2010 | OriginalPaper | Chapter
The Algebra f
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1.1.1. A
Carton datum
is a pair (
l
, ·) consisting of a finite set
I
and a symmetric bilinear form
$$v,{\rm{ }}v\prime \mapsto v{\rm{ }} \cdot {\rm{ }}v\prime$$
on the free abelian group Z[
I
], with values in Z. It is assumed that:
(a)
i · i
∈ {2,4,6,… } for any
i
∈
I;
(b)
$$2\frac{{i \cdot j}}{{i \cdot i}} \in \{ 0, - 1, - 2,\cdots\} $$
for any
i≠j in I.
Two Cartan data (
I
, ·) and (
I
, ○) are said to be
proportional
if there exist integers
a
,
b >
1 such that
ai
○
j = bi · j
for all
i,j ∈ I
.