Clifford Algebras#
AUTHORS:
Travis Scrimshaw (2013-09-06): Initial version
Trevor K. Karn (2022-07-27): Rewrite basis indexing using FrozenBitset
- class sage.algebras.clifford_algebra.CliffordAlgebra(Q, names, category=None)#
Bases:
sage.combinat.free_module.CombinatorialFreeModule
The Clifford algebra of a quadratic form.
Let \(Q : V \to \mathbf{k}\) denote a quadratic form on a vector space \(V\) over a field \(\mathbf{k}\). The Clifford algebra \(Cl(V, Q)\) is defined as \(T(V) / I_Q\) where \(T(V)\) is the tensor algebra of \(V\) and \(I_Q\) is the two-sided ideal generated by all elements of the form \(v \otimes v - Q(v)\) for all \(v \in V\).
We abuse notation to denote the projection of a pure tensor \(x_1 \otimes x_2 \otimes \cdots \otimes x_m \in T(V)\) onto \(T(V) / I_Q = Cl(V, Q)\) by \(x_1 \wedge x_2 \wedge \cdots \wedge x_m\). This is motivated by the fact that \(Cl(V, Q)\) is the exterior algebra \(\wedge V\) when \(Q = 0\) (one can also think of a Clifford algebra as a quantization of the exterior algebra). See
ExteriorAlgebra
for the concept of an exterior algebra.From the definition, a basis of \(Cl(V, Q)\) is given by monomials of the form
\[\{ e_{i_1} \wedge \cdots \wedge e_{i_k} \mid 1 \leq i_1 < \cdots < i_k \leq n \},\]where \(n = \dim(V)\) and where \(\{ e_1, e_2, \cdots, e_n \}\) is any fixed basis of \(V\). Hence
\[\dim(Cl(V, Q)) = \sum_{k=0}^n \binom{n}{k} = 2^n.\]Note
The algebra \(Cl(V, Q)\) is a \(\ZZ / 2\ZZ\)-graded algebra, but not (in general) \(\ZZ\)-graded (in a reasonable way).
This construction satisfies the following universal property. Let \(i : V \to Cl(V, Q)\) denote the natural inclusion (which is an embedding). Then for every associative \(\mathbf{k}\)-algebra \(A\) and any \(\mathbf{k}\)-linear map \(j : V \to A\) satisfying
\[j(v)^2 = Q(v) \cdot 1_A\]for all \(v \in V\), there exists a unique \(\mathbf{k}\)-algebra homomorphism \(f : Cl(V, Q) \to A\) such that \(f \circ i = j\). This property determines the Clifford algebra uniquely up to canonical isomorphism. The inclusion \(i\) is commonly used to identify \(V\) with a vector subspace of \(Cl(V)\).
The Clifford algebra \(Cl(V, Q)\) is a \(\ZZ_2\)-graded algebra (where \(\ZZ_2 = \ZZ / 2 \ZZ\)); this grading is determined by placing all elements of \(V\) in degree \(1\). It is also an \(\NN\)-filtered algebra, with the filtration too being defined by placing all elements of \(V\) in degree \(1\). The
degree()
gives the \(\NN\)-filtration degree, and to get the super degree use insteadis_even_odd()
.The Clifford algebra also can be considered as a covariant functor from the category of vector spaces equipped with quadratic forms to the category of algebras. In fact, if \((V, Q)\) and \((W, R)\) are two vector spaces endowed with quadratic forms, and if \(g : W \to V\) is a linear map preserving the quadratic form, then we can define an algebra morphism \(Cl(g) : Cl(W, R) \to Cl(V, Q)\) by requiring that it send every \(w \in W\) to \(g(w) \in V\). Since the quadratic form \(R\) on \(W\) is uniquely determined by the quadratic form \(Q\) on \(V\) (due to the assumption that \(g\) preserves the quadratic form), this fact can be rewritten as follows: If \((V, Q)\) is a vector space with a quadratic form, and \(W\) is another vector space, and \(\phi : W \to V\) is any linear map, then we obtain an algebra morphism \(Cl(\phi) : Cl(W, \phi(Q)) \to Cl(V, Q)\) where \(\phi(Q) = \phi^T \cdot Q \cdot \phi\) (we consider \(\phi\) as a matrix) is the quadratic form \(Q\) pulled back to \(W\). In fact, the map \(\phi\) preserves the quadratic form because of
\[\phi(Q)(x) = x^T \cdot \phi^T \cdot Q \cdot \phi \cdot x = (\phi \cdot x)^T \cdot Q \cdot (\phi \cdot x) = Q(\phi(x)).\]Hence we have \(\phi(w)^2 = Q(\phi(w)) = \phi(Q)(w)\) for all \(w \in W\).
REFERENCES:
INPUT:
Q
– a quadratic formnames
– (default:'e'
) the generator names
EXAMPLES:
To create a Clifford algebra, all one needs to do is specify a quadratic form:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl = CliffordAlgebra(Q) sage: Cl The Clifford algebra of the Quadratic form in 3 variables over Integer Ring with coefficients: [ 1 2 3 ] [ * 4 5 ] [ * * 6 ]
We can also explicitly name the generators. In this example, the Clifford algebra we construct is an exterior algebra (since we choose the quadratic form to be zero):
sage: Q = QuadraticForm(ZZ, 4, [0]*10) sage: Cl.<a,b,c,d> = CliffordAlgebra(Q) sage: a*d a*d sage: d*c*b*a + a + 4*b*c a*b*c*d + 4*b*c + a
- Element#
alias of
sage.algebras.clifford_algebra_element.CliffordAlgebraElement
- algebra_generators()#
Return the algebra generators of
self
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.algebra_generators() Finite family {'x': x, 'y': y, 'z': z}
- center_basis()#
Return a list of elements which correspond to a basis for the center of
self
.This assumes that the ground ring can be used to compute the kernel of a matrix.
