Clifford algebra elements

AUTHORS:

• Travis Scrimshaw (2013-09-06): Initial version

• Trevor Karn (2022-07-10): Rewrite multiplication using bitsets

class sage.algebras.clifford_algebra_element.CliffordAlgebraElement

Bases: IndexedFreeModuleElement

An element in a Clifford algebra.

clifford_conjugate()

Return the Clifford conjugate of self.

The Clifford conjugate of an element $x$ of a Clifford algebra is defined as

$$\overline{x}:=\alpha (x^t)=\alpha (x{)}^t$$

where $\alpha$ denotes the reflection automorphism and $t$ the transposition.

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y + x*z
sage: c = elt.conjugate(); c
-x*z - 5*x - y + 3
sage: c.conjugate() == elt
True

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> c = elt.conjugate(); c
-x*z - 5*x - y + 3
>>> c.conjugate() == elt
True

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y + x*z
c = elt.conjugate(); c
c.conjugate() == elt

conjugate()

Return the Clifford conjugate of self.

The Clifford conjugate of an element $x$ of a Clifford algebra is defined as

$$\overline{x}:=\alpha (x^t)=\alpha (x{)}^t$$

where $\alpha$ denotes the reflection automorphism and $t$ the transposition.

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y + x*z
sage: c = elt.conjugate(); c
-x*z - 5*x - y + 3
sage: c.conjugate() == elt
True

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> c = elt.conjugate(); c
-x*z - 5*x - y + 3
>>> c.conjugate() == elt
True

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y + x*z
c = elt.conjugate(); c
c.conjugate() == elt

degree_negation()

Return the image of the reflection automorphism on self.

The reflection automorphism of a Clifford algebra is defined as the linear endomorphism of this algebra which maps

$$x_1\wedge x_2\wedge \cdots \wedge x_m\mapsto (-1{)}^m{x}_1\wedge x_2\wedge \cdots \wedge x_m.$$

It is an algebra automorphism of the Clifford algebra.

degree_negation() is an alias for reflection().

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y + x*z
sage: r = elt.reflection(); r
x*z - 5*x - y
sage: r.reflection() == elt
True

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> r = elt.reflection(); r
x*z - 5*x - y
>>> r.reflection() == elt
True

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y + x*z
r = elt.reflection(); r
r.reflection() == elt

list()

Return the list of monomials and their coefficients in self (as a list of $2$-tuples, each of which has the form (monomial, coefficient)).

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y
sage: elt.list()
[(1, 5), (01, 1)]

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y
>>> elt.list()
[(1, 5), (01, 1)]

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y
elt.list()

reflection()

Return the image of the reflection automorphism on self.

The reflection automorphism of a Clifford algebra is defined as the linear endomorphism of this algebra which maps

$$x_1\wedge x_2\wedge \cdots \wedge x_m\mapsto (-1{)}^m{x}_1\wedge x_2\wedge \cdots \wedge x_m.$$

It is an algebra automorphism of the Clifford algebra.

degree_negation() is an alias for reflection().

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y + x*z
sage: r = elt.reflection(); r
x*z - 5*x - y
sage: r.reflection() == elt
True

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> r = elt.reflection(); r
x*z - 5*x - y
>>> r.reflection() == elt
True

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y + x*z
r = elt.reflection(); r
r.reflection() == elt

support()

Return the support of self.

This is the list of all monomials which appear with nonzero coefficient in self.

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y
sage: elt.support()
[1, 01]

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y
>>> elt.support()
[1, 01]

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y
elt.support()

transpose()

Return the transpose of self.

The transpose is an anti-algebra involution of a Clifford algebra and is defined (using linearity) by

$$x_1\wedge x_2\wedge \cdots \wedge x_m\mapsto x_m\wedge \cdots \wedge x_2\wedge x_1.$$

EXAMPLES:

Sage

sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
sage: Cl.<x,y,z> = CliffordAlgebra(Q)
sage: elt = 5*x + y + x*z
sage: t = elt.transpose(); t
-x*z + 5*x + y + 3
sage: t.transpose() == elt
True
sage: Cl.one().transpose()
1

Python

>>> from sage.all import *
>>> Q = QuadraticForm(ZZ, Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> Cl = CliffordAlgebra(Q, names=('x', 'y', 'z',)); (x, y, z,) = Cl._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> t = elt.transpose(); t
-x*z + 5*x + y + 3
>>> t.transpose() == elt
True
>>> Cl.one().transpose()
1

Sage Live

Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
Cl.<x,y,z> = CliffordAlgebra(Q)
elt = 5*x + y + x*z
t = elt.transpose(); t
t.transpose() == elt
Cl.one().transpose()

class sage.algebras.clifford_algebra_element.CohomologyRAAGElement

Bases: CliffordAlgebraElement

An element in the cohomology of a right-angled Artin group.

