X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=da3f6001e2d3878f6c5311eb309f8b2a22676a00;hb=9b6acc401eb02e9565db6212698662c9844c4239;hp=b8c7a912b1ded8149f7aefdf14b90b99d1056b2b;hpb=2a0cca4f62e8335db7069e04f0837a96b331614a;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index b8c7a91..da3f600 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -21,8 +21,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -46,18 +45,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names=names, category=cat, rank=rank, - natural_basis=natural_basis, - inner_product=inner_product) + natural_basis=natural_basis) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): """ EXAMPLES: @@ -73,7 +71,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank self._natural_basis = natural_basis - self._inner_product = inner_product + self._multiplication_table = mult_table fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -89,11 +87,182 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return fmt.format(self.degree(), self.base_ring()) + def _a_regular_element(self): + """ + Guess a regular element. Needed to compute the basis for our + characteristic polynomial coefficients. + """ + gs = self.gens() + z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) + if not z.is_regular(): + raise ValueError("don't know a regular element") + return z + + + @cached_method + def _charpoly_basis_space(self): + """ + Return the vector space spanned by the basis used in our + characteristic polynomial coefficients. This is used not only to + compute those coefficients, but also any time we need to + evaluate the coefficients (like when we compute the trace or + determinant). + """ + z = self._a_regular_element() + V = z.vector().parent().ambient_vector_space() + V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) ) + b = (V1.basis() + V1.complement().basis()) + return V.span_of_basis(b) + + + @cached_method + def _charpoly_coeff(self, i): + """ + Return the coefficient polynomial "a_{i}" of this algebra's + general characteristic polynomial. + + Having this be a separate cached method lets us compute and + store the trace/determinant (a_{r-1} and a_{0} respectively) + separate from the entire characteristic polynomial. + """ + (A_of_x, x, xr, detA) = self._charpoly_matrix_system() + R = A_of_x.base_ring() + if i >= self.rank(): + # Guaranteed by theory + return R.zero() + + # Danger: the in-place modification is done for performance + # reasons (reconstructing a matrix with huge polynomial + # entries is slow), but I don't know how cached_method works, + # so it's highly possible that we're modifying some global + # list variable by reference, here. In other words, you + # probably shouldn't call this method twice on the same + # algebra, at the same time, in two threads + Ai_orig = A_of_x.column(i) + A_of_x.set_column(i,xr) + numerator = A_of_x.det() + A_of_x.set_column(i,Ai_orig) + + # We're relying on the theory here to ensure that each a_i is + # indeed back in R, and the added negative signs are to make + # the whole charpoly expression sum to zero. + return R(-numerator/detA) + + + @cached_method + def _charpoly_matrix_system(self): + """ + Compute the matrix whose entries A_ij are polynomials in + X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector + corresponding to `x^r` and the determinent of the matrix A = + [A_ij]. In other words, all of the fixed (cachable) data needed + to compute the coefficients of the characteristic polynomial. + """ + r = self.rank() + n = self.dimension() + + # Construct a new algebra over a multivariate polynomial ring... + names = ['X' + str(i) for i in range(1,n+1)] + R = PolynomialRing(self.base_ring(), names) + J = FiniteDimensionalEuclideanJordanAlgebra(R, + self._multiplication_table, + rank=r) + + idmat = identity_matrix(J.base_ring(), n) + + W = self._charpoly_basis_space() + W = W.change_ring(R.fraction_field()) + + # Starting with the standard coordinates x = (X1,X2,...,Xn) + # and then converting the entries to W-coordinates allows us + # to pass in the standard coordinates to the charpoly and get + # back the right answer. Specifically, with x = (X1,X2,...,Xn), + # we have + # + # W.coordinates(x^2) eval'd at (standard z-coords) + # = + # W-coords of (z^2) + # = + # W-coords of (standard coords of x^2 eval'd at std-coords of z) + # + # We want the middle equivalent thing in our matrix, but use + # the first equivalent thing instead so that we can pass in + # standard coordinates. + x = J(vector(R, R.gens())) + l1 = [column_matrix(W.coordinates((x**k).vector())) for k in range(r)] + l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)] + A_of_x = block_matrix(R, 1, n, (l1 + l2)) + xr = W.coordinates((x**r).vector()) + return (A_of_x, x, xr, A_of_x.det()) + + + @cached_method + def characteristic_polynomial(self): + """ + + .. WARNING:: + + This implementation doesn't guarantee that the polynomial + denominator in the coefficients is not identically zero, so + theoretically it could crash. The way that this is handled + in e.g. Faraut and Koranyi is to use a basis that guarantees + the denominator is non-zero. But, doing so requires knowledge + of at least one regular element, and we don't even know how + to do that. The trade-off is that, if we use the standard basis, + the resulting polynomial will accept the "usual" coordinates. In + other words, we don't have to do a change of basis before e.g. + computing the trace or determinant. + + EXAMPLES: + + The characteristic polynomial in the spin algebra is given in + Alizadeh, Example 11.11:: + + sage: J = JordanSpinEJA(3) + sage: p = J.characteristic_polynomial(); p + X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 + sage: xvec = J.one().vector() + sage: p(*xvec) + t^2 - 2*t + 1 + + """ + r = self.rank() + n = self.dimension() + + # The list of coefficient polynomials a_1, a_2, ..., a_n. + a = [ self._charpoly_coeff(i) for i in range(n) ] + + # We go to a bit of trouble here to reorder the + # indeterminates, so that it's easier to evaluate the + # characteristic polynomial at x's coordinates and get back + # something in terms of t, which is what we want. + R = a[0].parent() + S = PolynomialRing(self.base_ring(),'t') + t = S.gen(0) + S = PolynomialRing(S, R.variable_names()) + t = S(t) + + # Note: all entries past the rth should be zero. The + # coefficient of the highest power (x^r) is 1, but it doesn't + # appear in the solution vector which contains coefficients + # for the other powers (to make them sum to x^r). + if (r < n): + a[r] = 1 # corresponds to x^r + else: + # When the rank is equal to the dimension, trying to + # assign a[r] goes out-of-bounds. + a.append(1) # corresponds to x^r + + return sum( a[k]*(t**k) for k in range(len(a)) ) + + def inner_product(self, x, y): """ The inner product associated with this Euclidean Jordan algebra. - Will default to the trace inner product if nothing else. + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. EXAMPLES: @@ -111,10 +280,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ if (not x in self) or (not y in self): raise TypeError("arguments must live in this algebra") - if self._inner_product is None: - return x.trace_inner_product(y) - else: - return self._inner_product(x,y) + return x.trace_inner_product(y) def natural_basis(self): @@ -132,7 +298,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() @@ -143,7 +309,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() @@ -180,14 +346,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity in `S^n` is converted to the identity in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: I = identity_matrix(QQ,3) sage: J(I) == J.one() True This skew-symmetric matrix can't be represented in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: A = matrix(QQ,3, lambda i,j: i-j) sage: J(A) Traceback (most recent call last): @@ -270,19 +436,82 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return A( (self.operator_matrix()**(n-1))*self.vector() ) + def apply_univariate_polynomial(self, p): + """ + Apply the univariate polynomial ``p`` to this element. + + A priori, SageMath won't allow us to apply a univariate + polynomial to an element of an EJA, because we don't know + that EJAs are rings (they are usually not associative). Of + course, we know that EJAs are power-associative, so the + operation is ultimately kosher. This function sidesteps + the CAS to get the answer we want and expect. + + EXAMPLES:: + + sage: R = PolynomialRing(QQ, 't') + sage: t = R.gen(0) + sage: p = t^4 - t^3 + 5*t - 2 + sage: J = RealCartesianProductEJA(5) + sage: J.one().apply_univariate_polynomial(p) == 3*J.one() + True + + TESTS: + + We should always get back an element of the algebra:: + + sage: set_random_seed() + sage: p = PolynomialRing(QQ, 't').random_element() + sage: J = random_eja() + sage: x = J.random_element() + sage: x.apply_univariate_polynomial(p) in J + True + + """ + if len(p.variables()) > 1: + raise ValueError("not a univariate polynomial") + P = self.parent() + R = P.base_ring() + # Convert the coeficcients to the parent's base ring, + # because a priori they might live in an (unnecessarily) + # larger ring for which P.sum() would fail below. + cs = [ R(c) for c in p.coefficients(sparse=False) ] + return P.sum( cs[k]*(self**k) for k in range(len(cs)) ) + + def characteristic_polynomial(self): """ - Return my characteristic polynomial (if I'm a regular - element). + Return the characteristic polynomial of this element. + + EXAMPLES: + + The rank of `R^3` is three, and the minimal polynomial of + the identity