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## 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 • Note: Bilinear and Alternating implies anticommutative.**A Misconception**Marius Sophus Lie • Norwegian Mathematician • Geometry and differential equations**Lie Groups**• Differentiable manifold • Such that the operations are compatible with the smooth structure. Examples**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.**Hermann Weyl**• Introduced in 1930 • Character Formula • Symmetry**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 • Note: Bilinear and Alternating implies anticommutative.**Example**• Consider any associative algebra, over some field . • Now we can define our Lie Bracket to be**Check that this is a Lie Algebra**• Let a, b Bilinear Alternating Jacobi Identity +**Example**• General Linear Group:**Example: Cross Product**• with multiplication defined by the cross product • Bilinear? • Alternating? • Jacob Identity?**Example**Abelian Lie Algebra • If every bracket product is zero • Question: can you name this Abelian Lie Algebra?**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 • Note: Bilinear and Alternating implies anticommutative.**Conclusion**• Every Lie Group has a corresponding Lie Algebra. • Lie Algebras help us understand Lie Groups