See also

CohomologyRAAG

class sage.algebras.clifford_algebra_element.ExteriorAlgebraElement

Bases: CliffordAlgebraElement

An element of an exterior algebra.

antiderivation(x)

Return the interior product (also known as antiderivation) of self with respect to x (that is, the element ${\iota }_x(\text{self})$ of the exterior algebra).

If $V$ is an $R$-module, and if $\alpha$ is a fixed element of $V^{\ast }$, then the interior product with respect to $\alpha$ is an $R$-linear map $i_{\alpha }:\mathrm{\Lambda }(V)\to \mathrm{\Lambda }(V)$, determined by the following requirements:

• $i_{\alpha }(v)=\alpha (v)$ for all $v\in V={\mathrm{\Lambda }}^1(V)$,

• it is a graded derivation of degree $-1$: all $x$ and $y$ in $\mathrm{\Lambda }(V)$ satisfy

$$i_{\alpha }(x\wedge y)=(i_{\alpha }x)\wedge y+(-1{)}^{\mathrm{deg}x}x\wedge (i_{\alpha }y).$$

It can be shown that this map $i_{\alpha }$ is graded of degree $-1$ (that is, sends ${\mathrm{\Lambda }}^k(V)$ into ${\mathrm{\Lambda }}^{k-1}(V)$ for every $k$).

When $V$ is a finite free $R$-module, the interior product can also be defined by

$$(i_{\alpha }\omega )(u_1,\dots ,u_k)=\omega (\alpha ,u_1,\dots ,u_k),$$

where $\omega \in {\mathrm{\Lambda }}^k(V)$ is thought of as an alternating multilinear mapping from $V^{\ast }\times \cdots \times V^{\ast }$ to $R$.

Since Sage is only dealing with exterior powers of modules of the form $R^d$ for some nonnegative integer $d$, the element $\alpha \in V^{\ast }$ can be thought of as an element of $V$ (by identifying the standard basis of $V=R^d$ with its dual basis). This is how $\alpha$ should be passed to this method.

We then extend the interior product to all $\alpha \in \mathrm{\Lambda }(V^{\ast })$ by

$$i_{\beta \wedge \gamma }=i_{\gamma }\circ i_{\beta }.$$

INPUT:

• x – element of (or coercing into) ${\mathrm{\Lambda }}^1(V)$ (for example, an element of $V$); this plays the role of $\alpha$ in the above definition

EXAMPLES:

Sage

sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: x.interior_product(x)
1
sage: (x + x*y).interior_product(2*y)
-2*x
sage: (x*z + x*y*z).interior_product(2*y - x)
-2*x*z - y*z - z
sage: x.interior_product(E.one())
x
sage: E.one().interior_product(x)
0
sage: x.interior_product(E.zero())
0
sage: E.zero().interior_product(x)
0

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> x.interior_product(x)
1
>>> (x + x*y).interior_product(Integer(2)*y)
-2*x
>>> (x*z + x*y*z).interior_product(Integer(2)*y - x)
-2*x*z - y*z - z
>>> x.interior_product(E.one())
x
>>> E.one().interior_product(x)
0
>>> x.interior_product(E.zero())
0
>>> E.zero().interior_product(x)
0

Sage Live

E.<x,y,z> = ExteriorAlgebra(QQ)
x.interior_product(x)
(x + x*y).interior_product(2*y)
(x*z + x*y*z).interior_product(2*y - x)
x.interior_product(E.one())
E.one().interior_product(x)
x.interior_product(E.zero())
E.zero().interior_product(x)

REFERENCES:

• Wikipedia article Exterior_algebra#Interior_product

constant_coefficient()

Return the constant coefficient of self.

Todo

Define a similar method for general Clifford algebras once the morphism to exterior algebras is implemented.

