Eric Forgy algebra of projections

Definition

Given a set SS, the algebra of projections on SS is the commutative algebra 𝒜\mathcal{A} spanned by the projections

{π s:SS|sS}\{\pi_s:S\to S|s\in S\}

satisfying

sSπ s=1\sum_{s\in S} \pi_s = 1

whose bilinear product m:𝒜𝒜𝒜m:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A} is given by

m(π sπ s)=δ s,sπ s.m(\pi_s\otimes\pi_{s'}) = \delta_{s,s'} \pi_s.

Note to Self

See Section 3.1 of

category: maths

Revised on April 10, 2011 at 16:18:22 by Eric Forgy