See also
supercenter_basis()
, http://math.stackexchange.com/questions/129183/center-of-clifford-algebra-depending-on-the-parity-of-dim-vTodo
Deprecate this in favor of a method called \(center()\) once subalgebras are properly implemented in Sage.
EXAMPLES:
sage: Q = QuadraticForm(QQ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Z = Cl.center_basis(); Z (1, -2/5*x*y*z + x - 3/5*y + 2/5*z) sage: all(z*b - b*z == 0 for z in Z for b in Cl.basis()) True sage: Q = QuadraticForm(QQ, 3, [1,-2,-3, 4, 2, 1]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Z = Cl.center_basis(); Z (1, -x*y*z + x + 3/2*y - z) sage: all(z*b - b*z == 0 for z in Z for b in Cl.basis()) True sage: Q = QuadraticForm(QQ, 2, [1,-2,-3]) sage: Cl.<x,y> = CliffordAlgebra(Q) sage: Cl.center_basis() (1,) sage: Q = QuadraticForm(QQ, 2, [-1,1,-3]) sage: Cl.<x,y> = CliffordAlgebra(Q) sage: Cl.center_basis() (1,)
A degenerate case:
sage: Q = QuadraticForm(QQ, 3, [4,4,-4,1,-2,1]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.center_basis() (1, x*y*z + x - 2*y - 2*z, x*y + x*z - 2*y*z)
The most degenerate case (the exterior algebra):
sage: Q = QuadraticForm(QQ, 3) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.center_basis() (1, x*y, x*z, y*z, x*y*z)
- degree_on_basis(m)#
Return the degree of the monomial indexed by
m
.We are considering the Clifford algebra to be \(\NN\)-filtered, and the degree of the monomial
m
is the length ofm
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.degree_on_basis((0,)) 1 sage: Cl.degree_on_basis((0,1)) 2
- dimension()#
Return the rank of
self
as a free module.Let \(V\) be a free \(R\)-module of rank \(n\); then, \(Cl(V, Q)\) is a free \(R\)-module of rank \(2^n\).
EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.dimension() 8
- free_module()#
Return the underlying free module \(V\) of
self
.This is the free module on which the quadratic form that was used to construct
self
is defined.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.free_module() Ambient free module of rank 3 over the principal ideal domain Integer Ring
- gen(i)#
Return the
i
-th standard generator of the algebraself
.This is the
i
-th basis vector of the vector space on which the quadratic form definingself
is defined, regarded as an element ofself
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: [Cl.gen(i) for i in range(3)] [x, y, z]
- gens()#
Return the generators of
self
(as an algebra).EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.gens() (x, y, z)
- graded_algebra()#
Return the associated graded algebra of
self
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.graded_algebra() The exterior algebra of rank 3 over Integer Ring
- is_commutative()#
Check if
self
is a commutative algebra.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.is_commutative() False
- lift_isometry(m, names=None)#
Lift an invertible isometry
m
of the quadratic form ofself
to a Clifford algebra morphism.Given an invertible linear map \(m : V \to W\) (here represented by a matrix acting on column vectors), this method returns the algebra morphism \(Cl(m)\) from \(Cl(V, Q)\) to \(Cl(W, m^{-1}(Q))\), where \(Cl(V, Q)\) is the Clifford algebra
self
and where \(m^{-1}(Q)\) is the pullback of the quadratic form \(Q\) to \(W\) along the inverse map \(m^{-1} : W \to V\). See the documentation ofCliffordAlgebra
for how this pullback and the morphism \(Cl(m)\) are defined.INPUT:
m
– an isometry of the quadratic form ofself
names
– (default:'e'
) the names of the generators of the Clifford algebra of the codomain of (the map represented by)m
OUTPUT:
The algebra morphism \(Cl(m)\) from
self
to \(Cl(W, m^{-1}(Q))\).EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: m = matrix([[1,1,2],[0,1,1],[0,0,1]]) sage: phi = Cl.lift_isometry(m, 'abc') sage: phi(x) a sage: phi(y) a + b sage: phi(x*y) a*b + 1 sage: phi(x) * phi(y) a*b + 1 sage: phi(z*y) a*b - a*c - b*c sage: phi(z) * phi(y) a*b - a*c - b*c sage: phi(x + z) * phi(y + z) == phi((x + z) * (y + z)) True
- lift_module_morphism(m, names=None)#
Lift the matrix
m
to an algebra morphism of Clifford algebras.Given a linear map \(m : W \to V\) (here represented by a matrix acting on column vectors), this method returns the algebra morphism \(Cl(m) : Cl(W, m(Q)) \to Cl(V, Q)\), where \(Cl(V, Q)\) is the Clifford algebra
self
and where \(m(Q)\) is the pullback of the quadratic form \(Q\) to \(W\). See the documentation ofCliffordAlgebra
for how this pullback and the morphism \(Cl(m)\) are defined.Note
This is a map into
self
.INPUT:
m
– a matrixnames
– (default:'e'
) the names of the generators of the Clifford algebra of the domain of (the map represented by)m
