Jordan Algebras

AUTHORS:

  • Travis Scrimshaw (2014-04-02): initial version

  • Travis Scrimshaw (2023-05-09): added the 27 dimensional exceptional Jordan algebra

class sage.algebras.jordan_algebra.ExceptionalJordanAlgebra(O)[source]

Bases: JordanAlgebra

The exceptional \(27\) dimensional Jordan algebra as self-adjoint \(3 \times 3\) matrix over an octonion algebra.

Let \(\mathbf{O}\) be the OctonionAlgebra over a commutative ring \(R\) of characteristic not equal to \(2\). The exceptional Jordan algebra \(\mathfrak{h}_3(\mathbf{O})\) is a \(27\) dimensional free \(R\)-module spanned by the matrices

\[\begin{split}\begin{bmatrix} \alpha & x & y \\ x^* & \beta & z \\ y^* & z^* & \gamma \end{bmatrix}\end{split}\]

for \(\alpha, \beta, \gamma \in R\) and \(x, y, z \in \mathbf{O}\), with multiplication given by the usual symmetrizer operation \(X \circ Y = \frac{1}{2}(XY + YX)\).

These are also known as Albert algebras due to the work of Abraham Adrian Albert on these algebras over \(\RR\).

EXAMPLES:

We construct an exceptional Jordan algebra over \(\QQ\) and perform some basic computations:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: gens = J.gens()
sage: gens[1]
[0 0 0]
[0 1 0]
[0 0 0]
sage: gens[3]
[0 1 0]
[1 0 0]
[0 0 0]
sage: gens[1] * gens[3]
[  0 1/2   0]
[1/2   0   0]
[  0   0   0]

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> gens = J.gens()
>>> gens[Integer(1)]
[0 0 0]
[0 1 0]
[0 0 0]
>>> gens[Integer(3)]
[0 1 0]
[1 0 0]
[0 0 0]
>>> gens[Integer(1)] * gens[Integer(3)]
[  0 1/2   0]
[1/2   0   0]
[  0   0   0]

The Lie algebra of derivations of the exceptional Jordan algebra is isomorphic to the simple Lie algebra of type \(F_4\). We verify that we the derivation module has the correct dimension:

sage: len(J.derivations_basis())  # long time
52
sage: LieAlgebra(QQ, cartan_type='F4').dimension()
52

>>> from sage.all import *
>>> len(J.derivations_basis())  # long time
52
>>> LieAlgebra(QQ, cartan_type='F4').dimension()
52

REFERENCES:

class Element(parent, data)[source]

Bases: AlgebraElement

An element of an exceptional Jordan algebra.

monomial_coefficients(copy=True)[source]

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: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: elt = sum(~QQ(ind) * b for ind, b in enumerate(J.basis()[::6], start=1)); elt
[              1           1/2*k  1/3*i + 1/4*lk]
[         -1/2*k               0          1/5*li]
[-1/3*i - 1/4*lk         -1/5*li               0]
sage: elt.monomial_coefficients()
{0: 1, 6: 1/2, 12: 1/3, 18: 1/4, 24: 1/5}

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> elt = sum(~QQ(ind) * b for ind, b in enumerate(J.basis()[::Integer(6)], start=Integer(1))); elt
[              1           1/2*k  1/3*i + 1/4*lk]
[         -1/2*k               0          1/5*li]
[-1/3*i - 1/4*lk         -1/5*li               0]
>>> elt.monomial_coefficients()
{0: 1, 6: 1/2, 12: 1/3, 18: 1/4, 24: 1/5}
algebra_generators()[source]

Return a basis of self.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: B = J.basis()
sage: B[::6]
([1 0 0]
 [0 0 0]
 [0 0 0],
 [ 0  k  0]
 [-k  0  0]
 [ 0  0  0],
 [ 0  0  i]
 [ 0  0  0]
 [-i  0  0],
 [  0   0  lk]
 [  0   0   0]
 [-lk   0   0],
 [  0   0   0]
 [  0   0  li]
 [  0 -li   0])
sage: len(B)
27

