Berkeley MATH 261A - MATH 261A Examples (3 pages)

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MATH 261A Examples



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MATH 261A Examples

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University of California, Berkeley
Course:
Math 261a - Lie Groups

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1 Examples A typical example of a Lie group is the group GL2 R of invertible 2 by 2 matrices and a Lie group is defined to be something that resembles this Its key properties are that it is a smooth manifold and a group and these structures are compatible So we define a Lie group to be a smooth manifold that is also a group such that the product and inverse are smooth maps All manifolds will be smooth and metrizable unless otherwise stated We start by trying to list all Lie groups Example 1 Any discrete group is a 0 dimensional Lie group This already shows that listing all Lie groups is hopeless as there are too many discrete groups However we can split a Lie group into two the component of the identity is a connected normal subgroup and the quotient is discrete Although a complete description of the discrete part is hopeless we can go quite far towards classifying the connected Lie groups Example 2 The real numbers under addition are a 1 dimensional commutative Lie group Similarly so is any finite dimensional real vector space under addition Example 3 The circle group S 1 of all complex numbers of absolute value 1 is a Lie group also abelian We have essentially found all the connected abelian Lie groups they are products of copies of the circle and the real numbers Foe example the nonzero complex numbers form a Lie group which via the exponential map and polar decomposition is isomorphic to the product of a circle and the reals Example 4 The general linear group GLn R is the archetypal example of a non commutative Lie group This has 2 components as the determinant can be positive or negative Similarly we can take the complex general linear group The classical groups are roughly the subgroups of general linear groups that preserve bilinear or hermitian forms The compact orthogonal groups On R preserve a positive definite symmetric bilinear form on a real vector space We do not have to restrict to positiver definite forms in special relativity we get the Lorentz



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