Weyl Algebras¶
AUTHORS:
Travis Scrimshaw (2013-09-06): Initial version
-
class
sage.algebras.weyl_algebra.
DifferentialWeylAlgebra
(R, names=None)¶ Bases:
sage.rings.ring.Algebra
,sage.structure.unique_representation.UniqueRepresentation
The differential Weyl algebra of a polynomial ring.
Let \(R\) be a commutative ring. The (differential) Weyl algebra \(W\) is the algebra generated by \(x_1, x_2, \ldots x_n, \partial_{x_1}, \partial_{x_2}, \ldots, \partial_{x_n}\) subject to the relations: \([x_i, x_j] = 0\), \([\partial_{x_i}, \partial_{x_j}] = 0\), and \(\partial_{x_i} x_j = x_j \partial_{x_i} + \delta_{ij}\). Therefore \(\partial_{x_i}\) is acting as the partial differential operator on \(x_i\).
The Weyl algebra can also be constructed as an iterated Ore extension of the polynomial ring \(R[x_1, x_2, \ldots, x_n]\) by adding \(x_i\) at each step. It can also be seen as a quantization of the symmetric algebra \(Sym(V)\), where \(V\) is a finite dimensional vector space over a field of characteristic zero, by using a modified Groenewold-Moyal product in the symmetric algebra.
The Weyl algebra (even for \(n = 1\)) over a field of characteristic 0 has many interesting properties.
It’s a non-commutative domain.
It’s a simple ring (but not in positive characteristic) that is not a matrix ring over a division ring.
It has no finite-dimensional representations.
It’s a quotient of the universal enveloping algebra of the Heisenberg algebra \(\mathfrak{h}_n\).
REFERENCES:
INPUT:
R
– a (polynomial) ringnames
– (default:None
) ifNone
andR
is a polynomial ring, then the variable names correspond to those ofR
; otherwise ifnames
is specified, thenR
is the base ring
EXAMPLES:
There are two ways to create a Weyl algebra, the first is from a polynomial ring:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R); W Differential Weyl algebra of polynomials in x, y, z over Rational Field
We can call
W.inject_variables()
to give the polynomial ring variables, now as elements ofW
, and the differentials:sage: W.inject_variables() Defining x, y, z, dx, dy, dz sage: (dx * dy * dz) * (x^2 * y * z + x * z * dy + 1) x*z*dx*dy^2*dz + z*dy^2*dz + x^2*y*z*dx*dy*dz + dx*dy*dz + x*dx*dy^2 + 2*x*y*z*dy*dz + dy^2 + x^2*z*dx*dz + x^2*y*dx*dy + 2*x*z*dz + 2*x*y*dy + x^2*dx + 2*x
Or directly by specifying a base ring and variable names:
sage: W.<a,b> = DifferentialWeylAlgebra(QQ); W Differential Weyl algebra of polynomials in a, b over Rational Field
Todo
Implement the
graded_algebra()
as a polynomial ring once they are considered to be graded rings (algebras).-
Element
¶ alias of
DifferentialWeylAlgebraElement
-
algebra_generators
()¶ Return the algebra generators of
self
.See also
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: W.algebra_generators() Finite family {'x': x, 'y': y, 'z': z, 'dx': dx, 'dy': dy, 'dz': dz}
-
basis
()¶ Return a basis of
self
.EXAMPLES:
sage: W.<x,y> = DifferentialWeylAlgebra(QQ) sage: B = W.basis() sage: it = iter(B) sage: [next(it) for i in range(20)] [1, x, y, dx, dy, x^2, x*y, x*dx, x*dy, y^2, y*dx, y*dy, dx^2, dx*dy, dy^2, x^3, x^2*y, x^2*dx, x^2*dy, x*y^2] sage: dx, dy = W.differentials() sage: sorted((dx*x).monomials(), key=str) [1, x*dx] sage: B[(x*y).support()[0]] x*y sage: sorted((dx*x).monomial_coefficients().items()) [(((0, 0), (0, 0)), 1), (((1, 0), (1, 0)), 1)]
-
degree_on_basis
(i)¶ Return the degree of the basis element indexed by
i
.EXAMPLES:
sage: W.<a,b> = DifferentialWeylAlgebra(QQ) sage: W.degree_on_basis( ((1, 3, 2), (0, 1, 3)) ) 10 sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: dx,dy,dz = W.differentials() sage: elt = y*dy - (3*x - z)*dx sage: elt.degree() 2
-
diff_action
()¶ Left action of this Weyl algebra on the underlying polynomial ring by differentiation.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: W = R.weyl_algebra() sage: dx, dy = W.differentials() sage: W.diff_action Left action by Differential Weyl algebra of polynomials in x, y over Rational Field on Multivariate Polynomial Ring in x, y over Rational Field sage: W.diff_action(dx^2 + dy + 1, x^3*y^3) x^3*y^3 + 3*x^3*y^2 + 6*x*y^3
-
differentials
()¶ Return the differentials of
self
.See also
EXAMPLES:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: W.differentials() Finite family {'dx': dx, 'dy': dy, 'dz': dz}
-
gen
(i)¶ Return the
i
-th generator ofself
.See also
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: [W.gen(i) for i in range(6)] [x, y, z, dx, dy, dz]
-
ngens
()¶ Return the number of generators of
self
.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: W.ngens() 6
-
one
()¶ Return the multiplicative identity element \(1\).