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> B = J.basis()
>>> B[::Integer(6)]
([1 0 0]
 [0 0 0]
 [0 0 0],
 [ 0  k  0]
 [-k  0  0]
 [ 0  0  0],
 [ 0  0  i]
 [ 0  0  0]
 [-i  0  0],
 [  0   0  lk]
 [  0   0   0]
 [-lk   0   0],
 [  0   0   0]
 [  0   0  li]
 [  0 -li   0])
>>> len(B)
27
basis()[source]

Return a basis of self.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: B = J.basis()
sage: B[::6]
([1 0 0]
 [0 0 0]
 [0 0 0],
 [ 0  k  0]
 [-k  0  0]
 [ 0  0  0],
 [ 0  0  i]
 [ 0  0  0]
 [-i  0  0],
 [  0   0  lk]
 [  0   0   0]
 [-lk   0   0],
 [  0   0   0]
 [  0   0  li]
 [  0 -li   0])
sage: len(B)
27

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> B = J.basis()
>>> B[::Integer(6)]
([1 0 0]
 [0 0 0]
 [0 0 0],
 [ 0  k  0]
 [-k  0  0]
 [ 0  0  0],
 [ 0  0  i]
 [ 0  0  0]
 [-i  0  0],
 [  0   0  lk]
 [  0   0   0]
 [-lk   0   0],
 [  0   0   0]
 [  0   0  li]
 [  0 -li   0])
>>> len(B)
27
gens()[source]

Return the generators of self.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: G = J.gens()
sage: G[0]
[1 0 0]
[0 0 0]
[0 0 0]
sage: G[5]
[ 0  j  0]
[-j  0  0]
[ 0  0  0]
sage: G[22]
[ 0  0  0]
[ 0  0  k]
[ 0 -k  0]

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> G = J.gens()
>>> G[Integer(0)]
[1 0 0]
[0 0 0]
[0 0 0]
>>> G[Integer(5)]
[ 0  j  0]
[-j  0  0]
[ 0  0  0]
>>> G[Integer(22)]
[ 0  0  0]
[ 0  0  k]
[ 0 -k  0]
one()[source]

Return multiplicative identity.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: J.one()
[1 0 0]
[0 1 0]
[0 0 1]
sage: all(J.one() * b == b for b in J.basis())
True

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> J.one()
[1 0 0]
[0 1 0]
[0 0 1]
>>> all(J.one() * b == b for b in J.basis())
True
some_elements()[source]

Return some elements of self.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: J.some_elements()
[[6/5   0   0]
 [  0 6/5   0]
 [  0   0 6/5],
 [1 0 0]
 [0 1 0]
 [0 0 1],
 [0 0 0]
 [0 0 0]
 [0 0 0],
 [0 0 0]
 [0 1 0]
 [0 0 0],
 [ 0  j  0]
 [-j  0  0]
 [ 0  0  0],
 [  0   0  lj]
 [  0   0   0]
 [-lj   0   0],
 [      0       0       0]
 [      0       1  1/2*lj]
 [      0 -1/2*lj       0],
 [        1         0  j + 2*li]
 [        0         1         0]
 [-j - 2*li         0         1],
 [      1  j + lk       l]
 [-j - lk       0  i + lj]
 [     -l -i - lj       0],
 [     1  3/2*l    2*k]
 [-3/2*l      0  5/2*j]
 [  -2*k -5/2*j      0]]

sage: O = OctonionAlgebra(GF(3))
sage: J = JordanAlgebra(O)
sage: J.some_elements()
[[-1  0  0]
 [ 0 -1  0]
 [ 0  0 -1],
 [1 0 0]
 [0 1 0]
 [0 0 1],
 [0 0 0]
 [0 0 0]
 [0 0 0],
 [0 0 0]
 [0 1 0]
 [0 0 0],
 [ 0  j  0]
 [-j  0  0]
 [ 0  0  0],
 [  0   0  lj]
 [  0   0   0]
 [-lj   0   0],
 [  0   0   0]
 [  0   1 -lj]
 [  0  lj   0],
 [      1       0  j - li]
 [      0       1       0]
 [-j + li       0       1],
 [      1  j + lk       l]
 [-j - lk       0  i + lj]
 [     -l -i - lj       0],
 [ 1  0 -k]
 [ 0  0  j]
 [ k -j  0]]

