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Wednesday, October 2, 2013

**Abstract:** I will give an overview of the Boson-Fermion correspondence which gives an isomorphism between the representation of the Heisenberg Lie algebra on $\mathbb{C}[x_{1},x_{2,}\cdots]$ and the representation of the Heisenberg Lie algebra on an infinite dimensional wedge space. I will briefly discuss how this isomorphism can be extended to isomorphisms between representations of larger Lie algebras. I will then discuss how to use the Boson-Fermion correspondence to understand the orbit of 1 under the action of the Lie group $GL_{\infty}$, and will mention the fact that elements in the orbit are solutions to an infinite set of differential equations. This parallels our previous discussion of representations of the Lie algebra $sl_{n}$ on the finite exterior algebra, $\Lambda(V)=\oplus_{k=0}^{n}\Lambda^{k}V$, which we used to obtain the Plücker relations. Our discussion of the Boson-Fermion correspondence and its uses will follow the treatment given in Kac and Raina's Highest Weight Representations of Infinite Dimensional Lie Algebras.