X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=f327bf51aada40b33fd05fb0d1c38c124d4b545e;hb=debb0160c47109a5f2060bffd2c618f2a3a19551;hp=0ef8bef1ab06bdb7dc08dd5785642f3c46b23369;hpb=29530845df671c7be5ca637f549e13993ee64efc;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 0ef8bef..f327bf5 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -54,7 +54,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def __init__(self, field, mult_table, - rank, prefix='e', category=None, natural_basis=None, @@ -91,7 +90,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # a real embedding. raise ValueError('field is not real') - self._rank = rank self._natural_basis = natural_basis if category is None: @@ -194,6 +192,24 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): coords = W.coordinate_vector(_mat2vec(elt)) return self.from_vector(coords) + @staticmethod + def _max_test_case_size(): + """ + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is quite vague -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is + far less than the dimension of the underlying vector space. + + We default to five in this class, which is safe in `R^n`. The + matrix algebra subclasses (or any class where the "size" is + interpreted to be far less than the dimension) should override + with a smaller number. + """ + return 5 def _repr_(self): """ @@ -219,168 +235,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def product_on_basis(self, i, j): return self._multiplication_table[i][j] - def _a_regular_element(self): - """ - Guess a regular element. Needed to compute the basis for our - characteristic polynomial coefficients. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS: - - Ensure that this hacky method succeeds for every algebra that we - know how to construct:: - - sage: set_random_seed() - sage: J = random_eja() - sage: J._a_regular_element().is_regular() - True - - """ - 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() - # Don't use the parent vector space directly here in case this - # happens to be a subalgebra. In that case, we would be e.g. - # two-dimensional but span_of_basis() would expect three - # coordinates. - V = VectorSpace(self.base_ring(), self.vector_space().dimension()) - basis = [ (z**k).to_vector() for k in range(self.rank()) ] - V1 = V.span_of_basis( basis ) - 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(): - return R.one() - 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() - - # Turn my vector space into a module so that "vectors" can - # have multivatiate polynomial entries. - names = tuple('X' + str(i) for i in range(1,n+1)) - R = PolynomialRing(self.base_ring(), names) - - # Using change_ring() on the parent's vector space doesn't work - # here because, in a subalgebra, that vector space has a basis - # and change_ring() tries to bring the basis along with it. And - # that doesn't work unless the new ring is a PID, which it usually - # won't be. - V = FreeModule(R,n) - - # Now let x = (X1,X2,...,Xn) be the vector whose entries are - # indeterminates... - x = V(names) - - # And figure out the "left multiplication by x" matrix in - # that setting. - lmbx_cols = [] - monomial_matrices = [ self.monomial(i).operator().matrix() - for i in range(n) ] # don't recompute these! - for k in range(n): - ek = self.monomial(k).to_vector() - lmbx_cols.append( - sum( x[i]*(monomial_matrices[i]*ek) - for i in range(n) ) ) - Lx = matrix.column(R, lmbx_cols) - - # Now we can compute powers of x "symbolically" - x_powers = [self.one().to_vector(), x] - for d in range(2, r+1): - x_powers.append( Lx*(x_powers[-1]) ) - - idmat = matrix.identity(R, 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_powers = [ W.coordinate_vector(xp) for xp in x_powers ] - l2 = [idmat.column(k-1) for k in range(r+1, n+1)] - A_of_x = matrix.column(R, n, (x_powers[:r] + l2)) - return (A_of_x, x, x_powers[r], A_of_x.det()) - - @cached_method - def characteristic_polynomial(self): + def characteristic_polynomial_of(self): """ - Return a characteristic polynomial that works for all elements - of this algebra. + Return the algebra's "characteristic polynomial of" function, + which is itself a multivariate polynomial that, when evaluated + at the coordinates of some algebra element, returns that + element's characteristic polynomial. The resulting polynomial has `n+1` variables, where `n` is the dimension of this algebra. The first `n` variables correspond to @@ -400,7 +261,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) - sage: p = J.characteristic_polynomial(); p + sage: p = J.characteristic_polynomial_of(); p X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 sage: xvec = J.one().to_vector() sage: p(*xvec) @@ -413,27 +274,28 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): any argument:: sage: J = TrivialEJA() - sage: J.characteristic_polynomial() + sage: J.characteristic_polynomial_of() 1 """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(r+1) ] + # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1). + a = self._charpoly_coefficients() # 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) + if r > 0: + R = a[0].parent() + S = PolynomialRing(S, R.variable_names()) + t = S(t) - return sum( a[k]*(t**k) for k in range(len(a)) ) + return (t**r + sum( a[k]*(t**k) for k in range(r) )) def inner_product(self, x, y): @@ -578,8 +440,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ Return the matrix space in which this algebra's natural basis elements live. + + Generally this will be an `n`-by-`1` column-vector space, + except when the algebra is trivial. There it's `n`-by-`n` + (where `n` is zero), to ensure that two elements of the + natural basis space (empty matrices) can be multiplied. """ - if self._natural_basis