X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=a0ba1c68bb30bf54383807c52961ad82e4a69f03;hb=cc75c4e093a920c44f1eaaef4a1baa525b5c5727;hp=835f76337f9449612bcd113de950afd011317107;hpb=5fcc3b1bdec67eec2d76cc39c79a25af00830e30;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 835f763..a0ba1c6 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -20,7 +20,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names='e', assume_associative=False, category=None, - rank=None): + rank=None, + natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -43,16 +44,34 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=assume_associative, names=names, category=cat, - rank=rank) + rank=rank, + natural_basis=natural_basis) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, - rank=None): + rank=None, + natural_basis=None): + """ + EXAMPLES: + + By definition, Jordan multiplication commutes:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x*y == y*x + True + + """ self._rank = rank + self._natural_basis = natural_basis + self._multiplication_table = mult_table fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -67,6 +86,172 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): fmt = "Euclidean Jordan algebra of degree {} over {}" return fmt.format(self.degree(), self.base_ring()) + + + @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) = self._charpoly_matrix() + R = A_of_x.base_ring() + A_cols = A_of_x.columns() + A_cols[i] = (x**self.rank()).vector() + numerator = column_matrix(A_of_x.base_ring(), A_cols).det() + denominator = A_of_x.det() + + # 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/denominator) + + + @cached_method + def _charpoly_matrix(self): + """ + Compute the matrix whose entries A_ij are polynomials in + X1,...,XN. This same matrix is used in more than one method and + it's not so fast to construct. + """ + 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) + + x = J(vector(R, R.gens())) + l1 = [column_matrix((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)) + return (A_of_x, x) + + + @cached_method + def characteristic_polynomial(self): + """ + 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. + + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. + + EXAMPLES: + + The inner product must satisfy its axiom for this algebra to truly + be 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).inner_product(z) == y.inner_product(x*z) + True + + """ + if (not x in self) or (not y in self): + raise TypeError("arguments must live in this algebra") + return x.trace_inner_product(y) + + + def natural_basis(self): + """ + Return a more-natural representation of this algebra's basis. + + Every finite-dimensional Euclidean Jordan Algebra is a direct + sum of five simple algebras, four of which comprise Hermitian + matrices. This method returns the original "natural" basis + for our underlying vector space. (Typically, the natural basis + is used to construct the multiplication table in the first place.) + + Note that this will always return a matrix. The standard basis + in `R^n` will be returned as `n`-by-`1` column matrices. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: J.basis() + Family (e0, e1, e2) + sage: J.natural_basis() + ( + [1 0] [0 1] [0 0] + [0 0], [1 0], [0 1] + ) + + :: + + sage: J = JordanSpinEJA(2) + sage: J.basis() + Family (e0, e1) + sage: J.natural_basis() + ( + [1] [0] + [0], [1] + ) + + """ + if self._natural_basis is None: + return tuple( b.vector().column() for b in self.basis() ) + else: + return self._natural_basis + + def rank(self): """ Return the rank of this EJA. @@ -80,20 +265,52 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): class Element(FiniteDimensionalAlgebraElement): """ An element of a Euclidean Jordan algebra. + """ - Since EJAs are commutative, the "right multiplication" matrix is - also the left multiplication matrix and must be symmetric:: + def __init__(self, A, elt=None): + """ + EXAMPLES: - sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_rn(5) - sage: J.random_element().matrix().is_symmetric() - True - sage: J = eja_ln(5) - sage: J.random_element().matrix().is_symmetric() - True + The identity in `S^n` is converted to the identity in the EJA:: - """ + 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 = RealSymmetricEJA(3) + sage: A = matrix(QQ,3, lambda i,j: i-j) + sage: J(A) + Traceback (most recent call last): + ... + ArithmeticError: vector is not in free module + + """ + # Goal: if we're given a matrix, and if it lives in our + # parent algebra's "natural ambient space," convert it + # into an algebra element. + # + # The catch is, we make a recursive call after converting + # the given matrix into a vector that lives in the algebra. + # This we need to try the parent class initializer first, + # to avoid recursing forever if we're given something that + # already fits into the algebra, but also happens to live + # in the parent's "natural ambient space" (this happens with + # vectors in R^n). + try: + FiniteDimensionalAlgebraElement.