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> J.some_elements()
[[6/5   0   0]
 [  0 6/5   0]
 [  0   0 6/5],
 [1 0 0]
 [0 1 0]
 [0 0 1],
 [0 0 0]
 [0 0 0]
 [0 0 0],
 [0 0 0]
 [0 1 0]
 [0 0 0],
 [ 0  j  0]
 [-j  0  0]
 [ 0  0  0],
 [  0   0  lj]
 [  0   0   0]
 [-lj   0   0],
 [      0       0       0]
 [      0       1  1/2*lj]
 [      0 -1/2*lj       0],
 [        1         0  j + 2*li]
 [        0         1         0]
 [-j - 2*li         0         1],
 [      1  j + lk       l]
 [-j - lk       0  i + lj]
 [     -l -i - lj       0],
 [     1  3/2*l    2*k]
 [-3/2*l      0  5/2*j]
 [  -2*k -5/2*j      0]]

>>> O = OctonionAlgebra(GF(Integer(3)))
>>> J = JordanAlgebra(O)
>>> J.some_elements()
[[-1  0  0]
 [ 0 -1  0]
 [ 0  0 -1],
 [1 0 0]
 [0 1 0]
 [0 0 1],
 [0 0 0]
 [0 0 0]
 [0 0 0],
 [0 0 0]
 [0 1 0]
 [0 0 0],
 [ 0  j  0]
 [-j  0  0]
 [ 0  0  0],
 [  0   0  lj]
 [  0   0   0]
 [-lj   0   0],
 [  0   0   0]
 [  0   1 -lj]
 [  0  lj   0],
 [      1       0  j - li]
 [      0       1       0]
 [-j + li       0       1],
 [      1  j + lk       l]
 [-j - lk       0  i + lj]
 [     -l -i - lj       0],
 [ 1  0 -k]
 [ 0  0  j]
 [ k -j  0]]
zero()[source]

Return the additive identity.

EXAMPLES:

sage: O = OctonionAlgebra(QQ)
sage: J = JordanAlgebra(O)
sage: J.zero()
[0 0 0]
[0 0 0]
[0 0 0]

>>> from sage.all import *
>>> O = OctonionAlgebra(QQ)
>>> J = JordanAlgebra(O)
>>> J.zero()
[0 0 0]
[0 0 0]
[0 0 0]
class sage.algebras.jordan_algebra.JordanAlgebra[source]

Bases: UniqueRepresentation, Parent

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

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F); J
Jordan algebra of Free Algebra on 3 generators (x, y, z) over Rational Field
>>> a,b,c = map(J, F.gens())
>>> a*b
1/2*x*y + 1/2*y*x
>>> 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

>>> from sage.all import *
>>> (a*b)*c
1/4*x*y*z + 1/4*y*x*z + 1/4*z*x*y + 1/4*z*y*x
>>> 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

>>> from sage.all import *
>>> (a*b)*(a*a) == a*(b*(a*a))
True
>>> x = a + c
>>> y = b - Integer(2)*a
>>> (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)

>>> from sage.all import *
>>> m = matrix([[-Integer(2),Integer(3)],[Integer(3),Integer(4)]])
>>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3); J
Jordan algebra over Integer Ring given by the symmetric bilinear form:
[-2  3]
[ 3  4]
>>> a
1 + (0, 0)
>>> b
0 + (1, 0)
>>> x = Integer(3)*a - Integer(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)

>>> from sage.all import *
>>> (x*b)*b
-6 + (7, 0)
>>> 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

