""" Euclidean Jordan Algebras. These are formally-real Jordan Algebras; specifically those where u^2 + v^2 = 0 implies that u = v = 0. They are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ from sage.categories.magmatic_algebras import MagmaticAlgebras from sage.structure.element import is_Matrix from sage.structure.category_object import normalize_names from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): @staticmethod def __classcall_private__(cls, field, mult_table, names='e', assume_associative=False, category=None, rank=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: b.set_immutable() if not (is_Matrix(b) and b.dimensions() == (n, n)): raise ValueError("input is not a multiplication table") mult_table = tuple(mult_table) cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis() cat.or_subcategory(category) if assume_associative: cat = cat.Associative() names = normalize_names(n, names) fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls) return fda.__classcall__(cls, field, mult_table, assume_associative=assume_associative, names=names, category=cat, rank=rank) def __init__(self, field, mult_table, names='e', assume_associative=False, category=None, rank=None): self._rank = rank fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, names=names, category=category) def _repr_(self): """ Return a string representation of ``self``. """ fmt = "Euclidean Jordan algebra of degree {} over {}" return fmt.format(self.degree(), self.base_ring()) def rank(self): """ Return the rank of this EJA. """ if self._rank is None: raise ValueError("no rank specified at genesis") else: return self._rank class Element(FiniteDimensionalAlgebraElement): """ An element of a Euclidean Jordan algebra. """ def __pow__(self, n): """ Return ``self`` raised to the power ``n``. Jordan algebras are always power-associative; see for example Faraut and Koranyi, Proposition II.1.2 (ii). .. WARNING: We have to override this because our superclass uses row vectors instead of column vectors! We, on the other hand, assume column vectors everywhere. EXAMPLES: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.matrix()*x.vector() == (x**2).vector() True """ A = self.parent() if n == 0: return A.one() elif n == 1: return self else: return A.element_class(A, (self.matrix()**(n-1))*self.vector()) def characteristic_polynomial(self): """ Return my characteristic polynomial (if I'm a regular element). Eventually this should be implemented in terms of the parent algebra's characteristic polynomial that works for ALL elements. """ if self.is_regular(): return self.minimal_polynomial() else: raise NotImplementedError('irregular element') def det(self): """ Return my determinant, the product of my eigenvalues. EXAMPLES:: sage: J = eja_ln(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 sage: J = eja_ln(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() -1 """ cs = self.characteristic_polynomial().coefficients(sparse=False) r = len(cs) - 1 if r >= 0: return cs[0] * (-1)**r else: raise ValueError('charpoly had no coefficients') def is_nilpotent(self): """ Return whether or not some power of this element is zero. The superclass method won't work unless we're in an associative algebra, and we aren't. However, we generate an assocoative subalgebra and we're nilpotent there if and only if we're nilpotent here (probably). TESTS: The identity element is never nilpotent:: sage: set_random_seed() sage: random_eja().one().is_nilpotent() False The additive identity is always nilpotent:: sage: set_random_seed() sage: random_eja().zero().is_nilpotent() True """ # The element we're going to call "is_nilpotent()" on. # Either myself, interpreted as an element of a finite- # dimensional algebra, or an element of an associative # subalgebra. elt = None if self.parent().is_associative(): elt = FiniteDimensionalAlgebraElement(self.parent(), self) else: V = self.span_of_powers() assoc_subalg = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! elt = assoc_subalg(V.coordinates(self.vector())) # Recursive call, but should work since elt lives in an # associative algebra. return elt.is_nilpotent() def is_regular(self): """ Return whether or not this is a regular element. EXAMPLES: The identity element always has degree one, but any element linearly-independent from it is regular:: sage: J = eja_ln(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity sage: for x in J.gens(): ....: (J.one() + x).is_regular() False True True True True """ return self.degree() == self.parent().rank() def degree(self): """ Compute the degree of this element the straightforward way according to the definition; by appending powers to a list and figuring out its dimension (that is, whether or not they're linearly dependent). EXAMPLES:: sage: J = eja_ln(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() sage: (e0 - e1).degree() 2 In the spin factor algebra (of rank two), all elements that aren't multiples of the identity are regular:: sage: set_random_seed() sage: n = ZZ.random_element(1,10).abs() sage: J = eja_ln(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True """ return self.span_of_powers().dimension() def matrix(self): """ Return the matrix that represents left- (or right-) multiplication by this element in the parent