__init__(self, A, elt) + except ValueError: + natural_basis = A.natural_basis() + if elt in natural_basis[0].matrix_space(): + # Thanks for nothing! Matrix spaces aren't vector + # spaces in Sage, so we have to figure out its + # natural-basis coordinates ourselves. + V = VectorSpace(elt.base_ring(), elt.nrows()**2) + W = V.span( _mat2vec(s) for s in natural_basis ) + coords = W.coordinates(_mat2vec(elt)) + FiniteDimensionalAlgebraElement.__init__(self, A, coords) def __pow__(self, n): """ @@ -108,12 +325,32 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): instead of column vectors! We, on the other hand, assume column vectors everywhere. - EXAMPLES: + EXAMPLES:: sage: set_random_seed() - sage: J = eja_ln(5) - sage: x = J.random_element() - sage: x.matrix()*x.vector() == (x**2).vector() + sage: x = random_eja().random_element() + sage: x.operator_matrix()*x.vector() == (x^2).vector() + True + + A few examples of power-associativity:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x*(x*x)*(x*x) == x^5 + True + sage: (x*x)*(x*x*x) == x^5 + True + + We also know that powers operator-commute (Koecher, Chapter + III, Corollary 1):: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: m = ZZ.random_element(0,10) + sage: n = ZZ.random_element(0,10) + sage: Lxm = (x^m).operator_matrix() + sage: Lxn = (x^n).operator_matrix() + sage: Lxm*Lxn == Lxn*Lxm True """ @@ -123,19 +360,398 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): elif n == 1: return self else: - return A.element_class(A, (self.matrix()**(n-1))*self.vector()) + return A( (self.operator_matrix()**(n-1))*self.vector() ) - def span_of_powers(self): + def apply_univariate_polynomial(self, p): """ - Return the vector space spanned by successive powers of - this element. + 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 + """ - # 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()) ) + 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 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 + + """ + p = self.parent().characteristic_polynomial() + return p(*self.vector()) + + + def inner_product(self, other): + """ + Return the parent algebra's inner product of myself and ``other``. + + EXAMPLES: + + The inner product in the Jordan spin algebra is the usual + 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 = JordanSpinEJA(3) + sage: x = vector(QQ,[1,2,3]) + sage: y = vector(QQ,[4,5,6]) + sage: x.inner_product(y) + 32 + sage: J(x).inner_product(J(y)) + 32 + + The inner product on `S^n` is ` = trace(X*Y)`, where + multiplication is the usual matrix multiplication in `S^n`, + so the inner product of the identity matrix with itself + should be the `n`:: + + sage: J = RealSymmetricEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + Likewise, the inner product on `C^n` is ` = + Re(trace(X*Y))`, where we must necessarily take the real + part because the product of Hermitian matrices may not be + Hermitian:: + + sage: J = ComplexHermitianEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + Ditto for the quaternions:: + + sage: J = QuaternionHermitianEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + TESTS: + + Ensure that we can always compute an inner product, and that + it gives us back a real number:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.inner_product(y) in RR + True + + """ + P = self.parent() + if not other in P: + raise TypeError("'other' must live in the same algebra") + + return P.inner_product(self, other) + + + def operator_commutes_with(self, other): + """ + Return whether or not this element operator-commutes + with ``other``. + + EXAMPLES: + + The definition of a Jordan algebra says that any element + operator-commutes with its square:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.operator_commutes_with(x^2) + True + + TESTS: + + Test Lemma 1 from Chapter III of Koecher:: + + sage: set_random_seed() + sage: J = random_eja() + sage: u = J.random_element() + sage: v = J.random_element() + sage: lhs = u.operator_commutes_with(u*v) + sage: rhs = v.operator_commutes_with(u^2) + sage: lhs == rhs + True + + """ + if not other in self.parent(): + raise TypeError("'other' must live in the same algebra") + + A = self.operator_matrix() + B = other.operator_matrix() + return (A*B == B*A) + + + def det(self): + """ + Return my determinant, the product of my eigenvalues. + + EXAMPLES:: + + sage: J = JordanSpinEJA(2) + sage: e0,e1 = J.gens() + sage: x = e0 + e1 + sage: x.det() + 0 + sage: J = JordanSpinEJA(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 inverse(self): + """ + Return the Jordan-multiplicative inverse of this element. + + We can't use the superclass method because it relies on the + algebra being associative. + + EXAMPLES: + + The inverse in the spin factor algebra is given in Alizadeh's + Example 11.11:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,10) + sage: J = JordanSpinEJA(n) + sage: x = J.random_element() + sage: while x.is_zero(): + ....: x = J.random_element() + sage: x_vec = x.vector() + sage: x0 = x_vec[0] + sage: x_bar = x_vec[1:] + sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar)) + sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) + sage: x_inverse = coeff*inv_vec + sage: x.inverse() == J(x_inverse) + True + + TESTS: + + The identity element is its own inverse:: + + sage: set_random_seed() + sage: J = random_eja() + 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:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: try: + ....: x.inverse()*x == J.one() + ....: except: + ....: True + True + + """ + 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') + + # We do this a little different than the usual recursive + # call to a finite-dimensional algebra element, because we + # wind up with an inverse that lives in the subalgebra and + # we need information about the parent to convert it back. + 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())) + + # This will be in the subalgebra's coordinates... + fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt) + subalg_inverse = fda_elt.inverse() + + # So we have to convert back... + basis = [ self.parent(v) for v in V.basis() ] + pairs = zip(subalg_inverse.vector(), basis) + return self.parent().linear_combination(pairs) + + + def is_invertible(self): + """ + Return whether or not this element is invertible. + + 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 + + """ + zero = self.parent().zero() + p = self.minimal_polynomial() + return not (p(zero) == zero) + + + 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 = JordanSpinEJA(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): @@ -147,38 +763,305 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(4) + sage: J = JordanSpinEJA(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:: + 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) + sage: J = JordanSpinEJA(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 minimal_polynomial(self): + """ + 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: 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() + 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) + 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: 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 minimal polynomial should always kill its element:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: p = x.minimal_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 + + """ + 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): + """ + Return a more-natural representation of this element. + + Every finite-dimensional Euclidean Jordan Algebra is a + direct sum of five simple algebras, four of which comprise + Hermitian matrices. This method returns the original + "natural" representation of this element as a Hermitian + matrix, if it has one. If not, you get the usual representation. + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3) + sage: J.one() + e0 + e5 + e8 + sage: J.one().natural_representation() + [1 0 0 0 0 0] + [0 1 0 0 0 0] + [0 0 1 0 0 0] + [0 0 0 1 0 0] + [0 0 0 0 1 0] + [0 0 0 0 0 1] + + :: + + sage: J = QuaternionHermitianEJA(3) + sage: J.one() + e0 + e9 + e14 + sage: J.one().natural_representation() + [1 0 0 0 0 0 0 0 0 0 0 0] + [0 1 0 0 0 0 0 0 0 0 0 0] + [0 0 1 0 0 0 0 0 0 0 0 0] + [0 0 0 1 0 0 0 0 0 0 0 0] + [0 0 0 0 1 0 0 0 0 0 0 0] + [0 0 0 0 0 1 0 0 0 0 0 0] + [0 0 0 0 0 0 1 0 0 0 0 0] + [0 0 0 0 0 0 0 1 0 0 0 0] + [0 0 0 0 0 0 0 0 1 0 0 0] + [0 0 0 0 0 0 0 0 0 1 0 0] + [0 0 0 0 0 0 0 0 0 0 1 0] + [0 0 0 0 0 0 0 0 0 0 0 1] + + """ + B = self.parent().natural_basis() + W = B[0].matrix_space() + return W.linear_combination(zip(self.vector(), B)) + + + def operator_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). + + EXAMPLES: + + Test the first polarization identity from my notes, Koecher Chapter + III, or from Baes (2.3):: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lxx = (x*x).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) + True + + Test the second polarization identity from my notes or from + Baes (2.4):: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: Lxz = (x*z).operator_matrix() + sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) + True + + Test the third polarization identity from my notes or from + Baes (2.5):: + + sage: set_random_seed() + sage: J = random_eja() + sage: u = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: Lu = u.