Terika Harris Lie Algebras 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 ID: 140092
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Elizabeth BolducTerika Harris
Lie AlgebrasSlide2
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:BilinearAlternatingJacobi identity Note: Bilinear and Alternating implies anticommutative.
Slide3
A Misconception
Marius Sophus
Lie
Norwegian Mathematician Geometry and differential equationsSlide4
Lie Groups
D
ifferentiable manifold
Such that the operations are compatible with the smooth structure. ExamplesSlide5
How 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. Slide6
Example:
Slide7
Hermann Weyl
Introduced in 1930
Character Formula
SymmetrySlide8
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:BilinearAlternatingJacobi identity Note: Bilinear and Alternating implies anticommutative.
Slide9
Example
Consider any associative algebra,
over some field
.Now we can define our Lie Bracket to be Slide10
Check that this is a Lie Algebra
Let a, b
Bilinear Alternating Jacobi Identity
+
Slide11
Example
General Linear Group:
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Example: Cross Product
with multiplication defined by the cross product
Bilinear?
Alternating?Jacob Identity? Slide13
Example
Abelian
Lie Algebra
If every bracket product is zeroQuestion: can you name this Abelian Lie Algebra?Slide14
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:BilinearAlternatingJacobi identity Note: Bilinear and Alternating implies anticommutative.
Slide15
Conclusion
Every Lie Group has a corresponding Lie Algebra.
Lie Algebras help us understand Lie Groups