OUTPUT:
The algebra morphism \(Cl(m)\) from \(Cl(W, m(Q))\) to
self
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: m = matrix([[1,-1,-1],[0,1,-1],[1,1,1]]) sage: phi = Cl.lift_module_morphism(m, 'abc') sage: phi Generic morphism: From: The Clifford algebra of the Quadratic form in 3 variables over Integer Ring with coefficients: [ 10 17 3 ] [ * 11 0 ] [ * * 5 ] To: The Clifford algebra of the Quadratic form in 3 variables over Integer Ring with coefficients: [ 1 2 3 ] [ * 4 5 ] [ * * 6 ] sage: a,b,c = phi.domain().gens() sage: phi(a) x + z sage: phi(b) -x + y + z sage: phi(c) -x - y + z sage: phi(a + 3*b) -2*x + 3*y + 4*z sage: phi(a) + 3*phi(b) -2*x + 3*y + 4*z sage: phi(a*b) x*y + 2*x*z - y*z + 7 sage: phi(b*a) -x*y - 2*x*z + y*z + 10 sage: phi(a*b + c) x*y + 2*x*z - y*z - x - y + z + 7 sage: phi(a*b) + phi(c) x*y + 2*x*z - y*z - x - y + z + 7
We check that the map is an algebra morphism:
sage: phi(a)*phi(b) x*y + 2*x*z - y*z + 7 sage: phi(a*b) x*y + 2*x*z - y*z + 7 sage: phi(a*a) 10 sage: phi(a)*phi(a) 10 sage: phi(b*a) -x*y - 2*x*z + y*z + 10 sage: phi(b) * phi(a) -x*y - 2*x*z + y*z + 10 sage: phi((a + b)*(a + c)) == phi(a + b) * phi(a + c) True
We can also lift arbitrary linear maps:
sage: m = matrix([[1,1],[0,1],[1,1]]) sage: phi = Cl.lift_module_morphism(m, 'ab') sage: a,b = phi.domain().gens() sage: phi(a) x + z sage: phi(b) x + y + z sage: phi(a*b) x*y - y*z + 15 sage: phi(a)*phi(b) x*y - y*z + 15 sage: phi(b*a) -x*y + y*z + 12 sage: phi(b)*phi(a) -x*y + y*z + 12 sage: m = matrix([[1,1,1,2], [0,1,1,1], [0,1,1,1]]) sage: phi = Cl.lift_module_morphism(m, 'abcd') sage: a,b,c,d = phi.domain().gens() sage: phi(a) x sage: phi(b) x + y + z sage: phi(c) x + y + z sage: phi(d) 2*x + y + z sage: phi(a*b*c + d*a) -x*y - x*z + 21*x + 7 sage: phi(a*b*c*d) 21*x*y + 21*x*z + 42
- ngens()#
Return the number of algebra generators of
self
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.ngens() 3
- one_basis()#
Return the basis index of the element
1
. The element1
is indexed by the emptyset, which is represented by thesage.data_structures.bitset.Bitset
0
.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.one_basis() 0
- pseudoscalar()#
Return the unit pseudoscalar of
self
.Given the basis \(e_1, e_2, \ldots, e_n\) of the underlying \(R\)-module, the unit pseudoscalar is defined as \(e_1 \cdot e_2 \cdots e_n\).
This depends on the choice of basis.
EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.pseudoscalar() x*y*z sage: Q = QuadraticForm(ZZ, 0, []) sage: Cl = CliffordAlgebra(Q) sage: Cl.pseudoscalar() 1
REFERENCES:
- quadratic_form()#
Return the quadratic form of
self
.This is the quadratic form used to define
self
. The quadratic form onself
is yet to be implemented.EXAMPLES:
sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.quadratic_form() Quadratic form in 3 variables over Integer Ring with coefficients: [ 1 2 3 ] [ * 4 5 ] [ * * 6 ]
- supercenter_basis()#
Return a list of elements which correspond to a basis for the supercenter of
self
.This assumes that the ground ring can be used to compute the kernel of a matrix.
See also
center_basis()
, http://math.stackexchange.com/questions/129183/center-of-clifford-algebra-depending-on-the-parity-of-dim-vTodo
Deprecate this in favor of a method called \(supercenter()\) once subalgebras are properly implemented in Sage.
EXAMPLES:
sage: Q = QuadraticForm(QQ, 3, [1,2,3,4,5,6]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: SZ = Cl.supercenter_basis(); SZ (1,) sage: all(z.supercommutator(b) == 0 for z in SZ for b in Cl.basis()) True sage: Q = QuadraticForm(QQ, 3, [1,-2,-3, 4, 2, 1]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1,) sage: Q = QuadraticForm(QQ, 2, [1,-2,-3]) sage: Cl.<x,y> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1,) sage: Q = QuadraticForm(QQ, 2, [-1,1,-3]) sage: Cl.<x,y> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1,)
Singular vectors of a quadratic form generate in the supercenter:
sage: Q = QuadraticForm(QQ, 3, [1/2,-2,4,256/249,3,-185/8]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1, x + 249/322*y + 22/161*z) sage: Q = QuadraticForm(QQ, 3, [4,4,-4,1,-2,1]) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1, x + 2*z, y + z, x*y + x*z - 2*y*z)
The most degenerate case:
sage: Q = QuadraticForm(QQ, 3) sage: Cl.<x,y,z> = CliffordAlgebra(Q) sage: Cl.supercenter_basis() (1, x, y, z, x*y, x*z, y*z, x*y*z)
- class sage.algebras.clifford_algebra.CliffordAlgebraIndices(Qdim)#
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
A facade parent for the indices of Clifford algebra. Users should not create instances of this class directly.
