lie algebras
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Lie Algebras. Elizabeth Bolduc Terika Harris. Lie Algebras: Definition. Let be a vector space over some field, and let and . is a Lie Algebra with a binary operation, known as the Lie Bracket if the following are true: Bilinear Alternating Jacobi identity - PowerPoint PPT PresentationTRANSCRIPT
Lie Algebras
Elizabeth BolducTerika HarrisLie AlgebrasLie Algebras: DefinitionA MisconceptionMarius Sophus Lie
Norwegian Mathematician
Geometry and differential equations
Sophus Lie, whom Lie Algebras are named after, was not the one who actually invited Lie Algebras. Sophus Lie was a Norwegian mathematician in the late 19th century studying geometry and differential equations. He is responsible for the discovery of Lie Groups.
3Lie GroupsDifferentiable manifold Such that the operations are compatible with the smooth structure. Examples
Tuesday in class we learned what a manifold was, its simply a topological space which in a small enough scale looks like a Euclidean space. Remember we said that the unit circle was S1 ie it looked like a straight line. This property of differentiable simply means that we can preform calculus. Recall smooth means we have as many derivatives as needed.Lie Groups are important in studying geometry and quantum mechanics. 4How do Lie Algebras Relate to Lie Groups?Lie Algebras help us understand Lie Groups.
If G is a Lie Group, the Lie Algebra of G is defined as the tangent space of the identity element of G.
Lie Algebras help us gain understanding about Lie Groups.In fact, Lie Algebras werent discovered by Sophus Lie. They were actually discovered 30 years after his death! By a man named Hermann Weyl. 56Hermann WeylIntroduced in 1930
Character Formula
Symmetry
Hermann, a German mathematician/physicist introduced lie algebras in the 1930, 30 years after Lie died. At the time he was studying compact groups and in the case of compact lie groups he solved the character formula which led to lie algebras. This research that he was doing at the time was important in understanding the symmetry structure in quantum mechanics. 7Lie Algebras: DefinitionExampleExampleIn the last example we saw how every within every associative algebra, there is a Lie Algebra if we define our Lie Bracket in that particular way. 11Example: Cross Product12 ExampleAbelian Lie Algebra
If every bracket product is zero
Question: can you name this Abelian Lie Algebra?Lie Algebras: DefinitionConclusionEvery Lie Group has a corresponding Lie Algebra.
Lie Algebras help us understand Lie Groups