algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on operads model structure on algebras over an operad A Jordan algebra is an algebra that may not be associative, but is commutative, subject to some further conditions which are modeled after the archetypical example: for any associative algebra, equipping it with the symmetrized product makes a Jordan algebra. It is this relation that originally motivated the notion in discussion of quantum mechanics, for the symmetrized product and hence the Jordan algebra structure of the algebra of observables of a quantum mechanical system is what remains when one ignores the otherwise all-important commutators and hence the Hamiltonian flows on observables. Later Jordan algebras were also studied for pure mathematical reasons, including their connection to self-dual convex homogeneous cones, hermitian symmetric spaces, 3-graded Lie algebras and the exceptional Lie algebras , and . In the process the Jordan algebra concept was generalized slightly to define quadratic Jordan algebras, Jordan triple systems and Jordan pairs. More recently, the Bohr topos associated to a noncommutative algebra of observables was found to depend on the underlying Jordan algebra structure. See at Bohr topos and poset of commutative subalgebras for more on this. A Jordan algebra is a commutative nonassociative algebra satisfying the Jordan identity for all in . It follows (via a nontrivial argument) that is power-associative, and the Jordan identity generalizes to for natural numbers (and, trivially, for