- cardinality()#
Return the cardinality of
self
.EXAMPLES:
sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices sage: idx = CliffordAlgebraIndices(7) sage: idx.cardinality() == 2^7 True sage: len(idx) == 2^7 True
- class sage.algebras.clifford_algebra.ExteriorAlgebra(R, names)#
Bases:
sage.algebras.clifford_algebra.CliffordAlgebra
An exterior algebra of a free module over a commutative ring.
Let \(V\) be a module over a commutative ring \(R\). The exterior algebra (or Grassmann algebra) \(\Lambda(V)\) of \(V\) is defined as the quotient of the tensor algebra \(T(V)\) of \(V\) modulo the two-sided ideal generated by all tensors of the form \(x \otimes x\) with \(x \in V\). The multiplication on \(\Lambda(V)\) is denoted by \(\wedge\) (so \(v_1 \wedge v_2 \wedge \cdots \wedge v_n\) is the projection of \(v_1 \otimes v_2 \otimes \cdots \otimes v_n\) onto \(\Lambda(V)\)) and called the “exterior product” or “wedge product”.
If \(V\) is a rank-\(n\) free \(R\)-module with a basis \(\{e_1, \ldots, e_n\}\), then \(\Lambda(V)\) is the \(R\)-algebra noncommutatively generated by the \(n\) generators \(e_1, \ldots, e_n\) subject to the relations \(e_i^2 = 0\) for all \(i\), and \(e_i e_j = - e_j e_i\) for all \(i < j\). As an \(R\)-module, \(\Lambda(V)\) then has a basis \((\bigwedge_{i \in I} e_i)\) with \(I\) ranging over the subsets of \(\{1, 2, \ldots, n\}\) (where \(\bigwedge_{i \in I} e_i\) is the wedge product of \(e_i\) for \(i\) running through all elements of \(I\) from smallest to largest), and hence is free of rank \(2^n\).
The exterior algebra of an \(R\)-module \(V\) can also be realized as the Clifford algebra of \(V\) for the quadratic form \(Q\) given by \(Q(v) = 0\) for all vectors \(v \in V\). See
CliffordAlgebra
for the notion of a Clifford algebra.The exterior algebra of an \(R\)-module \(V\) is a connected \(\ZZ\)-graded Hopf superalgebra. It is commutative in the super sense (i.e., the odd elements anticommute and square to \(0\)).
This class implements the exterior algebra \(\Lambda(R^n)\) for \(n\) a nonnegative integer.
INPUT:
R
– the base ring, or the free module whose exterior algebra is to be computednames
– a list of strings to name the generators of the exterior algebra; this list can either have one entry only (in which case the generators will be callede + '0'
,e + '1'
, …,e + 'n-1'
, withe
being said entry), or haven
entries (in which case these entries will be used directly as names for the generators)n
– the number of generators, i.e., the rank of the free module whose exterior algebra is to be computed (this doesn’t have to be provided if it can be inferred from the rest of the input)
REFERENCES:
- Element#
alias of
sage.algebras.clifford_algebra_element.ExteriorAlgebraElement
- antipode_on_basis(m)#
Return the antipode on the basis element indexed by
m
.Given a basis element \(\omega\), the antipode is defined by \(S(\omega) = (-1)^{\deg(\omega)} \omega\).
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.antipode_on_basis(()) 1 sage: E.antipode_on_basis((1,)) -y sage: E.antipode_on_basis((1,2)) y*z
- boundary(s_coeff)#
Return the boundary operator \(\partial\) defined by the structure coefficients
s_coeff
of a Lie algebra.For more on the boundary operator, see
ExteriorAlgebraBoundary
.INPUT:
s_coeff
– a dictionary whose keys are in \(I \times I\), where \(I\) is the index set of the underlying vector space \(V\), and whose values can be coerced into 1-forms (degree 1 elements) inE
(usually, these values will just be elements of \(V\))
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.boundary({(0,1): z, (1,2): x, (2,0): y}) Boundary endomorphism of The exterior algebra of rank 3 over Rational Field
- coboundary(s_coeff)#
Return the coboundary operator \(d\) defined by the structure coefficients
s_coeff
of a Lie algebra.For more on the coboundary operator, see
ExteriorAlgebraCoboundary
.INPUT:
s_coeff
– a dictionary whose keys are in \(I \times I\), where \(I\) is the index set of the underlying vector space \(V\), and whose values can be coerced into 1-forms (degree 1 elements) inE
(usually, these values will just be elements of \(V\))
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.coboundary({(0,1): z, (1,2): x, (2,0): y}) Coboundary endomorphism of The exterior algebra of rank 3 over Rational Field
- coproduct_on_basis(a)#
Return the coproduct on the basis element indexed by
a
.The coproduct is defined by
\[\Delta(e_{i_1} \wedge \cdots \wedge e_{i_m}) = \sum_{k=0}^m \sum_{\sigma \in Ush_{k,m-k}} (-1)^{\sigma} (e_{i_{\sigma(1)}} \wedge \cdots \wedge e_{i_{\sigma(k)}}) \otimes (e_{i_{\sigma(k+1)}} \wedge \cdots \wedge e_{i_{\sigma(m)}}),\]where \(Ush_{k,m-k}\) denotes the set of all \((k,m-k)\)-unshuffles (i.e., permutations in \(S_m\) which are increasing on the interval \(\{1, 2, \ldots, k\}\) and on the interval \(\{k+1, k+2, \ldots, k+m\}\)).
Warning
This coproduct is a homomorphism of superalgebras, not a homomorphism of algebras!
