Residue Auslander Algebra of the Pair Node

Let

M = B0 ⊕ R ⊕ B3

be the CM generator of the pair factor. Reducing the exact endomorphism order modulo 3 gives an 8-dimensional basic algebra with basis

e0, eR, e3, a, b, c, d, t,

where

a : B0 -> R,    b : R -> B0,
c : B3 -> R,    d : R -> B3,
t : R -> R

and t is the residue of the loop eta.

Quiver and relations

The ordinary quiver has three vertices and five arrows:

0 --a--> R --b--> 0,
3 --c--> R --d--> 3,
R --t--> R.

The defining residue relations are

b a = 0,
d c = 0,
b c = 0,
d a = 0,
a b = -t,
c d =  t,
t a = 0,
b t = 0,
t c = 0,
d t = 0,
t^2 = 0.

So t = c d = - a b, and the algebra is generated by the four branch arrows together with one square-zero loop at the node.

Radical structure

The Jacobson radical is generated by

a, b, c, d, t.

Its square is one-dimensional:

rad^2 = <t>,

and

rad^3 = 0.

So the residue Auslander algebra is a radical-cube-zero quiver algebra with exactly one nonzero length-2 path class.

Cartan matrix

With vertex order (B0,R,B3), the Cartan matrix is

[[1,1,0],
 [1,2,1],
 [0,1,1]].

Its singularity reflects the nodal defect already seen in the Hecke order.

This is the finite residue shadow of the exact Z_3 node: a three-vertex algebra whose only length-2 memory is the single nilpotent loop t at the center.