Deep Matrices and their Frankenstein Actions

The algebra of deep matrices E(X,A) is spanned over a coordinate algebra A by ``deep matrix units'' E_{kh} parameterized, not by single natural numbers like the standard matrix units E_{ij}, but by all ``deep indices'' or ``heads'' h,k (finite strings of natural numbers or some other infinite set X). This algebra has a natural Frankenstein action on the free right A-module V(X,A) with basis of all ``bodies'' b (infinite sequences or strings), where E_{kh} chops off head k from the body b and sews on a new head h (replaces an initial string k of b by h: E_{kh}(kd) = hd, E_{kh}(b) = 0 if b doesn't begin with the string k..

As with ordinary matrix algebras, the center and the ideals of the deep matrix algebra are just those of the coordinate algebra, because each nonzero element A is only ``distance 1'' away from a scalar: there exist a coordinate a and deep matrix units E,F such that EAF = a 1. In particular, over a simple coordinate algebra A the deep matrices form a simple unital algebra which acts irreducibly on each subspace of V(X,A), spanned by all b having the same ``tail,'' where two strings b,b' have the same tail if they become the same after chopping off suitable heads (of perhaps different sizes).

(Chapter 20 Non-Associative Algebra and its Applications CRC Press, Taylor & Francis Group 2006)

a href="mailto:kmm4m@virginia.edu">K. McCrimmon < kmm4m@virginia.edu >