Tensor

class sympy.tensor.tensor._TensorManager

Class to manage tensor properties.

Notes

Tensors belong to tensor commutation groups; each group has a label comm; there are predefined labels:

0 tensors commuting with any other tensor

1 tensors anticommuting among themselves

2 tensors not commuting, apart with those with comm=0

Other groups can be defined using set_comm; tensors in those groups commute with those with comm=0; by default they do not commute with any other group.

clear()

Clear the TensorManager.

comm_i2symbol(i)

Returns the symbol corresponding to the commutation group number.

comm_symbols2i(i)

get the commutation group number corresponding to i

i can be a symbol or a number or a string

If i is not already defined its commutation group number is set.

get_comm(i, j)

Return the commutation parameter for commutation group numbers i, j

see _TensorManager.set_comm

set_comm(i, j, c)

set the commutation parameter c for commutation groups i, j

Parameters

i, j : symbols representing commutation groups

c : group commutation number

Notes

i, j can be symbols, strings or numbers, apart from 0, 1 and 2 which are reserved respectively for commuting, anticommuting tensors and tensors not commuting with any other group apart with the commuting tensors. For the remaining cases, use this method to set the commutation rules; by default c=None.

The group commutation number c is assigned in correspondence to the group commutation symbols; it can be

0 commuting

1 anticommuting

None no commutation property

Examples

G and GH do not commute with themselves and commute with each other; A is commuting.

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensorManager
>>> Lorentz = TensorIndexType('Lorentz')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> G = tensorhead('G', [Lorentz], [[1]], 'Gcomm')
>>> GH = tensorhead('GH', [Lorentz], [[1]], 'GHcomm')
>>> TensorManager.set_comm('Gcomm', 'GHcomm', 0)
>>> (GH(i1)*G(i0)).canon_bp()
G(i0)*GH(i1)
>>> (G(i1)*G(i0)).canon_bp()
G(i1)*G(i0)
>>> (G(i1)*A(i0)).canon_bp()
A(i0)*G(i1)

set_comms(*args)

set the commutation group numbers c for symbols i, j

Parameters

args : sequence of (i, j, c)

class sympy.tensor.tensor.TensorIndexType

A TensorIndexType is characterized by its name and its metric.

Parameters

name : name of the tensor type

metric : metric symmetry or metric object or None

dim : dimension, it can be a symbol or an integer or None

eps_dim : dimension of the epsilon tensor

dummy_fmt : name of the head of dummy indices

Notes

The metric parameter can be: metric = False symmetric metric (in Riemannian geometry)

metric = True antisymmetric metric (for spinor calculus)

metric = None there is no metric

metric can be an object having name and antisym attributes.

If there is a metric the metric is used to raise and lower indices.

In the case of antisymmetric metric, the following raising and lowering conventions will be adopted:

psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)

g(-a, b) = delta(-a, b); g(b, -a) = -delta(a, -b)

where delta(-a, b) = delta(b, -a) is the Kronecker delta (see TensorIndex for the conventions on indices).

If there is no metric it is not possible to raise or lower indices; e.g. the index of the defining representation of SU(N) is ‘covariant’ and the conjugate representation is ‘contravariant’; for N > 2 they are linearly independent.

eps_dim is by default equal to dim, if the latter is an integer; else it can be assigned (for use in naive dimensional regularization); if eps_dim is not an integer epsilon is None.

Examples

>>> from sympy.tensor.tensor import TensorIndexType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> Lorentz.metric
metric(Lorentz,Lorentz)

Attributes

name
metric_name	(it is ‘metric’ or metric.name)
metric_antisym
metric	(the metric tensor)
delta	(Kronecker delta)
epsilon	(the Levi-Civita epsilon tensor)
dim
eps_dim
dummy_fmt
data	(a property to add ndarray values, to work in a specified basis.)

class sympy.tensor.tensor.TensorIndex

Represents an abstract tensor index.

Parameters

name : name of the index, or True if you want it to be automatically assigned

tensortype : TensorIndexType of the index

is_up : flag for contravariant index

Notes

Tensor indices are contracted with the Einstein summation convention.

An index can be in contravariant or in covariant form; in the latter case it is represented prepending a - to the index name.