EXAMPLES:

Sage

sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: elt = 5*x + y + x*z + 10
sage: elt.constant_coefficient()
10
sage: x.constant_coefficient()
0

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z + Integer(10)
>>> elt.constant_coefficient()
10
>>> x.constant_coefficient()
0

Sage Live

E.<x,y,z> = ExteriorAlgebra(QQ)
elt = 5*x + y + x*z + 10
elt.constant_coefficient()
x.constant_coefficient()

hodge_dual()

Return the Hodge dual of self.

The Hodge dual of an element $\alpha$ of the exterior algebra is defined as $i_{\alpha }\sigma$, where $\sigma$ is the volume form (volume_form()) and $i_{\alpha }$ denotes the antiderivation function with respect to $\alpha$ (see interior_product() for the definition of this).

Note

The Hodge dual of the Hodge dual of a homogeneous element $p$ of $\mathrm{\Lambda }(V)$ equals $(-1{)}^{k(n-k)}p$, where $n=\dim V$ and $k=\mathrm{deg}(p)=|p|$.

EXAMPLES:

Sage

sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: x.hodge_dual()
y*z
sage: (x*z).hodge_dual()
-y
sage: (x*y*z).hodge_dual()
1
sage: [a.hodge_dual().hodge_dual() for a in E.basis()]
[1, x, y, z, x*y, x*z, y*z, x*y*z]
sage: (x + x*y).hodge_dual()
y*z + z
sage: (x*z + x*y*z).hodge_dual()
-y + 1
sage: E = ExteriorAlgebra(QQ, 'wxyz')
sage: [a.hodge_dual().hodge_dual() for a in E.basis()]
[1, -w, -x, -y, -z, w*x, w*y, w*z, x*y, x*z, y*z,
 -w*x*y, -w*x*z, -w*y*z, -x*y*z, w*x*y*z]

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> x.hodge_dual()
y*z
>>> (x*z).hodge_dual()
-y
>>> (x*y*z).hodge_dual()
1
>>> [a.hodge_dual().hodge_dual() for a in E.basis()]
[1, x, y, z, x*y, x*z, y*z, x*y*z]
>>> (x + x*y).hodge_dual()
y*z + z
>>> (x*z + x*y*z).hodge_dual()
-y + 1
>>> E = ExteriorAlgebra(QQ, 'wxyz')
>>> [a.hodge_dual().hodge_dual() for a in E.basis()]
[1, -w, -x, -y, -z, w*x, w*y, w*z, x*y, x*z, y*z,
 -w*x*y, -w*x*z, -w*y*z, -x*y*z, w*x*y*z]

Sage Live

E.<x,y,z> = ExteriorAlgebra(QQ)
x.hodge_dual()
(x*z).hodge_dual()
(x*y*z).hodge_dual()
[a.hodge_dual().hodge_dual() for a in E.basis()]
(x + x*y).hodge_dual()
(x*z + x*y*z).hodge_dual()
E = ExteriorAlgebra(QQ, 'wxyz')
[a.hodge_dual().hodge_dual() for a in E.basis()]

interior_product(x)

Return the interior product (also known as antiderivation) of self with respect to x (that is, the element ${\iota }_x(\text{self})$ of the exterior algebra).

If $V$ is an $R$-module, and if $\alpha$ is a fixed element of $V^{\ast }$, then the interior product with respect to $\alpha$ is an $R$-linear map $i_{\alpha }:\mathrm{\Lambda }(V)\to \mathrm{\Lambda }(V)$, determined by the following requirements:

• $i_{\alpha }(v)=\alpha (v)$ for all $v\in V={\mathrm{\Lambda }}^1(V)$,

• it is a graded derivation of degree $-1$: all $x$ and $y$ in $\mathrm{\Lambda }(V)$ satisfy

$$i_{\alpha }(x\wedge y)=(i_{\alpha }x)\wedge y+(-1{)}^{\mathrm{deg}x}x\wedge (i_{\alpha }y).$$

It can be shown that this map $i_{\alpha }$ is graded of degree $-1$ (that is, sends ${\mathrm{\Lambda }}^k(V)$ into ${\mathrm{\Lambda }}^{k-1}(V)$ for every $k$).

When $V$ is a finite free $R$-module, the interior product can also be defined by

$$(i_{\alpha }\omega )(u_1,\dots ,u_k)=\omega (\alpha ,u_1,\dots ,u_k),$$

where $\omega \in {\mathrm{\Lambda }}^k(V)$ is thought of as an alternating multilinear mapping from $V^{\ast }\times \cdots \times V^{\ast }$ to $R$.