element is `(t-1)` from which it follows that + the characteristic polynomial should be `(t-1)^3`:: + + sage: J = RealCartesianProductEJA(3) + sage: J.one().characteristic_polynomial() + t^3 - 3*t^2 + 3*t - 1 + + Likewise, the characteristic of the zero element in the + rank-three algebra `R^{n}` should be `t^{3}`:: + + sage: J = RealCartesianProductEJA(3) + sage: J.zero().characteristic_polynomial() + t^3 + + The characteristic polynomial of an element should evaluate + to zero on that element:: + + sage: set_random_seed() + sage: x = RealCartesianProductEJA(3).random_element() + sage: p = x.characteristic_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 - 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') + p = self.parent().characteristic_polynomial() + return p(*self.vector()) def inner_product(self, other): @@ -295,7 +524,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): inner product on `R^n` (this example only works because the basis for the Jordan algebra is the standard basis in `R^n`):: - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: x = vector(QQ,[1,2,3]) sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) @@ -308,7 +537,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): so the inner product of the identity matrix with itself should be the `n`:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -317,13 +546,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): part because the product of Hermitian matrices may not be Hermitian:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 Ditto for the quaternions:: - sage: J = QuaternionHermitianSimpleEJA(3) + sage: J = QuaternionHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -390,24 +619,39 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() - sage: x = e0 + e1 + sage: x = sum( J.gens() ) sage: x.det() 0 - sage: J = JordanSpinSimpleEJA(3) + + :: + + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() - sage: x = e0 + e1 + e2 + sage: x = sum( J.gens() ) sage: x.det() -1 + TESTS: + + An element is invertible if and only if its determinant is + non-zero:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.is_invertible() == (x.det() != 0) + True + """ - 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') + P = self.parent() + r = P.rank() + p = P._charpoly_coeff(0) + # The _charpoly_coeff function already adds the factor of + # -1 to ensure that _charpoly_coeff(0) is really what + # appears in front of t^{0} in the charpoly. However, + # we want (-1)^r times THAT for the determinant. + return ((-1)**r)*p(*self.vector()) def inverse(self): @@ -424,7 +668,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: while x.is_zero(): ....: x = J.random_element() @@ -446,28 +690,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J.one().inverse() == J.one() True - If an element has an inverse, it acts like one. TODO: this - can be a lot less ugly once ``is_invertible`` doesn't crash - on irregular elements:: + If an element has an inverse, it acts like one:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() - sage: try: - ....: x.inverse()*x == J.one() - ....: except: - ....: True + sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) True """ + if not self.is_invertible(): + raise ValueError("element not invertible") + if self.parent().is_associative(): elt = FiniteDimensionalAlgebraElement(self.parent(), self) - return elt.inverse() - - # TODO: we can do better once the call to is_invertible() - # doesn't crash on irregular elements. - #if not self.is_invertible(): - # raise ValueError('element is not invertible') + # elt is in the right coordinates, but has the wrong class. + return self.parent()(elt.inverse().vector()) # We do this a little different than the usual recursive # call to a finite-dimensional algebra element, because we @@ -496,8 +734,36 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): We can't use the superclass method because it relies on the algebra being associative. + + ALGORITHM: + + The usual way to do this is to check if the determinant is + zero, but we need the characteristic polynomial for the + determinant. The minimal polynomial is a lot easier to get, + so we use Corollary 2 in Chapter V of Koecher to check + whether or not the paren't algebra's zero element is a root + of this element's minimal polynomial. + + TESTS: + + The identity element is always invertible:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.one().is_invertible() + True + + The zero element is never invertible:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.zero().is_invertible() + False + """ - return not self.det().is_zero() + zero = self.parent().zero() + p = self.minimal_polynomial() + return not (p(zero) == zero) def is_nilpotent(self): @@ -554,7 +820,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -579,7 +845,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -591,7 +857,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -602,14 +868,30 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): def minimal_polynomial(self): """ - EXAMPLES:: + Return the minimal polynomial of this element, + as a function of the variable `t`. + + ALGORITHM: + + We restrict ourselves to the associative subalgebra + generated by this element, and then return the minimal + polynomial of this element's operator matrix (in that + subalgebra). This works by Baes Proposition 2.3.16. + + TESTS: + + The minimal polynomial of the identity and zero elements are + always the same:: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True + sage: J = random_eja() + sage: J.one().minimal_polynomial() + t - 1 + sage: J.zero().minimal_polynomial() + t - :: + The degree of an element is (by one definition) the degree + of its minimal polynomial:: sage: set_random_seed() sage: x = random_eja().random_element() @@ -623,38 +905,38 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(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: t = PolynomialRing(J.base_ring(),'t').gen(0) + sage: expected = t^2 - 2*y0*t + (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 + The minimal polynomial should always kill its element:: - 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())) + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: p = x.minimal_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 - # Recursive call, but should work since elt lives in an - # associative algebra. - return elt.minimal_polynomial() + """ + 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())) + + # We get back a symbolic polynomial in 'x' but want a real + # polynomial in 't'. + p_of_x = elt.operator_matrix().minimal_polynomial() + return p_of_x.change_variable_name('t') def natural_representation(self): @@ -669,7 +951,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() @@ -682,7 +964,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = QuaternionHermitianSimpleEJA(3) + sage: J = QuaternionHermitianEJA(3) sage: J.one() e0 + e9 + e14 sage: J.one().natural_representation() @@ -783,7 +1065,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -804,38 +1086,77 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() + sage: Lx = x.operator_matrix() + sage: Lxx = (x*x).operator_matrix() + sage: Qx = x.quadratic_representation() + sage: Qy = y.quadratic_representation() + sage: Qxy = x.quadratic_representation(y) + sage: Qex = J.one().quadratic_representation(x) + sage: n = ZZ.random_element(10) + sage: Qxn = (x^n).quadratic_representation() Property 1: - sage: actual = x.quadratic_representation(y) - sage: expected = ( (x+y).quadratic_representation() - ....: -x.quadratic_representation() - ....: -y.quadratic_representation() ) / 2 - sage: actual == expected + sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy True Property 2: sage: alpha = QQ.random_element() - sage: actual = (alpha*x).quadratic_representation() - sage: expected = (alpha^2)*x.quadratic_representation() - sage: actual == expected + sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx + True + + Property 3: + + sage: not x.is_invertible() or ( + ....: Qx*x.inverse().vector() == x.vector() ) + True + + sage: not x.is_invertible() or ( + ....: Qx.inverse() + ....: == + ....: x.inverse().quadratic_representation() ) + True + + sage: Qxy*(J.one().vector()) == (x*y).vector() + True + + Property 4: + + sage: not x.is_invertible() or ( + ....: x.quadratic_representation(x.inverse())*Qx + ....: == Qx*x.quadratic_representation(x.inverse()) ) + True + + sage: not x.is_invertible() or ( + ....: x.quadratic_representation(x.inverse())*Qx + ....: == + ....: 2*x.operator_matrix()*Qex - Qx ) + True + + sage: 2*x.operator_matrix()*Qex - Qx == Lxx True Property 5: - sage: Qy = y.quadratic_representation() - sage: actual = J(Qy*x.vector()).quadratic_representation() - sage: expected = Qy*x.quadratic_representation()*Qy - sage: actual == expected + sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy True Property 6: - sage: k = ZZ.random_element(1,10) - sage: actual = (x^k).quadratic_representation() - sage: expected = (x.quadratic_representation())^k - sage: actual == expected + sage: Qxn == (Qx)^n + True + + Property 7: + + sage: not x.is_invertible() or ( + ....: Qx*x.inverse().operator_matrix() == Lx ) + True + + Property 8: + + sage: not x.operator_commutes_with(y) or ( + ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) ) True """ @@ -857,7 +1178,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): # 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() + # + # We do the extra ambient_vector_space() in case we're messing + # with polynomials and the