Jordan Algebras¶
AUTHORS:
Travis Scrimshaw (2014-04-02): initial version
- class sage.algebras.jordan_algebra.JordanAlgebra¶
Bases:
sage.structure.parent.Parent
,sage.structure.unique_representation.UniqueRepresentation
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:
- class sage.algebras.jordan_algebra.JordanAlgebraSymmetricBilinear(R, form, names=None)¶
Bases:
sage.algebras.jordan_algebra.JordanAlgebra
A Jordan algebra given by a symmetric bilinear form \(m\).
- class Element(parent, s, v)¶
Bases:
sage.structure.element.AlgebraElement
An element of a Jordan algebra defined by a symmetric bilinear form.
- bar()¶
Return the result of the bar involution of
self
.The bar involution \(\bar{\cdot}\) is the \(R\)-linear endomorphism of \(M^*\) defined by \(\bar{1} = 1\) and \(\bar{x} = -x\) for \(x \in M\).
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J.<a,b,c> = JordanAlgebra(m) sage: x = 4*a - b + 3*c sage: x.bar() 4 + (1, -3)
We check that it is an algebra morphism:
sage: y = 2*a + 2*b - c sage: x.bar() * y.bar() == (x*y).bar() True
- monomial_coefficients(copy=True)¶
Return a dictionary whose keys are indices of basis elements in the support of
self
and whose values are the corresponding coefficients.INPUT:
copy
– ignored
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J.<a,b,c> = JordanAlgebra(m) sage: elt = a + 2*b - c sage: elt.monomial_coefficients() {0: 1, 1: 2, 2: -1}
- norm()¶
Return the norm of
self
.The norm of an element \(\alpha + x \in M^*\) is given by \(n(\alpha + x) = \alpha^2 - (x, x)\).
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J.<a,b,c> = JordanAlgebra(m) sage: x = 4*a - b + 3*c; x 4 + (-1, 3) sage: x.norm() 13
- trace()¶
Return the trace of
self
.The trace of an element \(\alpha + x \in M^*\) is given by \(t(\alpha + x) = 2 \alpha\).
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J.<a,b,c> = JordanAlgebra(m) sage: x = 4*a - b + 3*c sage: x.trace() 8
- algebra_generators()¶
Return a basis of
self
.The basis returned begins with the unity of \(R\) and continues with the standard basis of \(M\).
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J = JordanAlgebra(m) sage: J.basis() Family (1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
- basis()¶
Return a basis of
self
.The basis returned begins with the unity of \(R\) and continues with the standard basis of \(M\).
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J = JordanAlgebra(m) sage: J.basis() Family (1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
- gens()¶
Return the generators of
self
.EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J = JordanAlgebra(m) sage: J.basis() Family (1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
- one()¶
Return the element 1 if it exists.
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J = JordanAlgebra(m) sage: J.one() 1 + (0, 0)
- zero()¶
Return the element 0.
EXAMPLES:
sage: m = matrix([[0,1],[1,1]]) sage: J = JordanAlgebra(m) sage: J.zero() 0 + (0, 0)
- class sage.algebras.jordan_algebra.SpecialJordanAlgebra(A, names=None)¶
Bases:
sage.algebras.jordan_algebra.JordanAlgebra
A (special) Jordan algebra \(A^+\) from an associative algebra \(A\).
- class Element(parent, x)¶
Bases:
sage.structure.element.AlgebraElement
An element of a special Jordan algebra.
- monomial_coefficients(copy=True)¶
Return a dictionary whose keys are indices of basis elements in the support of
self
and whose values are the corresponding coefficients.INPUT:
copy
– (default:True
) ifself
is internally represented by a dictionaryd
, then make a copy ofd
; ifFalse
, then this can cause undesired behavior by mutatingd
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: a,b,c = map(J, F.gens()) sage: elt = a + 2*b - c sage: elt.monomial_coefficients() {x: 1, y: 2, z: -1}
- algebra_generators()¶
Return the basis of
self
.EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: J.basis() Lazy family (Term map(i))_{i in Free monoid on 3 generators (x, y, z)}
- basis()¶
Return the basis of
self
.EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: J.basis() Lazy family (Term map(i))_{i in Free monoid on 3 generators (x, y, z)}
- gens()¶
Return the generators of
self
.EXAMPLES:
sage: cat = Algebras(QQ).WithBasis().FiniteDimensional() sage: C = CombinatorialFreeModule(QQ, ['x','y','z'], category=cat) sage: J = JordanAlgebra(C) sage: J.gens() (B['x'], B['y'], B['z']) sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: J.gens() Traceback (most recent call last): ... NotImplementedError: infinite set
- one()¶
Return the element \(1\) if it exists.
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: J.one() 1
- zero()¶
Return the element \(0\).
EXAMPLES:
sage: F.<x,y,z> = FreeAlgebra(QQ) sage: J = JordanAlgebra(F) sage: J.zero() 0