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: W.one() 1
-
options
(*get_value, **set_value)¶ Sets the global options for elements of the differential Weyl algebra class. The default is to have the factored representations turned off.
OPTIONS:
factor_representation
– (default:False
) Controls whether to factor the differentials out or not in the output representations
If no parameters are set, then the function returns a copy of the options dictionary.
EXAMPLES:
sage: R.<t> = QQ[] sage: D = DifferentialWeylAlgebra(R) sage: t,dt = D.gens() sage: x = dt^3*t^3 + dt^2*t^4 sage: x t^3*dt^3 + t^4*dt^2 + 9*t^2*dt^2 + 8*t^3*dt + 18*t*dt + 12*t^2 + 6 sage: D.options.factor_representation = True sage: x (12*t^2 + 6) + (8*t^3 + 18*t)*dt + (t^4 + 9*t^2)*dt^2 + (t^3)*dt^3 sage: D.options._reset()
See
GlobalOptions
for more features of these options.
-
polynomial_ring
()¶ Return the associated polynomial ring of
self
.EXAMPLES:
sage: W.<a,b> = DifferentialWeylAlgebra(QQ) sage: W.polynomial_ring() Multivariate Polynomial Ring in a, b over Rational Field
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: W.polynomial_ring() == R True
-
variables
()¶ Return the variables of
self
.See also
EXAMPLES:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: W.variables() Finite family {'x': x, 'y': y, 'z': z}
-
zero
()¶ Return the additive identity element \(0\).
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: W = DifferentialWeylAlgebra(R) sage: W.zero() 0
-
class
sage.algebras.weyl_algebra.
DifferentialWeylAlgebraAction
(G)¶ Bases:
sage.categories.action.Action
Left action of a Weyl algebra on its underlying polynomial ring by differentiation.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: W = R.weyl_algebra() sage: dx, dy = W.differentials() sage: W.diff_action Left action by Differential Weyl algebra of polynomials in x, y over Rational Field on Multivariate Polynomial Ring in x, y over Rational Field
sage: g = dx^2 + x*dy sage: p = x^5 + x^3 + y^2*x^2 + 1 sage: W.diff_action(g, p) 2*x^3*y + 20*x^3 + 2*y^2 + 6*x
The action is a left action:
sage: h = dx*x + x*y sage: W.diff_action(h, W.diff_action(g, p)) == W.diff_action(h*g, p) True
The action endomorphism of a differential operator:
sage: dg = W.diff_action(g); dg Action of dx^2 + x*dy on Multivariate Polynomial Ring in x, y over Rational Field under Left action by Differential Weyl algebra... sage: dg(p) == W.diff_action(g, p) == g.diff(p) True
-
class
sage.algebras.weyl_algebra.
DifferentialWeylAlgebraElement
(parent, monomials)¶ Bases:
sage.structure.element.AlgebraElement
An element in a differential Weyl algebra.
-
diff
(p)¶ Apply this differential operator to a polynomial.
