The Yang-Baxter equation for a linear operator 
, where 
is a vector
space over a field, is
 |
(1)
|
This is an identity of linear operators on the triple vector space tensor product 
. Each 
applies 
to factors 
and 
and leaves the remaining factor unchanged. Explicitly, 
, 
, and 
, where 
is the identity
operator on 
and 
interchanges the two factors (Etingof et al. 1999).
In the set-theoretic form, 
is a map of sets,
and the same equation is an identity of maps on 
. Writing 
and 
for the identity function
on 
,
the maps are 
,
,
and
 |
(2)
|
Composing 
with the flip gives the braid convention 
, which satisfies
 |
(3)
|
Invertibility and nondegeneracy are extra restrictions on a set-theoretic solution and are not imposed here.
An automorphism of a set-theoretic solution is a bijection 
satisfying 
. Militaru (2026) proved that every group occurs as the automorphism
group of such a solution, and a finite group can be
realized on a finite set 
. Militaru (2026) reports Claude assistance with mathematical
constructions and proof development.
