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Calculations: Extended Loop Lie Algebra

By trdunlap2

Let G be a Lie group. We are interested in the infinite dimensional Lie group LG=\text{Map}(S^1;G) where composition is done pointwise. One way to understand a group is by understanding its representations. In this particular case our interest is quickly narrowed to smooth, projective, positive energy representations. It turns out that a better way object of study is the semidirect product \mathbb{T}\tilde\times\tilde{LG} where \tilde{LG} is a particular one dimensional central extension and \mathbb{T} acts on LG by rotating loops (that is precomposing \gamma\in\text{Map}(S^1;G) with a rotation).

This thing’s Lie Algebra will be (as a vectors space) \mathbb{C}_\text{rot}\oplus L\mathfrak{g}\oplus\mathbb{C}_\text{cent}. My charge, by this Sunday, is to calculate the Lie bracket. Suffice we will consider the (dense?) subalgebra \mathbb{C}_\text{rot} \oplus \mathfrak{g}[t^{-1},t] \oplus \mathbb{C}_\text{cent}

To begin with:
\left[(z_1,0,0),(z_2,0,0)\right]=(0,0,0)
because rotation is commutative. The centeral extension is, well, central so we have:
\left[(z,t^k\alpha,w_1),(0,0,w_2)\right]=(0,0,0)
What’s left are
\left[(z,0,0),(0,t^l\beta,0)\right]
\left[(0,t^k\alpha,0),(0,t^l\beta,0)\right]

Lets start with the first. We’ll take what I’ll call the “scenic route” doing as much explicit calculation as possible.

\left[(z,0),(0,t^l\beta)\right] =ad_{(z,0)}(0,t^l\beta)
=\frac {d}{d\rho}|_{\rho=0}Ad_{\gamma(\rho)}(0,t^l\beta)
where \gamma(0)=(1,id)
and \gamma'(0)=(z,0)		 e.g. \gamma(\rho)=(e^{\rho z},id)
=\frac {d}{d\rho}|_{\rho=0}\frac {d}{d\xi}|_{\xi=0} (e^{z\rho},id)(1,exp(\xi t^l\beta)(e^{-z\rho},id)
=\frac {d}{d\rho}|_{\rho=0}\frac {d}{d\xi}|_{\xi=0}(1,exp(\xi(e^{z\rho}t)^l\beta)
=\frac {d}{d\rho}|_{\rho=0}(0,(e^{z\rho}t)^l\beta)
=(0,k(e^{z\rho}t)^{k-1}\alpha\cdot te^{z\rho}z)|_{\rho=0}
=(0,kzt^k\alpha,0)

The second calculation is actually fixed for us by the particular type of central extension we’re using. I’ll explain tomorrow.

This entry was posted on June 19, 2008 at 11:43 pm and is filed under Examples/exercises, Open. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Calculations: Extended Loop Lie Algebra”

  1. trdunlap2 Says:
    June 24, 2008 at 4:03 pm | Reply

    oops…”tomorrow” has turned into “next week”

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