algebra. We have to override this because the superclass method returns a matrix that acts on row vectors (that is, on the right). """ fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) return fda_elt.matrix().transpose() def minimal_polynomial(self): """ EXAMPLES:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.degree() == x.minimal_polynomial().degree() True :: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.degree() == x.minimal_polynomial().degree() True The minimal polynomial and the characteristic polynomial coincide and are known (see Alizadeh, Example 11.11) for all elements of the spin factor algebra that aren't scalar multiples of the identity:: sage: set_random_seed() sage: n = ZZ.random_element(2,10).abs() sage: J = eja_ln(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() sage: y0 = y.vector()[0] sage: y_bar = y.vector()[1:] sage: actual = y.minimal_polynomial() sage: x = SR.symbol('x', domain='real') sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2) sage: bool(actual == expected) True """ # The element we're going to call "minimal_polynomial()" on. # Either myself, interpreted as an element of a finite- # dimensional algebra, or an element of an associative # subalgebra. elt = None if self.parent().is_associative(): elt = FiniteDimensionalAlgebraElement(self.parent(), self) else: V = self.span_of_powers() assoc_subalg = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! elt = assoc_subalg(V.coordinates(self.vector())) # Recursive call, but should work since elt lives in an # associative algebra. return elt.minimal_polynomial() def quadratic_representation(self): """ Return the quadratic representation of this element. EXAMPLES: The explicit form in the spin factor algebra is given by Alizadeh's Example 11.12:: sage: n = ZZ.random_element(1,10).abs() sage: J = eja_ln(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) sage: B = 2*x0*x_bar.row() sage: C = 2*x0*x_bar.column() sage: D = identity_matrix(QQ, n-1) sage: D = (x0^2 - x_bar.inner_product(x_bar))*D sage: D = D + 2*x_bar.tensor_product(x_bar) sage: Q = block_matrix(2,2,[A,B,C,D]) sage: Q == x.quadratic_representation() True """ return 2*(self.matrix()**2) - (self**2).matrix() def span_of_powers(self): """ Return the vector space spanned by successive powers of this element. """ # The dimension of the subalgebra can't be greater than # the big algebra, so just put everything into a list # and let span() get rid of the excess. V = self.vector().parent() return V.span( (self**d).vector() for d in xrange(V.dimension()) ) def subalgebra_generated_by(self): """ Return the associative subalgebra of the parent EJA generated by this element. TESTS:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.subalgebra_generated_by().is_associative() True Squaring in the subalgebra should be the same thing as squaring in the superalgebra:: sage: set_random_seed() sage: x = random_eja().random_element() sage: u = x.subalgebra_generated_by().random_element() sage: u.matrix()*u.vector() == (u**2).vector() True """ # First get the subspace spanned by the powers of myself... V = self.span_of_powers() F = self.base_ring() # Now figure out the entries of the right-multiplication # matrix for the successive basis elements b0, b1,... of # that subspace. mats = [] for b_right in V.basis(): eja_b_right = self.parent()(b_right) b_right_rows = [] # The first row of the right-multiplication matrix by # b1 is what we get if we apply that matrix to b1. The # second row of the right multiplication matrix by b1 # is what we get when we apply that matrix to b2... # # IMPORTANT: this assumes that all vectors are COLUMN # vectors, unlike our superclass (which uses row vectors). for b_left in V.basis(): eja_b_left = self.parent()(b_left) # Multiply in the original EJA, but then get the # coordinates from the subalgebra in terms of its # basis. this_row = V.coordinates((eja_b_left*eja_b_right).vector()) b_right_rows.append(this_row) b_right_matrix = matrix(F, b_right_rows) mats.append(b_right_matrix) # It's an algebra of polynomials in one element, and EJAs # are power-associative. # # TODO: choose generator names intelligently. return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f') def subalgebra_idempotent(self): """ Find an idempotent in the associative subalgebra I generate using Proposition 2.3.5 in Baes. TESTS:: sage: set_random_seed() sage: J = eja_rn(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True sage: J = eja_ln(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True """ if self.is_nilpotent(): raise ValueError("this only works with non-nilpotent elements!") V = self.span_of_powers() J = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! u = J(V.coordinates(self.vector())) # The image of the matrix of left-u^m-multiplication # will be minimal for some natural number s... s = 0 minimal_dim = V.dimension() for i in xrange(1, V.dimension()): this_dim = (u**i).matrix().image().dimension() if this_dim < minimal_dim: minimal_dim = this_dim s = i # Now minimal_matrix should correspond to the smallest # non-zero subspace in Baes's (or really, Koecher's) # proposition. # # However, we need to restrict the matrix to work on the # subspace... or do we? Can't we just solve, knowing that # A(c) = u^(s+1) should