is obvious, because they're matrices!
- S means "special", det
*A*= 1, so s must mean tr*a*= 0 (traceless). - O means "orthogonal",
*A*^{t}=*A*^{-1}, so o must mean*a*^{t}= -*a*(antisymmetric). - U means "unitary",
*A*^{*}=*A*^{-1}, so u must mean*a*^{*}= -*a*(antiHermitian). - Sp means "symplectic", which is the same as unitary only for quaternions.

Therefore:

- GL(
*n*,**R**) is the Lie group of invertible real matrices of dimension*n*, and gl(*n*,**R**) is its Lie algebra of all real matrices of dimension*n*. - GL(
*n*,**C**) is the Lie group of invertible complex matrices of dimension*n*, and gl(*n*,**C**) is its Lie algebra of all complex matrices of dimension*n*. - GL(
*n*,**H**) is the Lie group of invertible quaternionic matrices of dimension*n*, and gl(*n*,**H**) is its Lie algebra of all quaternionic matrices of dimension*n*. - SL(
*n*,**R**) is the Lie group of special real matrices of dimension*n*, and sl(*n*,**R**) is its Lie algebra of traceless real matrices of dimension*n*. - SL(
*n*,**C**) is the Lie group of special complex matrices of dimension*n*, and sl(*n*,**C**) is its Lie algebra of traceless complex matrices of dimension*n*. - O(
*n*) is the Lie group of real orthogonal matrices of dimension*n*, and o(*n*) is its Lie algebra of real antisymmetric matrices of dimension*n*. - SO(
*n*) is the Lie group of special real orthogonal matrices of dimension*n*, and so(*n*) is its Lie algebra of traceless real antisymmetric matrices of dimension*n*. - U(
*n*) is the Lie group of complex unitary matrices of dimension*n*, and u(*n*) is its Lie algebra of complex antiHermitian matrices of dimension*n*. - SU(
*n*) is the Lie group of special complex unitary matrices of dimension*n*, and su(*n*) is its Lie algebra of traceless complex antiHermitian matrices of dimension*n*. - Sp(
*n*) is the Lie group of quaternionic unitary matrices of dimension*n*, and sp(*n*) is its Lie algebra of quaternionic antiHermitian matrices of dimension*n*.

Examples:

- U(1) is the complex numbers of unit magnitude, and u(1) is the purely imaginary complex numbers. Topologically, U(1) is a circle, and u(1) is a line.
- SU(2) is the quaternions of unit magnitude, and su(2) is the purely imaginary quaternions. Topologically, SU(2) is a 3D sphere, and su(2) is 3space.

A Lie bracket [,] is defined on the lie algebra **g**
and produces another element of the set **g**.
For matrices, this is
[*a*,*b*] = *a**b*-*b**a*.

Same examples:

- The Lie bracket on u(1) is trivial;
[
*a*,*b*] = 0 always. - The Lie bracket on su(2) is just the vector cross product.

Complete definitions:

- A Lie group must have a notion of multiplication
and must be a smooth manifold.
- If
*A*and*B*are in*G*, then*A**B*is in*G*.- Multiplication is associative:
(
*A**B*)*C*=*A*(*B**C*). - Multiplication has an identity: 1
*A*=*A*=*A*1. - Multiplication has an inverse:
*A**A*^{-1}= 1 =*A*^{-1}*A*.

- Multiplication is associative:
(
- The multiplication and inverse operations must be smooth.

- If
- A Lie algebra must have notions of
addition, scalar multiplication, and a Lie bracket.
- If
*a*and*b*are in**g**, then*a*+*b*is in**g**.- Addition is associative:
(
*a*+*b*) +*c*=*a*+ (*b*+*c*). - Addition has an identity:
*a*+ 0 =*a*= 0 +*a*. - Addition has an inverse:
*a*+ -*a*= 0 = -*a*+*a*. - Addition is commutative:
*a*+*b*=*b*+*a*.

- Addition is associative:
(
- If
*a*is in**g**and*k*is a real number, then*k**a*is in**g**.- Scalar multiplication is associative:
(
*k**l*)*a*=*k*(*l**a*). - Scalar multiplication has an identity: 1
*a*=*a*. - Scalar multiplication is linear on the left:
(
*k*+*l*)*a*=*k**a*+*l**a*. - Scalar multiplication is linear on the right:
*k*(*a*+*b*) =*k**a*+*k**b*.

- Scalar multiplication is associative:
(
- If
*a*is in**g**and*b*is in**g**, then [*a*,*b*] is in**g**.- The Lie bracket is linear on the left:
[
*a*+*b*,*c*] = [*a*,*c*] + [*b*,*c*]. - The Lie bracket is linear on the right:
[
*a*,*b*+*c*] = [*a*,*b*] + [*a*,*c*]. - The Lie bracket is alternating: [
*a*,*a*] = 0. - The Lie bracket is antisymmetric:
[
*a*,*b*] = -[*b*,*a*]. - The Lie bracket satisfies the Jacobi identity on the left:
[[
*a*,*b*],*c*] + [[*b*,*c*],*a*] + [[*c*,*a*],*b*] = 0. - The Lie bracket satisfies the Jacobi identity on the right:
[
*a*,[*b*,*c*]] + [*b*,[*c*,*a*]] + [*c*,[*a*,*b*]] = 0. - The Lie bracket satisfies the Leibniz law on the left:
[
*a*,[*b*,*c*]] = [[*a*,*b*],*c*] + [*b*,[*a*,*c*]]. - The Lie bracket satisfies the Leibniz law on the right:
[[
*a*,*b*],*c*] = [[*a*,*c*],*b*] + [*a*,[*b*,*c*]]. - (Some of the requirements for the Lie bracket are redundant.)

- The Lie bracket is linear on the left:
[

- If

- Q
- Given a Lie group G,
*which*specific Lie algebra**g**is*the*Lie algebra of the Lie group*G*? - A
- The space tangent to the identity element 1 of
*G*.

Go back to Oz's crib sheet.

This web page was written between 2000 and 2002 by Toby Bartels, Oz, and probably some other people, but I forget just who. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`https://tobybartels.name/Oz/Lie/`

.