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.coproduct_on_basis((0,)) 1 # x + x # 1 sage: E.coproduct_on_basis((0,1)) 1 # x*y + x # y - y # x + x*y # 1 sage: E.coproduct_on_basis((0,1,2)) 1 # x*y*z + x # y*z - y # x*z + x*y # z + z # x*y - x*z # y + y*z # x + x*y*z # 1
- counit(x)#
Return the counit of
x
.The counit of an element \(\omega\) of the exterior algebra is its constant coefficient.
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: elt = x*y - 2*x + 3 sage: E.counit(elt) 3
- degree_on_basis(m)#
Return the degree of the monomial indexed by
m
.The degree of
m
in the \(\ZZ\)-grading ofself
is defined to be the length ofm
.EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.degree_on_basis(()) 0 sage: E.degree_on_basis((0,)) 1 sage: E.degree_on_basis((0,1)) 2
- interior_product_on_basis(a, b)#
Return the interior product \(\iota_b a\) of
a
with respect tob
.See
interior_product()
for more information.In this method,
a
andb
are supposed to be basis elements (seeinterior_product()
for a method that computes interior product of arbitrary elements), and to be input as their keys.This depends on the choice of basis of the vector space whose exterior algebra is
self
.EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.interior_product_on_basis((0,), (0,)) 1 sage: E.interior_product_on_basis((0,2), (0,)) z sage: E.interior_product_on_basis((1,), (0,2)) 0 sage: E.interior_product_on_basis((0,2), (1,)) 0 sage: E.interior_product_on_basis((0,1,2), (0,2)) -y
- lift_morphism(phi, names=None)#
Lift the matrix
m
to an algebra morphism of exterior algebras.Given a linear map \(\phi : V \to W\) (here represented by a matrix acting on column vectors over the base ring of \(V\)), this method returns the algebra morphism \(\Lambda(\phi) : \Lambda(V) \to \Lambda(W)\). This morphism is defined on generators \(v_i \in \Lambda(V)\) by \(v_i \mapsto \phi(v_i)\).
Note
This is the map going out of
self
as opposed tolift_module_morphism()
for general Clifford algebras.INPUT:
phi
– a linear map \(\phi\) from \(V\) to \(W\), encoded as a matrixnames
– (default:'e'
) the names of the generators of the Clifford algebra of the domain of (the map represented by)phi
OUTPUT:
The algebra morphism \(\Lambda(\phi)\) from
self
to \(\Lambda(W)\).EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: phi = matrix([[0,1],[1,1],[1,2]]); phi [0 1] [1 1] [1 2] sage: L = E.lift_morphism(phi, ['a','b','c']); L Generic morphism: From: The exterior algebra of rank 2 over Rational Field To: The exterior algebra of rank 3 over Rational Field sage: L(x) b + c sage: L(y) a + b + 2*c sage: L.on_basis()((1,)) a + b + 2*c sage: p = L(E.one()); p 1 sage: p.parent() The exterior algebra of rank 3 over Rational Field sage: L(x*y) -a*b - a*c + b*c sage: L(x)*L(y) -a*b - a*c + b*c sage: L(x + y) a + 2*b + 3*c sage: L(x) + L(y) a + 2*b + 3*c sage: L(1/2*x + 2) 1/2*b + 1/2*c + 2 sage: L(E(3)) 3 sage: psi = matrix([[1, -3/2]]); psi [ 1 -3/2] sage: Lp = E.lift_morphism(psi, ['a']); Lp Generic morphism: From: The exterior algebra of rank 2 over Rational Field To: The exterior algebra of rank 1 over Rational Field sage: Lp(x) a sage: Lp(y) -3/2*a sage: Lp(x + 2*y + 3) -2*a + 3
- lifted_bilinear_form(M)#
Return the bilinear form on the exterior algebra
self
\(= \Lambda(V)\) which is obtained by lifting the bilinear form \(f\) on \(V\) given by the matrixM
.Let \(V\) be a module over a commutative ring \(R\), and let \(f : V \times V \to R\) be a bilinear form on \(V\). Then, a bilinear form \(\Lambda(f) : \Lambda(V) \times \Lambda(V) \to R\) on \(\Lambda(V)\) can be canonically defined as follows: For every \(n \in \NN\), \(m \in \NN\), \(v_1, v_2, \ldots, v_n, w_1, w_2, \ldots, w_m \in V\), we define
\[\begin{split}\Lambda(f) ( v_1 \wedge v_2 \wedge \cdots \wedge v_n , w_1 \wedge w_2 \wedge \cdots \wedge w_m ) := \begin{cases} 0, &\mbox{if } n \neq m ; \\ \det G, & \mbox{if } n = m \end{cases} ,\end{split}\]where \(G\) is the \(n \times m\)-matrix whose \((i, j)\)-th entry is \(f(v_i, w_j)\). This bilinear form \(\Lambda(f)\) is known as the bilinear form on \(\Lambda(V)\) obtained by lifting the bilinear form \(f\). Its restriction to the \(1\)-st homogeneous component \(V\) of \(\Lambda(V)\) is \(f\).
The bilinear form \(\Lambda(f)\) is symmetric if \(f\) is.
INPUT:
M
– a matrix over the same base ring asself
, whose \((i, j)\)-th entry is \(f(e_i, e_j)\), where \((e_1, e_2, \ldots, e_N)\) is the standard basis of the module \(V\) for whichself
\(= \Lambda(V)\) (so that \(N = \dim(V)\)), and where \(f\) is the bilinear form which is to be lifted.
