Lie Algebra

 The set of all infinitessimal generators is the Lie algebra of the group G.  Linear transformation: Let A be a matrixLet A be a matrix G = {a; a  I +  A;  << 1}G = {a; a  I +  A;  << 1}  Product maps to a sum. S1S1 (I+  A)(I+  B) = I +  (A+B) +  2 AB a  A b  B  I +  (A+B) ab  A+B

Lie Commutator  Group commutator: aba -1 b -1 Use power seriesUse power series Discard high order termsDiscard high order terms Non-abelian groupNon-abelian group  Lie bracket [A, B] = AB – BA  Lie algebra includes addition, subtraction and bracket operations S1S1 (I+  A)(I+  B)(I+  A ) -1 (I+  B ) -1 = (I+  (A+B)+  2 AB)(I–  A+  2 A 2 )(I–  B+  2 B 2 )  I +   (AB - BA) (I+  A) -1 = I –  A +  2 A 2 – o(  3 )

Bracket Properties  Distributive  Antisymmetric  Jacobi identity S1S1 [A + B, C] =[A, C] + [B, C] [  A, B] =  [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

Lie Derivative  The Lie bracket between two vector fields. Measures the gap in an infinitessimal shift on a surface.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.Use the local coordinates of the vector field.  p     next