Eric Forgy
differential envelope
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Definition
Let π \mathcal{A} m : π β π β π m:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}
m ( a β b ) = ab . m(a\otimes b) = ab. The universal differential envelope Ξ© Λ ( π ) \tilde\Omega(\mathcal{A}) graded differential algebra
Ξ© Λ ( π ) = β¨ r = 0 n Ξ© Λ r ( π ) , \tilde\Omega(\mathcal{A}) = \bigoplus_{r=0}^n \tilde\Omega^r(\mathcal{A}), where
Ξ© Λ 0 ( π ) = π , Ξ© Λ 1 ( π ) = Ker m , Ξ© Λ r ( π ) = β¨ r times Ξ© Λ 1 ( π ) . \begin{aligned}
\tilde\Omega^0(\mathcal{A}) &= \mathcal{A}, \\
\tilde\Omega^1(\mathcal{A}) &= \Ker{m}, \\
\tilde\Omega^r(\mathcal{A}) &= \bigotimes_{r\:\text{times}} \tilde\Omega^1(\mathcal{A}).
\end{aligned}
The derivation
d : Ξ© Λ r ( π ) β Ξ© Λ r + 1 ( π ) d:\tilde\Omega^r(\mathcal{A})\to\tilde\Omega^{r+1}(\mathcal{A}) is given by the graded commutator
d Ξ± = [ G Λ , Ξ± ] , d\alpha= [\tilde G,\alpha], where G Λ β π β π \tilde G\in\mathcal{A}\otimes\mathcal{A} universal graph operator
G Λ = 1 β 1 . \tilde G = 1\otimes 1. Note that although G Λ \tilde G Ξ© Λ 1 ( π ) \tilde\Omega^1(\mathcal{A}) a β Ξ© Λ 0 ( π ) a\in\tilde\Omega^0(\mathcal{A})
[ G Λ , a ] = 1 β a β a β 1 [\tilde G,a] = 1\otimes a - a\otimes 1 is in Ξ© Λ 1 ( π ) \tilde\Omega^1(\mathcal{A})
m ( 1 β a β a β 1 ) = 0 . m(1\otimes a - a\otimes 1) = 0. Any differential graded algebra may be derived as a quotient of the universal differential envelope by a two-sided differential ideal .
Example: Directed Graphs
Given a directed graph G G π \mathcal{A} algebra of projections on G 0 G_0
1 = β v Ο v . 1 = \sum_v \pi_v. The universal graph operator is then given by
G Λ = β v , v β² β G 0 Ο v β Ο v β² . \tilde G = \sum_{v,v'\in G_0} \pi_v\otimes\pi_{v'}. Interpreting the element Ο v β Ο v β² \pi_v\otimes\pi_{v'} v β v β² v\to v' complete directed graph with vertices G 0 G_0
Example: Smooth Manifolds
Given a smooth manifold β³ \mathcal{M} π \mathcal{A} β³ \mathcal{M}
References
Revised on September 25, 2009 at 04:00:47
by
Eric Forgy