Page 460 -

The Category WT(R) and the Weierstrass Hopf Algebroid 435

Substituting, we have

2

2
2
2
2
x = v x + r = v (v x + r ) + r = (vv ) x + (v r + r),
and

3

2
2

2
2

3
y = v y + v sx + t = v (v y + v s x + t ) + v s(v x + r ) + t

3

3
2
2

3
2

3
= (vv ) y + (v v s + (vv ) s)x + (v t + v sr + t).
If we introduce the following three by three matrix
 
1 r t
 0 2 2
M(φ r,s,t,v ) = v v s 
0 0 v 3
with variables r, s, t,v ∈ R, then the substitution rule takes the form of the following
matrix identity
     2 3 2 
1 r t 1 r t 1 r v + r t v + r sv + t
2
2
2
2

 0 v 2 s v   0 v 2 sv  =  0 (vv ) s (vv )  .
3
0 0 v 3 0 0 v 3 0 0 (vv )

where v = v v and three other relations

2
3

r = v r + r, v 2 3 2 2 and t = v t + v sr + t.

s = v v s + (vv ) s,
2

Here the matrix multiplication is in the opposite order from composition, but with
the transpose matrices we have a matrix formula for composition
tr tr tr
M (φ r,s,t,v )M (φ r ,s ,t ,v ) = M (φ r ,s ,t ).

Also the relation between the variables x, y and x , y canbeexpressedbythe matrix
multiplication formula
 
1 r t
2
(1, x, y) = (1, x , y )M(φ r,s,t,v ) = (1, x , y ) 0 v 2 v s  .


0 0 v 3

For v = v = 1 the transpose matrices satisfy the multiplicative relation in the
opposite order
     
1 0 0 1 0 0 1 0 0
 r 1 0  =  r 1 0   r 1 0  .
t s 1 t s 1 t s 1
(4.3) Remark. If there is a morphism φ r,s,t,v : F(a ) → F(a), then we can express

the constants a in terms of the constants a i and r, s, t,v. For this we begin with the
i
relation