Eric Forgy
differential envelope

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Definition

Let π’œ\mathcal{A} be a commutative algebra with unit 1 and define a bilinear product m:π’œβŠ—π’œβ†’π’œm:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A} by

m(aβŠ—b)=ab.m(a\otimes b) = ab.

The universal differential envelope Ω˜(π’œ)\tilde\Omega(\mathcal{A}) is the graded differential algebra

Ω˜(π’œ)=⨁ r=0 nΩ˜ r(π’œ),\tilde\Omega(\mathcal{A}) = \bigoplus_{r=0}^n \tilde\Omega^r(\mathcal{A}),

where

Ω˜ 0(π’œ) =π’œ, Ω˜ 1(π’œ) =Kerm, Ω˜ r(π’œ) =⨂ rtimesΩ˜ 1(π’œ).\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:Ω˜ r(π’œ)β†’Ξ©Λœ r+1(π’œ)d:\tilde\Omega^r(\mathcal{A})\to\tilde\Omega^{r+1}(\mathcal{A})

is given by the graded commutator

dα=[G˜,α],d\alpha= [\tilde G,\alpha],

where GΛœβˆˆπ’œβŠ—π’œ\tilde G\in\mathcal{A}\otimes\mathcal{A} is the universal graph operator

G˜=1βŠ—1.\tilde G = 1\otimes 1.

Note that although G˜\tilde G is not in Ω˜ 1(π’œ)\tilde\Omega^1(\mathcal{A}), given any a∈Ω˜ 0(π’œ)a\in\tilde\Omega^0(\mathcal{A})

[G˜,a]=1βŠ—aβˆ’aβŠ—1[\tilde G,a] = 1\otimes a - a\otimes 1

is in Ω˜ 1(π’œ)\tilde\Omega^1(\mathcal{A}) since

m(1βŠ—aβˆ’aβŠ—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 GG, let π’œ\mathcal{A} be the algebra of projections on G 0G_0 in which case

1=βˆ‘ vΟ€ v.1 = \sum_v \pi_v.

The universal graph operator is then given by

G˜=βˆ‘ v,vβ€²βˆˆG 0Ο€ vβŠ—Ο€ vβ€².\tilde G = \sum_{v,v'\in G_0} \pi_v\otimes\pi_{v'}.

Interpreting the element Ο€ vβŠ—Ο€ vβ€²\pi_v\otimes\pi_{v'} as (being dual to) a directed edge vβ†’vβ€²v\to v', the universal graph operator is seen to correspond to the complete directed graph with vertices G 0G_0.

Example: Smooth Manifolds

Given a smooth manifold β„³\mathcal{M}, let π’œ\mathcal{A} be the algebra of 0-forms on β„³\mathcal{M}.

References

category: maths

Revised on September 25, 2009 at 04:00:47 by Eric Forgy