Oz's crib sheet: Lie theory
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] = ab-ba.

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 AB is in G.
    • Multiplication is associative: (AB)C = A(BC).
    • Multiplication has an identity: 1A = A = A1.
    • Multiplication has an inverse: AA^-1 = 1 = A^-1A.
  • The multiplication and inverse operations must be smooth.
• 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.
  • If a is in g and k is a real number, then ka is in g.
    • Scalar multiplication is associative: (kl)a = k(la).
    • Scalar multiplication has an identity: 1a = a.
    • Scalar multiplication is linear on the left: (k + l)a = ka + la.
    • Scalar multiplication is linear on the right: k(a + b) = ka + kb.
  • 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.)

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.

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