INPUT:
p
– polynomial of the underlying polynomial ring
OUTPUT:
The result of the left action of the Weyl algebra on the polynomial ring via differentiation.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: W = R.weyl_algebra() sage: dx, dy = W.differentials() sage: dx.diff(x^3) 3*x^2 sage: (dx*dy).diff(W(x^3*y^3)) 9*x^2*y^2 sage: (x*dx + dy + 1).diff(x^4*y^4 + 1) 5*x^4*y^4 + 4*x^4*y^3 + 1
-
factor_differentials
()¶ Return a dict representing
self
with the differentials factored out.EXAMPLES:
sage: R.<t> = QQ[] sage: D = DifferentialWeylAlgebra(R) sage: t, dt = D.gens() sage: x = dt^3*t^3 + dt^2*t^4 sage: x t^3*dt^3 + t^4*dt^2 + 9*t^2*dt^2 + 8*t^3*dt + 18*t*dt + 12*t^2 + 6 sage: x.factor_differentials() {(0,): 12*t^2 + 6, (1,): 8*t^3 + 18*t, (2,): t^4 + 9*t^2, (3,): t^3} sage: D.zero().factor_differentials() {} sage: R.<x,y,z> = QQ[] sage: D = DifferentialWeylAlgebra(R) sage: x, y, z, dx, dy, dz = D.gens() sage: elt = dx^3*x^3 + (y^3-z*x)*dx^3 + dy^3*x^3 + dx*dy*dz*x*y*z sage: elt x^3*dy^3 + x*y*z*dx*dy*dz + y^3*dx^3 + x^3*dx^3 - x*z*dx^3 + y*z*dy*dz + x*z*dx*dz + x*y*dx*dy + 9*x^2*dx^2 + z*dz + y*dy + 19*x*dx + 7 sage: elt.factor_differentials() {(0, 0, 0): 7, (0, 0, 1): z, (0, 1, 0): y, (0, 1, 1): y*z, (0, 3, 0): x^3, (1, 0, 0): 19*x, (1, 0, 1): x*z, (1, 1, 0): x*y, (1, 1, 1): x*y*z, (2, 0, 0): 9*x^2, (3, 0, 0): x^3 + y^3 - x*z}
-
list
()¶ Return
self
as a list.This list consists of pairs \((m, c)\), where \(m\) is a pair of tuples indexing a basis element of
self
, and \(c\) is the coordinate ofself
corresponding to this basis element. (Only nonzero coordinates are shown.)EXAMPLES:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: dx,dy,dz = W.differentials() sage: elt = dy - (3*x - z)*dx sage: elt.list() [(((0, 0, 0), (0, 1, 0)), 1), (((0, 0, 1), (1, 0, 0)), 1), (((1, 0, 0), (1, 0, 0)), -3)]
-
monomial_coefficients
(copy=True)¶ Return a dictionary which has the basis keys in the support of
self
as keys and their corresponding coefficients as values.INPUT:
copy
– (default:True
) ifself
is internally represented by a dictionaryd
, then make a copy ofd
; ifFalse
, then this can cause undesired behavior by mutatingd
EXAMPLES:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: dx,dy,dz = W.differentials() sage: elt = (dy - (3*x - z)*dx) sage: sorted(elt.monomial_coefficients().items()) [(((0, 0, 0), (0, 1, 0)), 1), (((0, 0, 1), (1, 0, 0)), 1), (((1, 0, 0), (1, 0, 0)), -3)]
-
support
()¶ Return the support of
self
.EXAMPLES:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ) sage: dx,dy,dz = W.differentials() sage: elt = dy - (3*x - z)*dx + 1 sage: sorted(elt.support()) [((0, 0, 0), (0, 0, 0)), ((0, 0, 0), (0, 1, 0)), ((0, 0, 1), (1, 0, 0)), ((1, 0, 0), (1, 0, 0))]
-
-
sage.algebras.weyl_algebra.