>>> from sage.all import *
>>> y = -a + Integer(4)*b - c
>>> (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]

>>> from sage.all import *
>>> J = JordanAlgebra(m, QQ, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3); J
Jordan algebra over Rational Field given by the symmetric bilinear form:
[-2  3]
[ 3  4]
>>> J = JordanAlgebra(QQ, m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3); 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)[source]

Bases: JordanAlgebra

A Jordan algebra given by a symmetric bilinear form \(m\).

class Element(parent, s, v)[source]

Bases: AlgebraElement

An element of a Jordan algebra defined by a symmetric bilinear form.

bar()[source]

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)

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3)
>>> x = Integer(4)*a - b + Integer(3)*c
>>> 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

>>> from sage.all import *
>>> y = Integer(2)*a + Integer(2)*b - c
>>> x.bar() * y.bar() == (x*y).bar()
True
monomial_coefficients(copy=True)[source]

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}

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3)
>>> elt = a + Integer(2)*b - c
>>> elt.monomial_coefficients()
{0: 1, 1: 2, 2: -1}
norm()[source]

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

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3)
>>> x = Integer(4)*a - b + Integer(3)*c; x
4 + (-1, 3)
>>> x.norm()
13
trace()[source]

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

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3)
>>> x = Integer(4)*a - b + Integer(3)*c
>>> x.trace()
8
algebra_generators()[source]

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))

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m)
>>> J.basis()
Family (1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
basis()[source]

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))

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m)
>>> J.basis()
Family (1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
gens()[source]

Return the generators of self.

EXAMPLES:

sage: m = matrix([[0,1],[1,1]])
sage: J = JordanAlgebra(m)
sage: J.gens()
(1 + (0, 0), 0 + (1, 0), 0 + (0, 1))

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m)
>>> J.gens()
(1 + (0, 0), 0 + (1, 0), 0 + (0, 1))
one()[source]

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)

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m)
>>> J.one()
1 + (0, 0)
zero()[source]

Return the element 0.

EXAMPLES:

sage: m = matrix([[0,1],[1,1]])
sage: J = JordanAlgebra(m)
sage: J.zero()
0 + (0, 0)

>>> from sage.all import *
>>> m = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> J = JordanAlgebra(m)
>>> J.zero()
0 + (0, 0)
class sage.algebras.jordan_algebra.SpecialJordanAlgebra(A, names=None)[source]

Bases: JordanAlgebra

A (special) Jordan algebra \(A^+\) from an associative algebra \(A\).

class Element(parent, x)[source]

Bases: AlgebraElement

An element of a special Jordan algebra.

monomial_coefficients(copy=True)[source]

Return a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

  • copy – boolean (default: True); if self is internally represented by a dictionary d, then make a copy of d; if False, then this can cause undesired behavior by mutating d

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}

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> a,b,c = map(J, F.gens())
>>> elt = a + Integer(2)*b - c
>>> elt.monomial_coefficients()
{x: 1, y: 2, z: -1}
algebra_generators()[source]

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)}

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> J.basis()
Lazy family (Term map(i))_{i in Free monoid on 3 generators (x, y, z)}
basis()[source]

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)}

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> J.basis()
Lazy family (Term map(i))_{i in Free monoid on 3 generators (x, y, z)}
gens()[source]

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

>>> from sage.all import *
>>> cat = Algebras(QQ).WithBasis().FiniteDimensional()
>>> C = CombinatorialFreeModule(QQ, ['x','y','z'], category=cat)
>>> J = JordanAlgebra(C)
>>> J.gens()
(B['x'], B['y'], B['z'])

>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> J.gens()
Traceback (most recent call last):
...
NotImplementedError: infinite set
one()[source]

Return the element \(1\) if it exists.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: J = JordanAlgebra(F)
sage: J.one()
1

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> J.one()
1
zero()[source]

Return the element \(0\).

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: J = JordanAlgebra(F)
sage: J.zero()
0

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> J = JordanAlgebra(F)
>>> J.zero()
0