is None or len(self._natural_basis) == 0: + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + elif self._natural_basis is None or len(self._natural_basis) == 0: return MatrixSpace(self.base_ring(), self.dimension(), 1) else: return self._natural_basis[0].matrix_space() @@ -712,6 +581,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Vector space of degree 6 and dimension 2... sage: J1 Euclidean Jordan algebra of dimension 3... + sage: J0.one().natural_representation() + [0 0 0] + [0 0 0] + [0 0 1] + sage: orig_df = AA.options.display_format + sage: AA.options.display_format = 'radical' + sage: J.from_vector(J5.basis()[0]).natural_representation() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + sage: J.from_vector(J5.basis()[1]).natural_representation() + [ 0 0 0] + [ 0 0 1/2*sqrt(2)] + [ 0 1/2*sqrt(2) 0] + sage: AA.options.display_format = orig_df + sage: J1.one().natural_representation() + [1 0 0] + [0 1 0] + [0 0 0] TESTS: @@ -726,9 +614,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J1.superalgebra() == J and J1.dimension() == J.dimension() True - The identity elements in the two subalgebras are the - projections onto their respective subspaces of the - superalgebra's identity element:: + The decomposition is into eigenspaces, and its components are + therefore necessarily orthogonal. Moreover, the identity + elements in the two subalgebras are the projections onto their + respective subspaces of the superalgebra's identity element:: sage: set_random_seed() sage: J = random_eja() @@ -738,6 +627,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): ....: x = J.random_element() sage: c = x.subalgebra_idempotent() sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: ipsum = 0 + sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()): + ....: w = w.superalgebra_element() + ....: y = J.from_vector(y) + ....: z = z.superalgebra_element() + ....: ipsum += w.inner_product(y).abs() + ....: ipsum += w.inner_product(z).abs() + ....: ipsum += y.inner_product(z).abs() + sage: ipsum + 0 sage: J1(c) == J1.one() True sage: J0(J.one() - c) == J0.one() @@ -773,10 +672,61 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return (J0, J5, J1) - def random_elements(self, count): + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. + + Our algebra superclass method only returns a linear + combination of at most two basis elements. We instead + want the vector space "random element" method that + returns a more diverse selection. + + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + element when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + + """ + # For a general base ring... maybe we can trust this to do the + # right thing? Unlikely, but. + V = self.vector_space() + v = V.random_element() + + if self.base_ring() is AA: + # The "random element" method of the algebraic reals is + # stupid at the moment, and only returns integers between + # -2 and 2, inclusive: + # + # https://trac.sagemath.org/ticket/30875 + # + # Instead, we implement our own "random vector" method, + # and then coerce that into the algebra. We use the vector + # space degree here instead of the dimension because a + # subalgebra could (for example) be spanned by only two + # vectors, each with five coordinates. We need to + # generate all five coordinates. + if thorough: + v *= QQbar.random_element().real() + else: + v *= QQ.random_element() + + return self.from_vector(V.coordinate_vector(v)) + + def random_elements(self, count, thorough=False): """ Return ``count`` random elements as a tuple. + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + elements when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA @@ -791,23 +741,85 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in range(count) ) + return tuple( self.random_element(thorough) + for idx in range(count) ) + @classmethod + def random_instance(cls, field=AA, **kwargs): + """ + Return a random instance of this type of algebra. - def rank(self): + Beware, this will crash for "most instances" because the + constructor below looks wrong. """ - Return the rank of this EJA. + if cls is TrivialEJA: + # The TrivialEJA class doesn't take an "n" argument because + # there's only one. + return cls(field) + + n = ZZ.random_element(cls._max_test_case_size() + 1) + return cls(n, field, **kwargs) + + @cached_method + def _charpoly_coefficients(self): + r""" + The `r` polynomial coefficients of the "characteristic polynomial + of" function. + """ + n = self.dimension() + var_names = [ "X" + str(z) for z in range(1,n+1) ] + R = PolynomialRing(self.base_ring(), var_names) + vars = R.gens() + F = R.fraction_field() + + def L_x_i_j(i,j): + # From a result in my book, these are the entries of the + # basis representation of L_x. + return sum( vars[k]*self.monomial(k).operator().matrix()[i,j] + for k in range(n) ) + + L_x = matrix(F, n, n, L_x_i_j) + + r = None + if self.rank.is_in_cache(): + r = self.rank() + # There's no need to pad the system with redundant + # columns if we *know* they'll be redundant. + n = r + + # Compute an extra power in case the rank is equal to + # the dimension (otherwise, we would stop at x^(r-1)). + x_powers = [ (L_x**k)*self.one().to_vector() + for k in range(n+1) ] + A = matrix.column(F, x_powers[:n]) + AE = A.extended_echelon_form() + E = AE[:,n:] + A_rref = AE[:,:n] + if r is None: + r = A_rref.rank() + b = x_powers[r] + + # The theory says that only the first "r" coefficients are + # nonzero, and they actually live in the original polynomial + # ring and not the fraction field. We negate them because + # in the actual characteristic polynomial, they get moved + # to the other side where x^r lives. + return -A_rref.solve_right(E*b).change_ring(R)[:r] - ALGORITHM: + @cached_method + def rank(self): + r""" + Return the rank of this EJA. - The author knows of no algorithm to compute the rank of an EJA - where only the multiplication table is known. In lieu of one, we - require the rank to be specified when the algebra is created, - and simply pass along that number here. + This is a cached method because we know the rank a priori for + all of the algebras we can construct. Thus we can avoid the + expensive ``_charpoly_coefficients()`` call unless we truly + need to compute the whole characteristic polynomial. SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, @@ -848,8 +860,43 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: r > 0 or (r == 0 and J.is_trivial()) True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + + sage: J = HadamardEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 4 + + :: + + sage: J = JordanSpinEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = RealSymmetricEJA(3) + sage: J.rank.clear_cache() + sage: J.rank() + 3 + + :: + + sage: J = ComplexHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 """ - return self._rank + return len(self._charpoly_coefficients()) def vector_space(self): @@ -873,61 +920,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement -class KnownRankEJA(object): - """ - A class for algebras that we actually know we can construct. The - main issue is that, for most of our methods to make sense, we need - to know the rank of our algebra. Thus we can't simply generate a - "random" algebra, or even check that a given basis and product - satisfy the axioms; because even if everything looks OK, we wouldn't - know the rank we need to actuallty build the thing. - - Not really a subclass of FDEJA because doing that causes method - resolution errors, e.g. - - TypeError: Error when calling the metaclass bases - Cannot create a consistent method resolution - order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA - - """ - @staticmethod - def _max_test_case_size(): - """ - Return an integer "size" that is an upper bound on the size of - this algebra when it is used in a random test - case. Unfortunately, the term "size" is quite vague -- when - dealing with `R^n` under either the Hadamard or Jordan spin - product, the "size" refers to the dimension `n`. When dealing - with a matrix algebra (real symmetric or complex/quaternion - Hermitian), it refers to the size of the matrix, which is - far less than the dimension of the underlying vector space. - - We default to five in this class, which is safe in `R^n`. The - matrix algebra subclasses (or any class where the "size" is - interpreted to be far less than the dimension) should override - with a smaller number. - """ - return 5 - - @classmethod - def random_instance(cls, field=AA, **kwargs): - """ - Return a random instance of this type of algebra. - - Beware, this will crash for "most instances" because the - constructor below looks wrong. - """ - if cls is TrivialEJA: - # The TrivialEJA class doesn't take an "n" argument because - # there's only one. - return cls(field) - - n = ZZ.random_element(cls._max_test_case_size()) + 1 - return cls(n, field, **kwargs) - - -class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -973,7 +966,8 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): for i in range(n) ] fdeja = super(HadamardEJA, self) - return fdeja.__init__(field, mult_table, rank=n, **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(n) def inner_product(self, x, y): """ @@ -1000,7 +994,7 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): return x.to_vector().inner_product(y.to_vector()) -def random_eja(field=AA, nontrivial=False): +def random_eja(field=AA): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. @@ -1014,17 +1008,17 @@ def random_eja(field=AA, nontrivial=False): Euclidean Jordan algebra of dimension... """ - eja_classes = KnownRankEJA.__subclasses__() - if nontrivial: - eja_classes.remove(TrivialEJA) - classname = choice(eja_classes) + classname = choice([TrivialEJA, + HadamardEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) return classname.random_instance(field=field) - - class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): @staticmethod def _max_test_case_size(): @@ -1032,20 +1026,20 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # field can have dimension 4 (quaternions) too. return 2 - def __init__(self, field, basis, rank, normalize_basis=True, **kwargs): + def __init__(self, field, basis, normalize_basis=True, **kwargs): """ Compared to the superclass constructor, we take a basis instead of a multiplication table because the latter can be computed in terms of the former when the product is known (like it is here). """ - # Used in this class's fast _charpoly_coeff() override. + # Used in this class's fast _charpoly_coefficients() override. self._basis_normalizers = None # We're going to loop through this a few times, so now's a good # time to ensure that it isn't a generator expression. basis = tuple(basis) - if rank > 1 and normalize_basis: + if len(basis) > 1 and normalize_basis: # We'll need sqrt(2) to normalize the basis, and this # winds up in the multiplication table, so the whole # algebra needs to be over the field extension. @@ -1062,45 +1056,50 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Qs = self.multiplication_table_from_matrix_basis(basis) fdeja = super(MatrixEuclideanJordanAlgebra, self) - return fdeja.__init__(field, - Qs, - rank=rank, - natural_basis=basis, - **kwargs) + fdeja.