OUTPUT:
A bivariate function which takes two elements \(p\) and \(q\) of
self
to \(\Lambda(f)(p, q)\).Note
This takes a bilinear form on \(V\) as matrix, and returns a bilinear form on
self
as a function in two arguments. We do not return the bilinear form as a matrix since this matrix can be huge and one often needs just a particular value.Todo
Implement a class for bilinear forms and rewrite this method to use that class.
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: M = Matrix(QQ, [[1, 2, 3], [2, 3, 4], [3, 4, 5]]) sage: Eform = E.lifted_bilinear_form(M) sage: Eform Bilinear Form from The exterior algebra of rank 3 over Rational Field (+) The exterior algebra of rank 3 over Rational Field to Rational Field sage: Eform(x*y, y*z) -1 sage: Eform(x*y, y) 0 sage: Eform(x*(y+z), y*z) -3 sage: Eform(x*(y+z), y*(z+x)) 0 sage: N = Matrix(QQ, [[3, 1, 7], [2, 0, 4], [-1, -3, -1]]) sage: N.determinant() -8 sage: Eform = E.lifted_bilinear_form(N) sage: Eform(x, E.one()) 0 sage: Eform(x, x*z*y) 0 sage: Eform(E.one(), E.one()) 1 sage: Eform(E.zero(), E.one()) 0 sage: Eform(x, y) 1 sage: Eform(z, y) -3 sage: Eform(x*z, y*z) 20 sage: Eform(x+x*y+x*y*z, z+z*y+z*y*x) 11
Todo
Another way to compute this bilinear form seems to be to map \(x\) and \(y\) to the appropriate Clifford algebra and there compute \(x^t y\), then send the result back to the exterior algebra and return its constant coefficient. Or something like this. Once the maps to the Clifford and back are implemented, check if this is faster.
- volume_form()#
Return the volume form of
self
.Given the basis \(e_1, e_2, \ldots, e_n\) of the underlying \(R\)-module, the volume form is defined as \(e_1 \wedge e_2 \wedge \cdots \wedge e_n\).
This depends on the choice of basis.
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: E.volume_form() x*y*z
- class sage.algebras.clifford_algebra.ExteriorAlgebraBoundary(E, s_coeff)#
Bases:
sage.algebras.clifford_algebra.ExteriorAlgebraDifferential
The boundary \(\partial\) of an exterior algebra \(\Lambda(L)\) defined by the structure coefficients of \(L\).
Let \(L\) be a Lie algebra. We give the exterior algebra \(E = \Lambda(L)\) a chain complex structure by considering a differential \(\partial : \Lambda^{k+1}(L) \to \Lambda^k(L)\) defined by
\[\partial(x_1 \wedge x_2 \wedge \cdots \wedge x_{k+1}) = \sum_{i < j} (-1)^{i+j+1} [x_i, x_j] \wedge x_1 \wedge \cdots \wedge \hat{x}_i \wedge \cdots \wedge \hat{x}_j \wedge \cdots \wedge x_{k+1}\]where \(\hat{x}_i\) denotes a missing index. The corresponding homology is the Lie algebra homology.
INPUT:
E
– an exterior algebra of a vector space \(L\)s_coeff
– a dictionary whose keys are in \(I \times I\), where \(I\) is the index set of the basis of the vector space \(L\), and whose values can be coerced into 1-forms (degree 1 elements) inE
; this dictionary will be used to define the Lie algebra structure on \(L\) (indeed, the \(i\)-th coordinate of the Lie bracket of the \(j\)-th and \(k\)-th basis vectors of \(L\) for \(j < k\) is set to be the value at the key \((j, k)\) if this key appears ins_coeff
, or otherwise the negated of the value at the key \((k, j)\))
Warning
The values of
s_coeff
are supposed to be coercible into 1-forms inE
; but they can also be dictionaries themselves (in which case they are interpreted as giving the coordinates of vectors inL
). In the interest of speed, these dictionaries are not sanitized or checked.Warning
For any two distinct elements \(i\) and \(j\) of \(I\), the dictionary
s_coeff
must have only one of the pairs \((i, j)\) and \((j, i)\) as a key. This is not checked.EXAMPLES:
We consider the differential given by Lie algebra given by the cross product \(\times\) of \(\RR^3\):
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: par = E.boundary({(0,1): z, (1,2): x, (2,0): y}) sage: par(x) 0 sage: par(x*y) z sage: par(x*y*z) 0 sage: par(x+y-y*z+x*y) -x + z sage: par(E.zero()) 0
We check that \(\partial \circ \partial = 0\):
sage: p2 = par * par sage: all(p2(b) == 0 for b in E.basis()) True
Another example: the Lie algebra \(\mathfrak{sl}_2\), which has a basis \(e,f,h\) satisfying \([h,e] = 2e\), \([h,f] = -2f\), and \([e,f] = h\):
sage: E.<e,f,h> = ExteriorAlgebra(QQ) sage: par = E.boundary({(0,1): h, (2,1): -2*f, (2,0): 2*e}) sage: par(E.zero()) 0 sage: par(e) 0 sage: par(e*f) h sage: par(f*h) 2*f sage: par(h*f) -2*f sage: C = par.chain_complex(); C Chain complex with at most 4 nonzero terms over Rational Field sage: ascii_art(C) [ 0 -2 0] [0] [ 0 0 2] [0] [0 0 0] [ 1 0 0] [0] 0 <-- C_0 <-------- C_1 <----------- C_2 <---- C_3 <-- 0 sage: C.homology() {0: Vector space of dimension 1 over Rational Field, 1: Vector space of dimension 0 over Rational Field, 2: Vector space of dimension 0 over Rational Field, 3: Vector space of dimension 1 over Rational Field}
Over the integers:
sage: C = par.chain_complex(R=ZZ); C Chain complex with at most 4 nonzero terms over Integer Ring sage: ascii_art(C) [ 0 -2 0] [0] [ 0 0 2] [0] [0 0 0] [ 1 0 0] [0] 0 <-- C_0 <-------- C_1 <----------- C_2 <---- C_3 <-- 0 sage: C.homology() {0: Z, 1: C2 x C2, 2: 0, 3: Z}
REFERENCES:
- chain_complex(R=None)#
Return the chain complex over
R
determined byself
.INPUT:
R
– the base ring; the default is the base ring of the exterior algebra
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: par = E.boundary({(0,1): z, (1,2): x, (2,0): y}) sage: C = par.chain_complex(); C Chain complex with at most 4 nonzero terms over Rational Field sage: ascii_art(C) [ 0 0 1] [0] [ 0 -1 0] [0] [0 0 0] [ 1 0 0] [0] 0 <-- C_0 <-------- C_1 <----------- C_2 <---- C_3 <-- 0
- class sage.algebras.clifford_algebra.ExteriorAlgebraCoboundary(E, s_coeff)#
Bases:
sage.algebras.clifford_algebra.ExteriorAlgebraDifferential
The coboundary \(d\) of an exterior algebra \(\Lambda(L)\) defined by the structure coefficients of a Lie algebra \(L\).
