Lecture Notes for Applied Representation Theory

Poisson Brackets are Lie Brackets

Poisson Brackets are Lie Brackets

From elementary mechanics, we develop the basics of Hamiltonian Mechanics and describe the Lie algebra of real functions over phase space using the Poisson Bracket. This is the first installment of my ongoing exploration of the intersection of Representation Theory and Physics.

Poisson Brackets, Time Evolution and the Exponential Map

Time Evolution and the Exponential Map

We explore the Exponential Map from the Lie algebra of functions over phase space to its associated Lie group that acts on that algebra. Specifically we explore the group element associated with the Energy function, and demonstrate how it generates displacements in time through specific examples.

Linear Symplectomorphisms and SL(2,R)

We describe the subgroup of linear transformations of phase space that preserves the Poisson bracket, and show that it is essentially the special linear group of 2x2 matrices, SL(2,R). We introduce some gentle proofs to get the physicists in the audience on board with a more Mathematical style.

The Lie group Sp(2n,R)

We derive the symplectic condition on linear transformations on a general 2n-dimensional phase space that is, that they preserve the Poisson Bracket. This involves creating a more symmetric approach to modeling the Hamilton Equations and the Poisson Bracket. We continue our gentle introduction to proofs for our Physics friends.

The Lie algebra sp(2n,R)

We extend our analysis of the Lie group of linear symplectomorphisms of phase space, Sp(2n,R) to derive the conditions on the infinitesimal group elements - the Lie algebra sp(2n,R).We summarize these results as an explicit form of a 2nx2n-matrix, blocked into four nxn-matrices. We then consider a few representative examples of the Lie algebra elements that exponentiate up to group elements.

A Polynomial Representation of SL(2,R)

In this episode we describe a polynomial representation of sp(2,R) = sl(2,R) as a warm up before we study the general case. We first demonstrate that the quadratic polynomials form a subalgebra of the Lie algebra induced by the Poisson Bracket. We then prove the isomorphism with sl2(2,R) by brute force. Finally we finish up with some examples of how these polynomials generate group transformations in the Adjoint representation of SL(2,R).