Jordan Algebras¶
AUTHORS:
- Travis Scrimshaw (2014-04-02): initial version
-
sage.algebras.jordan_algebra.
JordanAlgebra
¶ A Jordan algebra.
A Jordan algebra is a magmatic algebra (over a commutative ring \(R\)) whose multiplication satisfies the following axioms:
- \(xy = yx\), and
- \((xy)(xx) = x(y(xx))\) (the Jordan identity).
See [Ja1971], [Ch2012], and [McC1978], for example.
These axioms imply that a Jordan algebra is power-associative and the following generalization of Jordan’s identity holds [Al1947]: \((x^m y) x^n = x^m (y x^n)\) for all \(m, n \in \ZZ_{>0}\).
Let \(A\) be an associative algebra over a ring \(R\) in which \(2\) is invertible. We construct a Jordan algebra \(A^+\) with ground set \(A\) by defining the multiplication as
\[x \circ y = \frac{xy + yx}{2}.\]Often the multiplication is written as \(x \circ y\) to avoid confusion with the product in the associative algebra \(A\). We note that if \(A\) is commutative then this reduces to the usual multiplication in \(A\).
Jordan algebras constructed in this fashion, or their subalgebras, are called special. All other Jordan algebras are called exceptional.
Jordan algebras can also be constructed from a module \(M\) over \(R\) with a symmetric bilinear form \((\cdot, \cdot) : M \times M \to R\). We begin with the module \(M^* = R \oplus M\) and define multiplication in \(M^*\) by
\[(\alpha + x) \circ (\beta + y) = \underbrace{\alpha \beta + (x,y)}_{\in R} + \underbrace{\beta x + \alpha y}_{\in M}\]where \(\alpha, \beta \in R\) and \(x,y \in M\).
INPUT:
Can be either an associative algebra \(A\) or a symmetric bilinear form given as a matrix (possibly followed by, or preceded by, a base ring argument)
EXAMPLES:
We let the base algebra \(A\) be the free algebra on 3 generators:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F); J Jordan algebra of Free Algebra on 3 generators (x, y, z) over Rational Field sage: a,b,c = map(J, F.gens()) sage: a*b 1/2*x*y + 1/2*y*x sage: b*a 1/2*x*y + 1/2*y*x
Jordan algebras are typically non-associative:
sage: (a*b)*c 1/4*x*y*z + 1/4*y*x*z + 1/4*z*x*y + 1/4*z*y*x sage: a*(b*c) 1/4*x*y*z + 1/4*x*z*y + 1/4*y*z*x + 1/4*z*y*x
We check the Jordan identity:
sage: (a*b)*(a*a) == a*(b*(a*a)) True sage: x = a + c sage: y = b - 2*a sage: (x*y)*(x*x) == x*(y*(x*x)) True
Next we construct a Jordan algebra from a symmetric bilinear form:
sage: m = matrix([[-2,3],[3,4]]) sage: J.<a,b,c> = JordanAlgebra(m); J Jordan algebra over Integer Ring given by the symmetric bilinear form: [-2 3] [ 3 4] sage: a 1 + (0, 0) sage: b 0 + (1, 0) sage: x = 3*a - 2*b + c; x 3 + (-2, 1)
We again show that Jordan algebras are usually non-associative:
sage: (x*b)*b -6 + (7, 0) sage: x*(b*b) -6 + (4, -2)
We verify the Jordan identity:
sage: y = -a + 4*b - c sage: (x*y)*(x*x) == x*(y*(x*x)) True
The base ring, while normally inferred from the matrix, can also be explicitly specified:
sage: J.<a,b,c> = JordanAlgebra(m, QQ); J Jordan algebra over Rational Field given by the symmetric bilinear form: [-2 3] [ 3 4] sage: J.<a,b,c> = JordanAlgebra(QQ, m); J # either order work Jordan algebra over Rational Field given by the symmetric bilinear form: [-2 3] [ 3 4]
REFERENCES:
-
sage.algebras.jordan_algebra.
JordanAlgebraSymmetricBilinear
¶ A Jordan algebra given by a symmetric bilinear form \(m\).
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sage.algebras.jordan_algebra.
SpecialJordanAlgebra
¶ A (special) Jordan algebra \(A^+\) from an associative algebra \(A\).