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Luy = (u*y).operator_matrix() + sage: Luz = (u*z).operator_matrix() + sage: Luyz = (u*(y*z)).operator_matrix() + sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz + sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly + sage: bool(lhs == rhs) + True + + """ + fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) + return fda_elt.matrix().transpose() + + + def quadratic_representation(self, other=None): + """ + 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: set_random_seed() + sage: n = ZZ.random_element(1,10) + sage: J = JordanSpinEJA(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 + + Test all of the properties from Theorem 11.2 in Alizadeh:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) + sage: J = random_eja() sage: x = J.random_element() - sage: x == x.coefficient(0)*J.one() or x.degree() == 2 + sage: y = J.random_element() + + Property 1: + + sage: actual = x.quadratic_representation(y) + sage: expected = ( (x+y).quadratic_representation() + ....: -x.quadratic_representation() + ....: -y.quadratic_representation() ) / 2 + sage: actual == expected True - """ - return self.span_of_powers().dimension() + Property 2: + sage: alpha = QQ.random_element() + sage: actual = (alpha*x).quadratic_representation() + sage: expected = (alpha^2)*x.quadratic_representation() + sage: actual == expected + 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 + 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 + True - def matrix(self): """ - Return the matrix that represents left- (or right-) - multiplication by this element in the parent algebra. + if other is None: + other=self + elif not other in self.parent(): + raise TypeError("'other' must live in the same algebra") - We have to override this because the superclass method - returns a matrix that acts on row vectors (that is, on - the right). + L = self.operator_matrix() + M = other.operator_matrix() + return ( L*M + M*L - (self*other).operator_matrix() ) + + + def span_of_powers(self): """ - fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) - return fda_elt.matrix().transpose() + 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. + # + # 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()) ) def subalgebra_generated_by(self): @@ -189,23 +1072,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_rn(n) - sage: x = J.random_element() - sage: x.subalgebra_generated_by().is_associative() - True - sage: J = eja_ln(n) - sage: x = J.random_element() + 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: J = eja_ln(5) - sage: x = J.random_element() + 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() + sage: u.operator_matrix()*u.vector() == (u**2).vector() True """ @@ -244,122 +1121,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f') - def minimal_polynomial(self): - """ - EXAMPLES:: - - sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_rn(n) - sage: x = J.random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True - - :: - - sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) - sage: x = J.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 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: n = ZZ.random_element(2,10).abs() - sage: J = eja_rn(n) - sage: J.one().is_nilpotent() - False - sage: J = eja_ln(n) - sage: J.one().is_nilpotent() - False - - The additive identity is always nilpotent:: - - sage: set_random_seed() - sage: n = ZZ.random_element(2,10).abs() - sage: J = eja_rn(n) - sage: J.zero().is_nilpotent() - True - sage: J = eja_ln(n) - sage: J.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 subalgebra_idempotent(self): """ Find an idempotent in the associative subalgebra I generate @@ -368,11 +1129,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) + sage: J = RealCartesianProductEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = eja_ln(5) + sage: J = JordanSpinEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -393,7 +1154,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): s = 0 minimal_dim = V.dimension() for i in xrange(1, V.dimension()): - this_dim = (u**i).matrix().image().dimension() + this_dim = (u**i).operator_matrix().image().dimension() if this_dim < minimal_dim: minimal_dim = this_dim s = i @@ -410,7 +1171,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): # Beware, solve_right() means that we're using COLUMN vectors. # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) - A = u_next.matrix() + A = u_next.operator_matrix() c_coordinates = A.solve_right(u_next.vector()) # Now c_coordinates is the idempotent we want, but it's in @@ -422,21 +1183,67 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return self.parent().linear_combination(zip(c_coordinates, basis)) + def trace(self): + """ + Return my trace, the sum of my eigenvalues. + + EXAMPLES:: + + 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 + + """ + 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 characteristic_polynomial(self): - return self.matrix().characteristic_polynomial() + + def trace_inner_product(self, other): + """ + Return the trace inner product of myself and ``other``. + """ + if not other in self.parent(): + raise TypeError("'other' must live in the same algebra") + + 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 @@ -452,27 +1259,630 @@ 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) ] + @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(): + """ + 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. + + * The ``n``-by-``n`` complex-rational Hermitian matrices embedded + in the space of ``2n``-by-``2n`` real symmetric matrices. + + * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded + in the space of ``4n``-by-``4n`` real