Since Sage is only dealing with exterior powers of modules of the form $R^d$ for some nonnegative integer $d$, the element $\alpha \in V^{\ast }$ can be thought of as an element of $V$ (by identifying the standard basis of $V=R^d$ with its dual basis). This is how $\alpha$ should be passed to this method.

We then extend the interior product to all $\alpha \in \mathrm{\Lambda }(V^{\ast })$ by

$$i_{\beta \wedge \gamma }=i_{\gamma }\circ i_{\beta }.$$

INPUT:

• x – element of (or coercing into) ${\mathrm{\Lambda }}^1(V)$ (for example, an element of $V$); this plays the role of $\alpha$ in the above definition

EXAMPLES:

Sage

sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: x.interior_product(x)
1
sage: (x + x*y).interior_product(2*y)
-2*x
sage: (x*z + x*y*z).interior_product(2*y - x)
-2*x*z - y*z - z
sage: x.interior_product(E.one())
x
sage: E.one().interior_product(x)
0
sage: x.interior_product(E.zero())
0
sage: E.zero().interior_product(x)
0

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> x.interior_product(x)
1
>>> (x + x*y).interior_product(Integer(2)*y)
-2*x
>>> (x*z + x*y*z).interior_product(Integer(2)*y - x)
-2*x*z - y*z - z
>>> x.interior_product(E.one())
x
>>> E.one().interior_product(x)
0
>>> x.interior_product(E.zero())
0
>>> E.zero().interior_product(x)
0

Sage Live

E.<x,y,z> = ExteriorAlgebra(QQ)
x.interior_product(x)
(x + x*y).interior_product(2*y)
(x*z + x*y*z).interior_product(2*y - x)
x.interior_product(E.one())
E.one().interior_product(x)
x.interior_product(E.zero())
E.zero().interior_product(x)

REFERENCES:

• Wikipedia article Exterior_algebra#Interior_product

reduce(I, left=True)

Reduce self with respect to the elements in I.

INPUT:

• I – list of exterior algebra elements or an ideal

• left – boolean; if reduce as a left ideal (True) or right ideal (False), ignored if I is an ideal

EXAMPLES:

Sage

sage: E.<a,b,c,d> = ExteriorAlgebra(QQ)
sage: f = (a + b*c) * d
sage: f.reduce([a + b*c], True)
2*a*d
sage: f.reduce([a + b*c], False)
0

sage: I = E.ideal([a + b*c])
sage: f.reduce(I)
0

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('a', 'b', 'c', 'd',)); (a, b, c, d,) = E._first_ngens(4)
>>> f = (a + b*c) * d
>>> f.reduce([a + b*c], True)
2*a*d
>>> f.reduce([a + b*c], False)
0

>>> I = E.ideal([a + b*c])
>>> f.reduce(I)
0

Sage Live

E.<a,b,c,d> = ExteriorAlgebra(QQ)
f = (a + b*c) * d
f.reduce([a + b*c], True)
f.reduce([a + b*c], False)
I = E.ideal([a + b*c])
f.reduce(I)

scalar(other)

Return the standard scalar product of self with other.

The standard scalar product of $x,y\in \mathrm{\Lambda }(V)$ is defined by $⟨x,y⟩=⟨x^{t}y⟩$, where $⟨a⟩$ denotes the degree-0 term of $a$, and where $x^t$ denotes the transpose (transpose()) of $x$.

Todo

Define a similar method for general Clifford algebras once the morphism to exterior algebras is implemented.

EXAMPLES:

Sage

sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: elt = 5*x + y + x*z
sage: elt.scalar(z + 2*x)
0
sage: elt.transpose() * (z + 2*x)
-2*x*y + 5*x*z + y*z

Python

>>> from sage.all import *
>>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> elt = Integer(5)*x + y + x*z
>>> elt.scalar(z + Integer(2)*x)
0
>>> elt.transpose() * (z + Integer(2)*x)
-2*x*y + 5*x*z + y*z

Sage Live

E.<x,y,z> = ExteriorAlgebra(QQ)
elt = 5*x + y + x*z
elt.scalar(z + 2*x)
elt.transpose() * (z + 2*x)

Make live