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Lie Algebra. Lie Algebra. The set of all infinitessimal generators is the Lie algebra of the group G. Linear transformation: Let A be a matrix G = { a ; a  I + e A ; e << 1} Product maps to a sum. S 1. a  A b  B. ( I + e A )( I + e B ) = I + e ( A + B ) + e 2 AB.

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lie algebra1
Lie Algebra
  • The set of all infinitessimal generators is the Lie algebra of the group G.
  • Linear transformation:
    • Let A be a matrix
    • G = {a; aI + eA;e << 1}
  • Product maps to a sum.

S1

a  A

b  B

(I+eA)(I+eB) = I + e(A+B) + e2AB

I + e(A+B)

ab A+B

lie commutator
Group commutator: aba-1b-1

Use power series

Discard high order terms

Non-abelian group

Lie bracket

[A, B] = AB – BA

Lie algebra includes addition, subtraction and bracket operations

Lie Commutator

S1

(I+eA)-1 = I – eA + e2A2– o(e3)

(I+eA)(I+eB)(I+eA )-1 (I+eB )-1 = (I+e(A+B)+e2AB)(I–eA+e2A2)(I–eB+e2B2)

I + e2(AB - BA)

bracket properties
Bracket Properties
  • Distributive
  • Antisymmetric
  • Jacobi identity

S1

[A + B, C] =[A, C] + [B, C]

[kA, B] = k[A, B]

[A, B] = -[B, A]

[A, [B, C]]+ [B, [C, A]]+ [C, [A, B]] = 0

vector field commutator
Two vector fields acting on a scalar one-form.

Obeys the three laws of a vector field.

Obeys rules for a Lie bracket

Vector Field Commutator
lie derivative
The Lie bracket between two vector fields.

Measures the gap in an infinitessimal shift on a surface.

Can be applied to functions and one-forms as well.

Use the local coordinates of the vector field.

Lie Derivative

eh

e2[x,h]

ex

ex

p

eh

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