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: 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 instead is_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:

• Wikipedia article Clifford_algebra

INPUT:

• Q – a quadratic form

• names – (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 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-v

Todo

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 of m.

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 algebra self.

This is the i-th basis vector of the vector space on which the quadratic form defining self is defined, regarded as an element of self.

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 of self 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 of CliffordAlgebra for how this pullback and the morphism \(Cl(m)\) are defined.

INPUT:

• m – an isometry of the quadratic form of self

• 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 of CliffordAlgebra for how this pullback and the morphism \(Cl(m)\) are defined.

Note

This is a map into self.

INPUT:

• m – a matrix

• names – (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 element 1 is indexed by the emptyset, which is represented by the sage.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:

• Wikipedia article Classification_of_Clifford_algebras#Unit_pseudoscalar

quadratic_form()

Return the quadratic form of self.

This is the quadratic form used to define self. The quadratic form on self 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-v

Todo

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: UniqueRepresentation, 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: 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 computed

• names – 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 called e + '0', e + '1', …, e + 'n-1', with e being said entry), or have n 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:

• Wikipedia article Exterior_algebra

Element

alias of 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) in E (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) in E (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 of self is defined to be the length of m.

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 to b.

See interior_product() for more information.

In this method, a and b are supposed to be basis elements (see interior_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: k = list(E.basis().keys())
sage: E.interior_product_on_basis(k[1], k[1])
1
sage: E.interior_product_on_basis(k[5], k[1])
z
sage: E.interior_product_on_basis(k[2], k[5])
0
sage: E.interior_product_on_basis(k[5], k[2])
0
sage: E.interior_product_on_basis(k[7], k[5])
-y

Check trac ticket #34694:

sage: E = ExteriorAlgebra(SR,'e',3)
sage: E.inject_variables()
Defining e0, e1, e2
sage: a = (e0*e1).interior_product(e0)
sage: a * e0
-e0*e1

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 to lift_module_morphism() for general Clifford algebras.

INPUT:

• phi – a linear map \(\phi\) from \(V\) to \(W\), encoded as a matrix

• names – (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 matrix M.

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 as self, 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 which self \(= \Lambda(V)\) (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: 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) in E; 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 in s_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 in E; but they can also be dictionaries themselves (in which case they are interpreted as giving the coordinates of vectors in L). 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:

• Wikipedia article Exterior_algebra#Lie_algebra_homology

chain_complex(R=None)

Return the chain complex over R determined by self.

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: 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) in E; 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 in s_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:

• Wikipedia article Exterior_algebra#Differential_geometry

chain_complex(R=None)

Return the chain complex over R determined by self.

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: ModuleMorphismByLinearity, 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 and ExteriorAlgebraCoboundary 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: 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 modulo self.

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