Let \(L\) be a Lie algebra. We endow its exterior algebra \(E = \Lambda(L)\) with a cochain complex structure by considering a differential \(d : \Lambda^k(L) \to \Lambda^{k+1}(L)\) defined by
\[d x_i = \sum_{j < k} s_{jk}^i x_j x_k,\]where \((x_1, x_2, \ldots, x_n)\) is a basis of \(L\), and where \(s_{jk}^i\) is the \(x_i\)-coordinate of the Lie bracket \([x_j, x_k]\).
The corresponding cohomology is the Lie algebra cohomology of \(L\).
This can also be thought of as the exterior derivative, in which case the resulting cohomology is the de Rham cohomology of a manifold whose exterior algebra of differential forms is
E
.INPUT:
E
– an exterior algebra of a vector space \(L\)s_coeff
– a dictionary whose keys are in \(I \times I\), where \(I\) is the index set of the basis of the vector space \(L\), and whose values can be coerced into 1-forms (degree 1 elements) inE
; this dictionary will be used to define the Lie algebra structure on \(L\) (indeed, the \(i\)-th coordinate of the Lie bracket of the \(j\)-th and \(k\)-th basis vectors of \(L\) for \(j < k\) is set to be the value at the key \((j, k)\) if this key appears ins_coeff
, or otherwise the negated of the value at the key \((k, j)\))
Warning
For any two distinct elements \(i\) and \(j\) of \(I\), the dictionary
s_coeff
must have only one of the pairs \((i, j)\) and \((j, i)\) as a key. This is not checked.EXAMPLES:
We consider the differential coming from the Lie algebra given by the cross product \(\times\) of \(\RR^3\):
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: d = E.coboundary({(0,1): z, (1,2): x, (0, 2): -y}) sage: d(x) y*z sage: d(y) -x*z sage: d(x+y-y*z) -x*z + y*z sage: d(x*y) 0 sage: d(E.one()) 0 sage: d(E.zero()) 0
We check that \(d \circ d = 0\):
sage: d2 = d * d sage: all(d2(b) == 0 for b in E.basis()) True
Another example: the Lie algebra \(\mathfrak{sl}_2\), which has a basis \(e,f,h\) satisfying \([h,e] = 2e\), \([h,f] = -2f\), and \([e,f] = h\):
sage: E.<e,f,h> = ExteriorAlgebra(QQ) sage: d = E.coboundary({(0,1): h, (2,1): -2*f, (2,0): 2*e}) sage: d(E.zero()) 0 sage: d(e) -2*e*h sage: d(f) 2*f*h sage: d(h) e*f sage: d(e*f) 0 sage: d(f*h) 0 sage: d(e*h) 0 sage: C = d.chain_complex(); C Chain complex with at most 4 nonzero terms over Rational Field sage: ascii_art(C) [ 0 0 1] [0] [-2 0 0] [0] [0 0 0] [ 0 2 0] [0] 0 <-- C_3 <-------- C_2 <----------- C_1 <---- C_0 <-- 0 sage: C.homology() {0: Vector space of dimension 1 over Rational Field, 1: Vector space of dimension 0 over Rational Field, 2: Vector space of dimension 0 over Rational Field, 3: Vector space of dimension 1 over Rational Field}
Over the integers:
sage: C = d.chain_complex(R=ZZ); C Chain complex with at most 4 nonzero terms over Integer Ring sage: ascii_art(C) [ 0 0 1] [0] [-2 0 0] [0] [0 0 0] [ 0 2 0] [0] 0 <-- C_3 <-------- C_2 <----------- C_1 <---- C_0 <-- 0 sage: C.homology() {0: Z, 1: 0, 2: C2 x C2, 3: Z}
REFERENCES:
- chain_complex(R=None)#
Return the chain complex over
R
determined byself
.INPUT:
R
– the base ring; the default is the base ring of the exterior algebra
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: d = E.coboundary({(0,1): z, (1,2): x, (2,0): y}) sage: C = d.chain_complex(); C Chain complex with at most 4 nonzero terms over Rational Field sage: ascii_art(C) [ 0 0 1] [0] [ 0 -1 0] [0] [0 0 0] [ 1 0 0] [0] 0 <-- C_3 <-------- C_2 <----------- C_1 <---- C_0 <-- 0
- class sage.algebras.clifford_algebra.ExteriorAlgebraDifferential(E, s_coeff)#
Bases:
sage.modules.with_basis.morphism.ModuleMorphismByLinearity
,sage.structure.unique_representation.UniqueRepresentation
Internal class to store the data of a boundary or coboundary of an exterior algebra \(\Lambda(L)\) defined by the structure coefficients of a Lie algebra \(L\).