symmetric matrices. + + Later this might be extended to return Cartesian products of the + EJAs above. + + TESTS:: + + sage: random_eja() + Euclidean Jordan algebra of degree... + + """ + + # 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) + + + +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 tuple(S) + + +def _complex_hermitian_basis(n, field=QQ): + """ + Returns a basis for the space of complex Hermitian n-by-n matrices. + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) + True + + """ + F = QuadraticField(-1, 'I') + I = F.gen() + + # This is like the symmetric case, but we need to be careful: + # + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. + # + 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 = _embed_complex_matrix(Eij) + S.append(Sij) + else: + # Beware, orthogonal but not normalized! The second one + # has a minus because it's conjugated. + Sij_real = _embed_complex_matrix(Eij + Eij.transpose()) + S.append(Sij_real) + Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) + S.append(Sij_imag) + return tuple(S) + + +def _quaternion_hermitian_basis(n, field=QQ): + """ + Returns a basis for the space of quaternion Hermitian n-by-n matrices. + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) + True + + """ + Q = QuaternionAlgebra(QQ,-1,-1) + I,J,K = Q.gens() + + # This is like the symmetric case, but we need to be careful: + # + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. + # + S = [] + for i in xrange(n): + for j in xrange(i+1): + Eij = matrix(Q, n, lambda k,l: k==i and l==j) + if i == j: + Sij = _embed_quaternion_matrix(Eij) + S.append(Sij) + else: + # Beware, orthogonal but not normalized! The second, + # third, and fourth ones have a minus because they're + # conjugated. + Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) + S.append(Sij_real) + Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) + S.append(Sij_I) + Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) + S.append(Sij_J) + Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) + S.append(Sij_K) + return tuple(S) + + +def _mat2vec(m): + return vector(m.base_ring(), m.list()) + +def _vec2mat(v): + return matrix(v.base_ring(), sqrt(v.degree()), v.list()) + +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. A reordered copy + of the basis is also returned to work around the fact that + the ``span()`` in this function will change the order of the basis + from what we think it is, to... something else. + """ + # 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() + + 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 = tuple( _vec2mat(b) for b in W.basis() ) + + Qs = [] + for s in S: + # 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 S: + this_row = _mat2vec((s*t + t*s)/2) + Q_rows.append(W.coordinates(this_row)) + Q = matrix(field, W.dimension(), Q_rows) + Qs.append(Q) + + return (Qs, S) + + +def _embed_complex_matrix(M): + """ + Embed the n-by-n complex matrix ``M`` into the space of real + matrices of size 2n-by-2n via the map the sends each entry `z = a + + bi` to the block matrix ``[[a,b],[-b,a]]``. + + EXAMPLES:: + + sage: F = QuadraticField(-1,'i') + sage: x1 = F(4 - 2*i) + sage: x2 = F(1 + 2*i) + sage: x3 = F(-i) + sage: x4 = F(6) + sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) + sage: _embed_complex_matrix(M) + [ 4 -2| 1 2] + [ 2 4|-2 1] + [-----+-----] + [ 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: + raise ValueError("the matrix 'M' must be square") + field = M.base_ring() + blocks = [] + for z in M.list(): + a = z.real() + b = z.imag() + blocks.append(matrix(field, 2, [[a,b],[-b,a]])) + + # We can drop the imaginaries here. + return block_matrix(field.base_ring(), n, blocks) + + +def _unembed_complex_matrix(M): + """ + The inverse of _embed_complex_matrix(). + + EXAMPLES:: + + sage: A = matrix(QQ,[ [ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [ 9, 10, 11, 12], + ....: [-10, 9, -12, 11] ]) + sage: _unembed_complex_matrix(A) + [ 2*i + 1 4*i + 3] + [ 10*i + 9 12*i + 11] + + TESTS: + + Unembedding is the inverse of embedding:: + + sage: set_random_seed() + sage: F = QuadraticField(-1, 'i') + sage: M = random_matrix(F, 3) + sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M + True + + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + if not n.mod(2).is_zero(): + raise ValueError("the matrix 'M' must be a complex embedding") + + F = QuadraticField(-1, 'i') + i = F.gen() + + # Go top-left to bottom-right (reading order), converting every + # 2-by-2 block we see to a single complex element. + elements = [] + for k in xrange(n/2): + for j in xrange(n/2): + submat = M[2*k:2*k+2,2*j:2*j+2] + if submat[0,0] != submat[1,1]: + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0]: + raise ValueError('bad off-diagonal submatrix') + z = submat[0,0] + submat[0,1]*i + elements.append(z) + + return matrix(F, n/2, elements) + + +def _embed_quaternion_matrix(M): + """ + Embed the n-by-n quaternion matrix ``M`` into the space of real + matrices of size 4n-by-4n by first sending each