Connection algebra: Difference between revisions

Revision as of 21:40, 5 April 2008

This article gives a global construction for a differential manifold. There exists a sheaf analog of it, that associates a similar construct to every open subset. This sheaf analog is termed: connection sheaf

Definition

Let $M$ be a differential manifold. The connection algebra of $M$ , denoted $C (M)$ , is defined as follows. Consider the Lie algebra of first-order differential operators on $M$ , and treat it as a $C^{\infty} (M)$ -bimodule. Take the tensor algebra generated by this as a $C^{\infty} (M)$ -bimodule, and quotient it by the two-sided ideal generated by $m (1) - 1$ . Here $m (1)$ is the differential operator obtained as multiplication by the constant function $1$ .

The quotient algebra we get is termed the connection algebra on $M$ .

The term connection algebra is also sometimes used for the connection sheaf, which is a sheaf that associates to every open subset, the connection algebra over that open subset.

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 ==Definition==
-Let <math>M</math> be a [[differential manifold]]. The '''connection algebra''' of <math>M</math>, denoted <math>\mathcal{C}(M)</math>, is defined as follows. Consider the [[Lie algebra of first-order differential operators]] on <math>M</math>, and treat it ''simply'' as a <math>\R</math>-vector space. Take the tensor algebra generated by this vector space, and quotient this tensor algebra by the two-sided ideal generated by <math>m(1) - 1</math>. Here <math>m(1)</math> is the differential operator obtained as multiplication by the constant function <math>1</math>.
+Let <math>M</math> be a [[differential manifold]]. The '''connection algebra''' of <math>M</math>, denoted <math>\mathcal{C}(M)</math>, is defined as follows. Consider the [[Lie algebra of first-order differential operators]] on <math>M</math>, and treat it as a <math>C^\infty(M)</math>-bimodule. Take the tensor algebra generated by this as a <math>C^\infty(M)</math>-bimodule, and quotient it by the two-sided ideal generated by <math>m(1) - 1</math>. Here <math>m(1)</math> is the differential operator obtained as multiplication by the constant function <math>1</math>.
 The quotient algebra we get is termed the '''connection algebra''' on <math>M</math>.
 The term '''connection algebra''' is also sometimes used for the [[connection sheaf]], which is a sheaf that associates to every open subset, the connection algebra over that open subset.