Jordan rings #
Let A
be a non-unital, non-associative ring. Then A
is said to be a (commutative, linear) Jordan
ring if the multiplication is commutative and satisfies a weak associativity law known as the
Jordan Identity: for all a
and b
in A
,
(a * b) * a^2 = a * (b * a^2)
i.e. the operators of multiplication by a
and a^2
commute.
A more general concept of a (non-commutative) Jordan ring can also be defined, as a
(non-commutative, non-associative) ring A
where, for each a
in A
, the operators of left and
right multiplication by a
and a^2
commute.
Every associative algebra can be equipped with a symmetrized multiplication (characterized by
SymAlg.sym_mul_sym
) making it into a commutative Jordan algebra (IsCommJordan
).
Jordan algebras arising this way are said to be special.
A real Jordan algebra A
can be introduced by
variable {A : Type*} [NonUnitalNonAssocCommRing A] [Module ℝ A] [SMulCommClass ℝ A A]
[IsScalarTower ℝ A A] [IsCommJordan A]
Main results #
two_nsmul_lie_lmul_lmul_add_add_eq_zero
: Linearisation of the commutative Jordan axiom
Implementation notes #
We shall primarily be interested in linear Jordan algebras (i.e. over rings of characteristic not two) leaving quadratic algebras to those better versed in that theory.
The conventional way to linearise the Jordan axiom is to equate coefficients (more formally, assume that the axiom holds in all field extensions). For simplicity we use brute force algebraic expansion and substitution instead.
Motivation #
Every Jordan algebra A
has a triple product defined, for a
b
and c
in A
by
There are also exceptional Jordan algebras which can be shown not to be the symmetrization of any associative algebra. The 3x3 matrices of octonions is the canonical example.
Non-commutative Jordan algebras have connections to the Vidav-Palmer theorem [cabreragarciarodriguezpalacios2014].
References #
- [Cabrera García and Rodríguez Palacios, Non-associative normed algebras. Volume 1] [cabreragarciarodriguezpalacios2014]
- [Hanche-Olsen and Størmer, Jordan Operator Algebras][hancheolsenstormer1984]
- [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
A (non-commutative) Jordan multiplication.
Instances
A (commutative) Jordan multiplication is also a Jordan multiplication
Semigroup multiplication satisfies the (non-commutative) Jordan axioms
The Jordan axioms can be expressed in terms of commuting multiplication operators.
The endomorphisms on an additive monoid AddMonoid.End
form a Ring
, and this may be equipped
with a Lie Bracket via Ring.bracket
.