have a solution in the big space, # too? # # Beware, solve_right() means that we're using COLUMN vectors. # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.matrix() c_coordinates = A.solve_right(u_next.vector()) # Now c_coordinates is the idempotent we want, but it's in # the coordinate system of the subalgebra. # # We need the basis for J, but as elements of the parent algebra. # basis = [self.parent(v) for v in V.basis()] return self.parent().linear_combination(zip(c_coordinates, basis)) def trace(self): """ Return my trace, the sum of my eigenvalues. EXAMPLES:: sage: J = eja_ln(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() 2 """ cs = self.characteristic_polynomial().coefficients(sparse=False) if len(cs) >= 2: return -1*cs[-2] else: raise ValueError('charpoly had fewer than 2 coefficients') def trace_inner_product(self, other): """ Return the trace inner product of myself and ``other``. """ if not other in self.parent(): raise ArgumentError("'other' must live in the same algebra") return (self*other).trace() def eja_rn(dimension, field=QQ): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. EXAMPLES: This multiplication table can be verified by hand:: sage: J = eja_rn(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 sage: e0*e1 0 sage: e0*e2 0 sage: e1*e1 e1 sage: e1*e2 0 sage: e2*e2 e2 """ # The FiniteDimensionalAlgebra constructor takes a list of # matrices, the ith representing right multiplication by the ith # basis element in the vector space. So if e_1 = (1,0,0), then # right (Hadamard) multiplication of x by e_1 picks out the first # component of x; and likewise for the ith basis element e_i. Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i)) for i in xrange(dimension) ] 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(): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. ALGORITHM: For now, we choose a random natural number ``n`` (greater than zero) and then give you back one of the following: * The cartesian product of the rational numbers ``n`` times; this is ``QQ^n`` with the Hadamard product. * The Jordan spin algebra on ``QQ^n``. * The ``n``-by-``n`` rational symmetric matrices with the symmetric product. Later this might be extended to return Cartesian products of the EJAs above. TESTS:: sage: random_eja() Euclidean Jordan algebra of degree... """ n = ZZ.random_element(1,10).abs() constructor = choice([eja_rn, eja_ln, eja_sn]) return constructor(dimension=n, field=QQ) def _real_symmetric_basis(n, field=QQ): """ Return a basis for the space of real symmetric n-by-n matrices. """ # The basis of symmetric matrices, as matrices, in their R^(n-by-n) # coordinates. S = [] for i in xrange(n): for j in xrange(i+1): Eij = matrix(field, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: # Beware, orthogonal but not normalized! Sij = Eij + Eij.transpose() S.append(Sij) return S def _multiplication_table_from_matrix_basis(basis): """ At least three of the five simple Euclidean Jordan algebras have the symmetric multiplication (A,B) |-> (AB + BA)/2, where the multiplication on the right is matrix multiplication. Given a basis for the underlying matrix space, this function returns a multiplication table (obtained by looping through the basis elements) for an algebra of those matrices. """ # In S^2, for example, we nominally have four coordinates even # though the space is of dimension three only. The vector space V # is supposed to hold the entire long vector, and the subspace W # of V will be spanned by the vectors that arise from symmetric # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. field = basis[0].base_ring() dimension = basis[0].nrows() def mat2vec(m): return vector(field, m.list()) def vec2mat(v): return matrix(field, dimension, v.list()) V = VectorSpace(field, dimension**2) W = V.span( mat2vec(s) for s in basis ) # Taking the span above reorders our basis (thanks, jerk!) so we # need to put our "matrix basis" in the same order as the # (reordered) vector basis. S = [ vec2mat(b) for b in W.basis() ] Qs = [] for s in basis: # Brute force the multiplication-by-s matrix by looping # through all elements of the basis and doing the computation # to find out what the corresponding row should be. BEWARE: # these multiplication tables won't be symmetric! It therefore # becomes REALLY IMPORTANT that the underlying algebra # constructor uses ROW vectors and not COLUMN vectors. That's # why we're computing rows here and not columns. Q_rows = [] for t in basis: this_row = mat2vec((s*t + t*s)/2) Q_rows.append(W.coordinates(this_row)) Q = matrix(field,Q_rows) Qs.append(Q) return Qs def RealSymmetricSimpleEJA(n): """ 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. """ pass def ComplexHermitianSimpleEJA(n): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, and the real-part-of-trace inner product. It has dimension `n^2 over the reals. """ pass def QuaternionHermitianSimpleEJA(n): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the real-part-of-trace inner product. It has dimension `2n^2 - n` over the reals. """ pass def OctonionHermitianSimpleEJA(n): """ This shit be crazy. It has dimension 27 over the reals. """ n = 3 pass def JordanSpinSimpleEJA(n): """ 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. """ pass