__init__(field, Qs, natural_basis=basis, **kwargs) + return @cached_method - def _charpoly_coeff(self, i): - """ + def _charpoly_coefficients(self): + r""" Override the parent method with something that tries to compute over a faster (non-extension) field. """ if self._basis_normalizers is None: # We didn't normalize, so assume that the basis we started # with had entries in a nice field. - return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i) + return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients() else: basis = ( (b/n) for (b,n) in zip(self.natural_basis(), self._basis_normalizers) ) # Do this over the rationals and convert back at the end. + # Only works because we know the entries of the basis are + # integers. J = MatrixEuclideanJordanAlgebra(QQ, basis, - self.rank(), normalize_basis=False) - (_,x,_,_) = J._charpoly_matrix_system() - p = J._charpoly_coeff(i) - # p might be missing some vars, have to substitute "optionally" - pairs = zip(x.base_ring().gens(), self._basis_normalizers) - substitutions = { v: v*c for (v,c) in pairs } - result = p.subs(substitutions) - - # The result of "subs" can be either a coefficient-ring - # element or a polynomial. Gotta handle both cases. - if result in QQ: - return self.base_ring()(result) - else: - return result.change_ring(self.base_ring()) + a = J._charpoly_coefficients() + + # Unfortunately, changing the basis does change the + # coefficients of the characteristic polynomial, but since + # these are really the coefficients of the "characteristic + # polynomial of" function, everything is still nice and + # unevaluated. It's therefore "obvious" how scaling the + # basis affects the coordinate variables X1, X2, et + # cetera. Scaling the first basis vector up by "n" adds a + # factor of 1/n into every "X1" term, for example. So here + # we simply undo the basis_normalizer scaling that we + # performed earlier. + # + # The a[0] access here is safe because trivial algebras + # won't have any basis normalizers and therefore won't + # make it to this "else" branch. + XS = a[0].parent().gens() + subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i] + for i in range(len(XS)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) @staticmethod @@ -1118,6 +1117,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # 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. + if len(basis) == 0: + return [] + field = basis[0].base_ring() dimension = basis[0].nrows() @@ -1197,7 +1199,7 @@ class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return M -class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): +class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1275,6 +1277,11 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod def _denormalized_basis(cls, n, field): @@ -1315,7 +1322,8 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) + super(RealSymmetricEJA, self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @@ -1477,7 +1485,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 -class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): +class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1547,6 +1555,11 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod @@ -1605,7 +1618,8 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) + super(ComplexHermitianEJA,self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @@ -1771,8 +1785,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 -class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, - KnownRankEJA): +class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the @@ -1842,6 +1855,11 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod def _denormalized_basis(cls, n, field): @@ -1901,10 +1919,11 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) + super(QuaternionHermitianEJA,self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) -class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): r""" The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the half-trace inner product and jordan product ``x*y = @@ -1984,7 +2003,8 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): # one-dimensional ambient space (because the rank is bounded # by the ambient dimension). fdeja = super(BilinearFormEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(min(n,2)) def inner_product(self, x, y): r""" @@ -2078,7 +2098,7 @@ class JordanSpinEJA(BilinearFormEJA): return super(JordanSpinEJA, self).__init__(n, field, **kwargs) -class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The trivial Euclidean Jordan algebra consisting of only a zero element. @@ -2112,4 +2132,52 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): fdeja = super(TrivialEJA, self) # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. - return fdeja.__init__(field, mult_table, rank=0, **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(0) + + +class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): + r""" + The external (orthogonal) direct sum of two other Euclidean Jordan + algebras. Essentially the Cartesian product of its two factors. + Every Euclidean Jordan algebra decomposes into an orthogonal + direct sum of simple Euclidean Jordan algebras, so no generality + is lost by providing only this construction. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(3) + sage: J = DirectSumEJA(J1,J2) + sage: J.dimension() + 8 + sage: J.rank() + 5 + + """ + def __init__(self, J1, J2, field=AA, **kwargs): + n1 = J1.dimension() + n2 = J2.dimension() + n = n1+n2 + V = VectorSpace(field, n) + mult_table = [ [ V.zero() for j in range(n) ] + for i in range(n) ] + for i in range(n1): + for j in range(n1): + p = (J1.monomial(i)*J1.monomial(j)).to_vector() + mult_table[i][j] = V(p.list() + [field.zero()]*n2) + + for i in range(n2): + for j in range(n2): + p = (J2.monomial(i)*J2.monomial(j)).to_vector() + mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list()) + + fdeja = super(DirectSumEJA, self) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(J1.rank() + J2.rank())