repr_factored
(w, latex_output=False)¶ Return a string representation of
w
with the \(dx_i\) generators factored on the right.EXAMPLES:
sage: from sage.algebras.weyl_algebra import repr_factored sage: R.<t> = QQ[] sage: D = DifferentialWeylAlgebra(R) sage: t, dt = D.gens() sage: x = dt^3*t^3 + dt^2*t^4 sage: x t^3*dt^3 + t^4*dt^2 + 9*t^2*dt^2 + 8*t^3*dt + 18*t*dt + 12*t^2 + 6 sage: print(repr_factored(x)) (12*t^2 + 6) + (8*t^3 + 18*t)*dt + (t^4 + 9*t^2)*dt^2 + (t^3)*dt^3 sage: repr_factored(x, True) (12 t^{2} + 6) + (8 t^{3} + 18 t) \frac{\partial}{\partial t} + (t^{4} + 9 t^{2}) \frac{\partial^{2}}{\partial t^{2}} + (t^{3}) \frac{\partial^{3}}{\partial t^{3}} sage: repr_factored(D.zero()) '0'
With multiple variables:
sage: R.<x,y,z> = QQ[] sage: D = DifferentialWeylAlgebra(R) sage: x, y, z, dx, dy, dz = D.gens() sage: elt = dx^3*x^3 + (y^3-z*x)*dx^3 + dy^3*x^3 + dx*dy*dz*x*y*z sage: elt x^3*dy^3 + x*y*z*dx*dy*dz + y^3*dx^3 + x^3*dx^3 - x*z*dx^3 + y*z*dy*dz + x*z*dx*dz + x*y*dx*dy + 9*x^2*dx^2 + z*dz + y*dy + 19*x*dx + 7 sage: print(repr_factored(elt)) (7) + (z)*dz + (y)*dy + (y*z)*dy*dz + (x^3)*dy^3 + (19*x)*dx + (x*z)*dx*dz + (x*y)*dx*dy + (x*y*z)*dx*dy*dz + (9*x^2)*dx^2 + (x^3 + y^3 - x*z)*dx^3 sage: repr_factored(D.zero(), True) 0
-
sage.algebras.weyl_algebra.
repr_from_monomials
(monomials, term_repr, use_latex=False)¶ Return a string representation of an element of a free module from the dictionary
monomials
.INPUT:
monomials
– a list of pairs[m, c]
wherem
is the index andc
is the coefficientterm_repr
– a function which returns a string given an index (can berepr
orlatex
, for example)use_latex
– (default:False
) ifTrue
then the output is in latex format
EXAMPLES:
sage: from sage.algebras.weyl_algebra import repr_from_monomials sage: R.<x,y,z> = QQ[] sage: d = [(z, 4/7), (y, sqrt(2)), (x, -5)] sage: repr_from_monomials(d, lambda m: repr(m)) '4/7*z + sqrt(2)*y - 5*x' sage: a = repr_from_monomials(d, lambda m: latex(m), True); a \frac{4}{7} z + \sqrt{2} y - 5 x sage: type(a) <class 'sage.misc.latex.LatexExpr'>
The zero element:
sage: repr_from_monomials([], lambda m: repr(m)) '0' sage: a = repr_from_monomials([], lambda m: latex(m), True); a 0 sage: type(a) <class 'sage.misc.latex.LatexExpr'>
A “unity” element:
sage: repr_from_monomials([(1, 1)], lambda m: repr(m)) '1' sage: a = repr_from_monomials([(1, 1)], lambda m: latex(m), True); a 1 sage: type(a) <class 'sage.misc.latex.LatexExpr'>
sage: repr_from_monomials([(1, -1)], lambda m: repr(m)) '-1' sage: a = repr_from_monomials([(1, -1)], lambda m: latex(m), True); a -1 sage: type(a) <class 'sage.misc.latex.LatexExpr'>
Leading minus signs are dealt with appropriately:
sage: d = [(z, -4/7), (y, -sqrt(2)), (x, -5)] sage: repr_from_monomials(d, lambda m: repr(m)) '-4/7*z - sqrt(2)*y - 5*x' sage: a = repr_from_monomials(d, lambda m: latex(m), True); a -\frac{4}{7} z - \sqrt{2} y - 5 x sage: type(a) <class 'sage.misc.latex.LatexExpr'>
Indirect doctests using a class that uses this function:
sage: R.<x,y> = QQ[] sage: A = CliffordAlgebra(QuadraticForm(R, 3, [x,0,-1,3,-4,5])) sage: a,b,c = A.gens() sage: a*b*c e0*e1*e2 sage: b*c e1*e2 sage: (a*a + 2) x + 2 sage: c*(a*a + 2)*b (-x - 2)*e1*e2 - 4*x - 8 sage: latex(c*(a*a + 2)*b) \left( -x - 2 \right) e_{1} e_{2} - 4 x - 8