From 486601c8b12cc184da19c7540011b88ee1f26282 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 5 Jul 2019 23:10:00 -0400 Subject: [PATCH 1/1] eja: replace eja_ln() and eja_sn() with their new names. --- mjo/eja/euclidean_jordan_algebra.py | 167 +++++++++++++--------------- 1 file changed, 77 insertions(+), 90 deletions(-) diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 6bc53a1..0a53667 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -133,12 +133,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(2) + sage: J = JordanSpinSimpleEJA(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 - sage: J = eja_ln(3) + sage: J = JordanSpinSimpleEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() @@ -207,7 +207,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = eja_ln(5) + sage: J = JordanSpinSimpleEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -232,7 +232,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(4) + sage: J = JordanSpinSimpleEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -244,7 +244,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -289,7 +289,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() @@ -333,7 +333,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Alizadeh's Example 11.12:: sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -433,7 +433,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = eja_ln(5) + sage: J = JordanSpinSimpleEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -489,7 +489,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(3) + sage: J = JordanSpinSimpleEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() @@ -549,82 +549,6 @@ def eja_rn(dimension, field=QQ): return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) -def eja_ln(dimension, field=QQ): - """ - Return the Jordan algebra corresponding to the Lorentz "ice cream" - cone of the given ``dimension``. - - EXAMPLES: - - This multiplication table can be verified by hand:: - - sage: J = eja_ln(4) - sage: e0,e1,e2,e3 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - e1 - sage: e0*e2 - e2 - sage: e0*e3 - e3 - sage: e1*e2 - 0 - sage: e1*e3 - 0 - sage: e2*e3 - 0 - - In one dimension, this is the reals under multiplication:: - - sage: J1 = eja_ln(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True - - """ - Qs = [] - id_matrix = identity_matrix(field,dimension) - for i in xrange(dimension): - ei = id_matrix.column(i) - Qi = zero_matrix(field,dimension) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += diagonal_matrix(dimension, [ei[0]]*dimension) - # The addition of the diagonal matrix adds an extra ei[0] in the - # upper-left corner of the matrix. - Qi[0,0] = Qi[0,0] * ~field(2) - Qs.append(Qi) - - # The rank of the spin factor algebra is two, UNLESS we're in a - # one-dimensional ambient space (the rank is bounded by the - # ambient dimension). - rank = min(dimension,2) - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank) - - -def eja_sn(dimension, field=QQ): - """ - Return the simple Jordan algebra of ``dimension``-by-``dimension`` - symmetric matrices over ``field``. - - EXAMPLES:: - - sage: J = eja_sn(2) - sage: e0, e1, e2 = J.gens() - sage: e0*e0 - e0 - sage: e1*e1 - e0 + e2 - sage: e2*e2 - e2 - - """ - S = _real_symmetric_basis(dimension, field=field) - Qs = _multiplication_table_from_matrix_basis(S) - - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) - def random_eja(): """ @@ -653,7 +577,10 @@ def random_eja(): """ n = ZZ.random_element(1,5).abs() - constructor = choice([eja_rn, eja_ln, eja_sn, ComplexHermitianSimpleEJA]) + constructor = choice([eja_rn, + JordanSpinSimpleEJA, + RealSymmetricSimpleEJA, + ComplexHermitianSimpleEJA]) return constructor(n, field=QQ) @@ -802,13 +729,28 @@ def _unembed_complex_matrix(M): return matrix(F, n/2, elements) -def RealSymmetricSimpleEJA(n): +def RealSymmetricSimpleEJA(n, field=QQ): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner product. It has dimension `(n^2 + n)/2` over the reals. + + EXAMPLES:: + + sage: J = RealSymmetricSimpleEJA(2) + sage: e0, e1, e2 = J.gens() + sage: e0*e0 + e0 + sage: e1*e1 + e0 + e2 + sage: e2*e2 + e2 """ - pass + S = _real_symmetric_basis(n, field=field) + Qs = _multiplication_table_from_matrix_basis(S) + + return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n) + def ComplexHermitianSimpleEJA(n, field=QQ): """ @@ -844,11 +786,56 @@ def OctonionHermitianSimpleEJA(n): n = 3 pass -def JordanSpinSimpleEJA(n): +def JordanSpinSimpleEJA(n, field=QQ): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = JordanSpinSimpleEJA(4) + sage: e0,e1,e2,e3 = J.gens() + sage: e0*e0 + e0 + sage: e0*e1 + e1 + sage: e0*e2 + e2 + sage: e0*e3 + e3 + sage: e1*e2 + 0 + sage: e1*e3 + 0 + sage: e2*e3 + 0 + + In one dimension, this is the reals under multiplication:: + + sage: J1 = JordanSpinSimpleEJA(1) + sage: J2 = eja_rn(1) + sage: J1 == J2 + True + """ - pass + Qs = [] + id_matrix = identity_matrix(field, n) + for i in xrange(n): + ei = id_matrix.column(i) + Qi = zero_matrix(field, n) + Qi.set_row(0, ei) + Qi.set_column(0, ei) + Qi += diagonal_matrix(n, [ei[0]]*n) + # The addition of the diagonal matrix adds an extra ei[0] in the + # upper-left corner of the matrix. + Qi[0,0] = Qi[0,0] * ~field(2) + Qs.append(Qi) + + # The rank of the spin factor algebra is two, UNLESS we're in a + # one-dimensional ambient space (the rank is bounded by the + # ambient dimension). + return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=min(n,2)) -- 2.44.2