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Department of Mathematics, University of California San Diego

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Proof of G-K theorem for the Lie algebra $A_n=gl(N+1)$

Abstract:

Algebraic quantization is one of several mathematical counterparts to the physical notion of quantization''. It's goal is to describe non-commutative algebras using generators satisfying nice commutation relations. \vskip .1in \noindent Let $\frak g$ be a Lie algebra. We denote by $U(\frak g)$ its enveloping algebra and by $D(\frak g)$ the quotient skew field (to be defined and explained). \vskip .1in \noindent $G-K$ theorem claims that the skew field $D(\frak g)$ is generated by $2n = N(N+1)$ elements $p_1,\,\dots ,\,p_n,\, q_1,\,\dots,\,q_n$ satisfying the canonical commutation relations $$[p_i,\,p_j] = [q_i,\,q_j] = 0,\quad [p_i,\,q_j] = \delta_{ij}$$ and by $N$ elements $z_1,\,\dots ,\,z_N$ which are in the center of $U(\frak g)$. \vskip .1in \noindent Let $\frak p_N$ be a parabolic subalgebra in $\frak g$ isomorphic to $\frak{gl}(N)\times \Bbb{R}^N$ (the stabilizer of a non-zero row vector in the standard realization). The crucial fact is that $D(\frak p_N)$ contains $2N$ elements satisfying canonical commutation relations such that their centralizer is isomorphic to $D(\frak p_{N-1})$.

Host: Nolan Wallach

AP&M 7218

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