See
ExteriorAlgebraBoundary
andExteriorAlgebraCoboundary
for the actual classes, which inherit from this.Warning
This is not a general class for differentials on the exterior algebra.
- homology(deg=None, **kwds)#
Return the homology determined by
self
.EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: par = E.boundary({(0,1): z, (1,2): x, (2,0): y}) sage: par.homology() {0: Vector space of dimension 1 over Rational Field, 1: Vector space of dimension 0 over Rational Field, 2: Vector space of dimension 0 over Rational Field, 3: Vector space of dimension 1 over Rational Field} sage: d = E.coboundary({(0,1): z, (1,2): x, (2,0): y}) sage: d.homology() {0: Vector space of dimension 1 over Rational Field, 1: Vector space of dimension 0 over Rational Field, 2: Vector space of dimension 0 over Rational Field, 3: Vector space of dimension 1 over Rational Field}
- class sage.algebras.clifford_algebra.ExteriorAlgebraIdeal(ring, gens, coerce=True, side='twosided')#
Bases:
sage.rings.noncommutative_ideals.Ideal_nc
An ideal of the exterior algebra.
EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: I = E.ideal(x*y); I Twosided Ideal (x*y) of The exterior algebra of rank 3 over Rational Field
We can also use it to build a quotient:
sage: Q = E.quotient(I); Q Quotient of The exterior algebra of rank 3 over Rational Field by the ideal (x*y) sage: Q.inject_variables() Defining xbar, ybar, zbar sage: xbar * ybar 0
- groebner_basis(term_order=None, reduced=True)#
Return the (reduced) Gröbner basis of
self
.INPUT:
term_order
– the term order used to compute the Gröbner basis; must be one of the following:"neglex"
– (default) negative (read right-to-left) lex order"degrevlex"
– degree reverse lex order"deglex"
– degree lex order
reduced
– (default:True
) whether or not to return the reduced Gröbner basis
EXAMPLES:
We compute an example:
sage: E.<a,b,c,d,e> = ExteriorAlgebra(QQ) sage: rels = [c*d*e - b*d*e + b*c*e - b*c*d, ....: c*d*e - a*d*e + a*c*e - a*c*d, ....: b*d*e - a*d*e + a*b*e - a*b*d, ....: b*c*e - a*c*e + a*b*e - a*b*c, ....: b*c*d - a*c*d + a*b*d - a*b*c] sage: I = E.ideal(rels) sage: I.groebner_basis() (-a*b*c + a*b*d - a*c*d + b*c*d, -a*b*c + a*b*e - a*c*e + b*c*e, -a*b*d + a*b*e - a*d*e + b*d*e, -a*c*d + a*c*e - a*d*e + c*d*e)
With different term orders:
sage: I.groebner_basis("degrevlex") (b*c*d - b*c*e + b*d*e - c*d*e, a*c*d - a*c*e + a*d*e - c*d*e, a*b*d - a*b*e + a*d*e - b*d*e, a*b*c - a*b*e + a*c*e - b*c*e) sage: I.groebner_basis("deglex") (-a*b*c + a*b*d - a*c*d + b*c*d, -a*b*c + a*b*e - a*c*e + b*c*e, -a*b*d + a*b*e - a*d*e + b*d*e, -a*c*d + a*c*e - a*d*e + c*d*e)
The example above was computed first using M2, which agrees with the
"degrevlex"
ordering:E = QQ[a..e, SkewCommutative => true] I = ideal( c*d*e - b*d*e + b*c*e - b*c*d, c*d*e - a*d*e + a*c*e - a*c*d, b*d*e - a*d*e + a*b*e - a*b*d, b*c*e - a*c*e + a*b*e - a*b*c, b*c*d - a*c*d + a*b*d - a*b*c) groebnerBasis(I) returns: o3 = | bcd-bce+bde-cde acd-ace+ade-cde abd-abe+ade-bde abc-abe+ace-bce |
By default, the Gröbner basis is reduced, but we can get non-reduced Gröber bases (which are not unique):
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: I = E.ideal([x+y*z]) sage: I.groebner_basis(reduced=False) (x*y, x*z, y*z + x, x*y*z) sage: I.groebner_basis(reduced=True) (x*y, x*z, y*z + x)
However, if we have already computed a reduced Gröbner basis (with a given term order), then we return that:
sage: I = E.ideal([x+y*z]) # A fresh ideal sage: I.groebner_basis() (x*y, x*z, y*z + x) sage: I.groebner_basis(reduced=False) (x*y, x*z, y*z + x)
- reduce(f)#
Reduce
f
moduloself
.EXAMPLES:
sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: I = E.ideal(x*y); sage: I.reduce(x*y + x*y*z + z) z sage: I.reduce(x*y + x + y) x + y sage: I.reduce(x*y + x*y*z) 0 sage: E.<a,b,c,d> = ExteriorAlgebra(QQ) sage: I = E.ideal([a+b*c]) sage: I.reduce(I.gen(0) * d) 0