quaternion entry + `z = a + bi + cj + dk` to the block-complex matrix + ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into + a real matrix. + + EXAMPLES:: + + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: i,j,k = Q.gens() + sage: x = 1 + 2*i + 3*j + 4*k + sage: M = matrix(Q, 1, [[x]]) + sage: _embed_quaternion_matrix(M) + [ 1 2 3 4] + [-2 1 -4 3] + [-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() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + + F = QuadraticField(-1, 'i') + i = F.gen() + + blocks = [] + for z in M.list(): + t = z.coefficient_tuple() + a = t[0] + b = t[1] + c = t[2] + d = t[3] + cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], + [-c + d*i, a - b*i]]) + blocks.append(_embed_complex_matrix(cplx_matrix)) + + # We should have real entries by now, so use the realest field + # we've got for the return value. + return block_matrix(quaternions.base_ring(), n, blocks) + + +def _unembed_quaternion_matrix(M): + """ + The inverse of _embed_quaternion_matrix(). + + EXAMPLES:: + + sage: M = matrix(QQ, [[ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [-3, 4, 1, -2], + ....: [-4, -3, 2, 1]]) + sage: _unembed_quaternion_matrix(M) + [1 + 2*i + 3*j + 4*k] + + TESTS: + + Unembedding is the inverse of embedding:: + + sage: set_random_seed() + sage: Q = QuaternionAlgebra(QQ, -1, -1) + sage: M = random_matrix(Q, 3) + sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M + True + + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + if not n.mod(4).is_zero(): + raise ValueError("the matrix 'M' must be a complex embedding") + + Q = QuaternionAlgebra(QQ,-1,-1) + i,j,k = Q.gens() + + # Go top-left to bottom-right (reading order), converting every + # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 + # quaternion block. + elements = [] + for l in xrange(n/4): + for m in xrange(n/4): + submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) + if submat[0,0] != submat[1,1].conjugate(): + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0].conjugate(): + raise ValueError('bad off-diagonal submatrix') + z = submat[0,0].real() + submat[0,0].imag()*i + z += submat[0,1].real()*j + submat[0,1].imag()*k + elements.append(z) + + return matrix(Q, n/4, elements) + + +# The usual inner product on R^n. +def _usual_ip(x,y): + return x.vector().inner_product(y.vector()) + +# The inner product used for the real symmetric simple EJA. +# We keep it as a separate function because e.g. the complex +# algebra uses the same inner product, except divided by 2. +def _matrix_ip(X,Y): + X_mat = X.natural_representation() + Y_mat = Y.natural_representation() + return (X_mat*Y_mat).trace() + + +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 + product. It has dimension `(n^2 + n)/2` over the reals. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: e0, e1, e2 = J.gens() + sage: e0*e0 + e0 + sage: e1*e1 + e0 + e2 + sage: e2*e2 + e2 + + TESTS: + + The degree of this algebra is `(n^2 + n) / 2`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealSymmetricEJA(n) + sage: J.degree() == (n^2 + n)/2 + True + + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealSymmetricEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + + """ + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _real_symmetric_basis(n, field=field) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) + def inner_product(self, x, y): + return _matrix_ip(x,y) -def eja_ln(dimension, 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, + and the real-part-of-trace inner product. It has dimension `n^2` over + the reals. + + TESTS: + + The degree of this algebra is `n^2`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = ComplexHermitianEJA(n) + sage: J.degree() == n^2 + True + + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = ComplexHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + + """ + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + 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 + + +class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): + """ + 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. + + TESTS: + + The degree of this algebra is `n^2`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = QuaternionHermitianEJA(n) + sage: J.degree() == 2*(n^2) - n + True + + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = QuaternionHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + + """ + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _quaternion_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(QuaternionHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + 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): """ - Return the Jordan algebra corresponding to the Lorentz "ice cream" - cone of the given ``dimension``. + 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 = eja_ln(4) + sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 @@ -489,29 +1899,27 @@ def eja_ln(dimension, field=QQ): 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) + @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)