direct parent is a module. + V = self.vector().parent().ambient_vector_space() return V.span( (self**d).vector() for d in xrange(V.dimension()) ) @@ -926,12 +1250,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) - sage: c = J.random_element().subalgebra_idempotent() - sage: c^2 == c - True - sage: J = JordanSpinSimpleEJA(5) - sage: c = J.random_element().subalgebra_idempotent() + sage: J = random_eja() + sage: x = J.random_element() + sage: while x.is_nilpotent(): + ....: x = J.random_element() + sage: c = x.subalgebra_idempotent() sage: c^2 == c True @@ -986,23 +1309,81 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(3) - sage: e0,e1,e2 = J.gens() - sage: x = e0 + e1 + e2 + sage: J = JordanSpinEJA(3) + sage: x = sum(J.gens()) sage: x.trace() 2 + :: + + sage: J = RealCartesianProductEJA(5) + sage: J.one().trace() + 5 + + TESTS: + + The trace of an element is a real number:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.random_element().trace() in J.base_ring() + True + """ - cs = self.characteristic_polynomial().coefficients(sparse=False) - if len(cs) >= 2: - return -1*cs[-2] - else: - raise ValueError('charpoly had fewer than 2 coefficients') + P = self.parent() + r = P.rank() + p = P._charpoly_coeff(r-1) + # The _charpoly_coeff function already adds the factor of + # -1 to ensure that _charpoly_coeff(r-1) is really what + # appears in front of t^{r-1} in the charpoly. However, + # we want the negative of THAT for the trace. + return -p(*self.vector()) def trace_inner_product(self, other): """ Return the trace inner product of myself and ``other``. + + TESTS: + + The trace inner product is commutative:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element(); y = J.random_element() + sage: x.trace_inner_product(y) == y.trace_inner_product(x) + True + + The trace inner product is bilinear:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: a = QQ.random_element(); + sage: actual = (a*(x+z)).trace_inner_product(y) + sage: expected = ( a*x.trace_inner_product(y) + + ....: a*z.trace_inner_product(y) ) + sage: actual == expected + True + sage: actual = x.trace_inner_product(a*(y+z)) + sage: expected = ( a*x.trace_inner_product(y) + + ....: a*x.trace_inner_product(z) ) + sage: actual == expected + True + + The trace inner product satisfies the compatibility + condition in the definition of a Euclidean Jordan algebra:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) + True + """ if not other in self.parent(): raise TypeError("'other' must live in the same algebra") @@ -1010,16 +1391,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return (self*other).trace() -def eja_rn(dimension, field=QQ): +class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + EXAMPLES: This multiplication table can be verified by hand:: - sage: J = eja_rn(3) + sage: J = RealCartesianProductEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -1035,19 +1420,21 @@ def eja_rn(dimension, field=QQ): 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, - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + # 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, n, n, lambda k,j: 1*(k == j == i)) + for i in xrange(n) ] + + fdeja = super(RealCartesianProductEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=n) + def inner_product(self, x, y): + return _usual_ip(x,y) def random_eja(): @@ -1082,12 +1469,17 @@ def random_eja(): Euclidean Jordan algebra of degree... """ - n = ZZ.random_element(1,5) - constructor = choice([eja_rn, - JordanSpinSimpleEJA, - RealSymmetricSimpleEJA, - ComplexHermitianSimpleEJA, - QuaternionHermitianSimpleEJA]) + + # The max_n component lets us choose different upper bounds on the + # value "n" that gets passed to the constructor. This is needed + # because e.g. R^{10} is reasonable to test, while the Hermitian + # 10-by-10 quaternion matrices are not. + (constructor, max_n) = choice([(RealCartesianProductEJA, 6), + (JordanSpinEJA, 6), + (RealSymmetricEJA, 5), + (ComplexHermitianEJA, 4), + (QuaternionHermitianEJA, 3)]) + n = ZZ.random_element(1, max_n) return constructor(n, field=QQ) @@ -1264,6 +1656,20 @@ def _embed_complex_matrix(M): [ 0 -1| 6 0] [ 1 0| 0 6] + TESTS: + + Embedding is a homomorphism (isomorphism, in fact):: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: F = QuadraticField(-1, 'i') + sage: X = random_matrix(F, n) + sage: Y = random_matrix(F, n) + sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) + sage: expected = _embed_complex_matrix(X*Y) + sage: actual == expected + True + """ n = M.nrows() if M.ncols() != n: @@ -1293,7 +1699,9 @@ def _unembed_complex_matrix(M): [ 2*i + 1 4*i + 3] [ 10*i + 9 12*i + 11] - TESTS:: + TESTS: + + Unembedding is the inverse of embedding:: sage: set_random_seed() sage: F = QuadraticField(-1, 'i') @@ -1347,6 +1755,18 @@ def _embed_quaternion_matrix(M): [-3 4 1 -2] [-4 -3 2 1] + Embedding is a homomorphism (isomorphism, in fact):: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: X = random_matrix(Q, n) + sage: Y = random_matrix(Q, n) + sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) + sage: expected = _embed_quaternion_matrix(X*Y) + sage: actual == expected + True + """ quaternions = M.base_ring() n = M.nrows() @@ -1385,7 +1805,9 @@ def _unembed_quaternion_matrix(M): sage: _unembed_quaternion_matrix(M) [1 + 2*i + 3*j + 4*k] - TESTS:: + TESTS: + + Unembedding is the inverse of embedding:: sage: set_random_seed() sage: Q = QuaternionAlgebra(QQ, -1, -1) @@ -1434,7 +1856,7 @@ def _matrix_ip(X,Y): return (X_mat*Y_mat).trace() -def RealSymmetricSimpleEJA(n, field=QQ): +class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1442,7 +1864,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1457,7 +1879,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + sage: J = RealSymmetricEJA(n) sage: J.degree() == (n^2 + n)/2 True @@ -1465,7 +1887,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + sage: J = RealSymmetricEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1478,17 +1900,23 @@ def RealSymmetricSimpleEJA(n, field=QQ): True """ - S = _real_symmetric_basis(n, field=field) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _real_symmetric_basis(n, field=field) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=_matrix_ip) + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + def inner_product(self, x, y): + return _matrix_ip(x,y) -def ComplexHermitianSimpleEJA(n, field=QQ): + +class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1501,7 +1929,7 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + sage: J = ComplexHermitianEJA(n) sage: J.degree() == n^2 True @@ -1509,7 +1937,7 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + sage: J = ComplexHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1522,26 +1950,30 @@ def ComplexHermitianSimpleEJA(n, field=QQ): True """ - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - # Since a+bi on the diagonal is represented as - # - # a + bi = [ a b ] - # [ -b a ], - # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - ip = lambda X,Y: _matrix_ip(X,Y)/2 + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) + def inner_product(self, x, y): + # Since a+bi on the diagonal is represented as + # + # a + bi = [ a b ] + # [ -b a ], + # + # we'll double-count the "a" entries if we take the trace of + # the embedding. + return _matrix_ip(x,y)/2 -def QuaternionHermitianSimpleEJA(n, field=QQ): +class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the @@ -1554,7 +1986,7 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianSimpleEJA(n) + sage: J = QuaternionHermitianEJA(n) sage: J.degree() == 2*(n^2) - n True @@ -1562,7 +1994,7 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianSimpleEJA(n) + sage: J = QuaternionHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1575,35 +2007,32 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): True """ - S = _quaternion_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) - - # Since a+bi+cj+dk on the diagonal is represented as - # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], - # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - ip = lambda X,Y: _matrix_ip(X,Y)/4 - - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) - + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _quaternion_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) -def OctonionHermitianSimpleEJA(n): - """ - This shit be crazy. It has dimension 27 over the reals. - """ - n = 3 - pass + fdeja = super(QuaternionHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) -def JordanSpinSimpleEJA(n, field=QQ): + def inner_product(self, x, y): + # Since a+bi+cj+dk on the diagonal is represented as + # + # a + bi +cj + dk = [ a b c d] + # [ -b a -d c] + # [ -c d a -b] + # [ -d -c b a], + # + # we'll quadruple-count the "a" entries if we take the trace of + # the embedding. + return _matrix_ip(x,y)/4 + + +class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = @@ -1614,7 +2043,7 @@ def JordanSpinSimpleEJA(n, field=QQ): This multiplication table can be verified by hand:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 @@ -1631,31 +2060,27 @@ def JordanSpinSimpleEJA(n, field=QQ): 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 - """ - 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), - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + 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 algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded by + # the ambient dimension). + fdeja = super(JordanSpinEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) + + def inner_product(self, x, y): + return _usual_ip(x,y)