Dummy indices have a name with head given by tensortype._dummy_fmt

Examples

>>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i = TensorIndex('i', Lorentz); i
i
>>> sym1 = TensorSymmetry(*get_symmetric_group_sgs(1))
>>> S1 = TensorType([Lorentz], sym1)
>>> A, B = S1('A,B')
>>> A(i)*B(-i)
A(L_0)*B(-L_0)

If you want the index name to be automatically assigned, just put True in the name field, it will be generated using the reserved character _ in front of its name, in order to avoid conflicts with possible existing indices:

>>> i0 = TensorIndex(True, Lorentz)
>>> i0
_i0
>>> i1 = TensorIndex(True, Lorentz)
>>> i1
_i1
>>> A(i0)*B(-i1)
A(_i0)*B(-_i1)
>>> A(i0)*B(-i0)
A(L_0)*B(-L_0)

Attributes

name
tensortype
is_up

sympy.tensor.tensor.tensor_indices(s, typ)

Returns list of tensor indices given their names and their types

Parameters

s : string of comma separated names of indices

typ : TensorIndexType of the indices

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)

class sympy.tensor.tensor.TensorSymmetry

Monoterm symmetry of a tensor

Parameters

bsgs : tuple (base, sgs) BSGS of the symmetry of the tensor

Notes

A tensor can have an arbitrary monoterm symmetry provided by its BSGS. Multiterm symmetries, like the cyclic symmetry of the Riemann tensor, are not covered.

Examples

Define a symmetric tensor

>>> from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')

See also

sympy.combinatorics.tensor_can.get_symmetric_group_sgs

Attributes

base	(base of the BSGS)
generators	(generators of the BSGS)
rank	(rank of the tensor)

sympy.tensor.tensor.tensorsymmetry(*args)

Return a TensorSymmetry object.

One can represent a tensor with any monoterm slot symmetry group using a BSGS.

args can be a BSGS args[0] base args[1] sgs

Usually tensors are in (direct products of) representations of the symmetric group; args can be a list of lists representing the shapes of Young tableaux

Notes

For instance: [[1]] vector [[1]*n] symmetric tensor of rank n [[n]] antisymmetric tensor of rank n [[2, 2]] monoterm slot symmetry of the Riemann tensor [[1],[1]] vector*vector [[2],[1],[1] (antisymmetric tensor)*vector*vector

Notice that with the shape [2, 2] we associate only the monoterm symmetries of the Riemann tensor; this is an abuse of notation, since the shape [2, 2] corresponds usually to the irreducible representation characterized by the monoterm symmetries and by the cyclic symmetry.

Examples

Symmetric tensor using a Young tableau

>>> from sympy.tensor.tensor import TensorIndexType, TensorType, tensorsymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')

Symmetric tensor using a BSGS (base, strong generator set)

>>> from sympy.tensor.tensor import get_symmetric_group_sgs
>>> sym2 = tensorsymmetry(*get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')

class sympy.tensor.tensor.TensorType

Class of tensor types.

Parameters

index_types : list of TensorIndexType of the tensor indices

symmetry : TensorSymmetry of the tensor

Examples

Define a symmetric tensor

>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')

Attributes

index_types
symmetry
types	(list of TensorIndexType without repetitions)

class sympy.tensor.tensor.TensorHead

Tensor head of the tensor

Parameters

name : name of the tensor

typ : list of TensorIndexType

comm : commutation group number

Notes

A TensorHead belongs to a commutation group, defined by a symbol on number comm (see _TensorManager.set_comm); tensors in a commutation group have the same commutation properties; by default comm is 0, the group of the commuting tensors.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> A = tensorhead('A', [Lorentz, Lorentz], [[1],[1]])

Examples with ndarray values, the components data assigned to the TensorHead object are assumed to be in a fully-contravariant representation. In case it is necessary to assign components data which represents the values of a non-fully covariant tensor, see the other examples.

>>> from sympy.tensor.tensor import tensor_indices, tensorhead
>>> from sympy import diag
>>> i0, i1 = tensor_indices('i0:2', Lorentz)

Specify a replacement dictionary to keep track of the arrays to use for replacements in the tensorial expression. The TensorIndexType is associated to the metric used for contractions (in fully covariant form):

>>> repl = {Lorentz: diag(1, -1, -1, -1)}

Let’s see some examples of working with components with the electromagnetic tensor:

>>> from sympy import symbols
>>> Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z')
>>> c = symbols('c', positive=True)

Let’s define \(F\), an antisymmetric tensor, we have to assign an antisymmetric matrix to it, because \([[2]]\) stands for the Young tableau representation of an antisymmetric set of two elements:

>>> F = tensorhead('F', [Lorentz, Lorentz], [[2]])

Let’s update the dictionary to contain the matrix to use in the replacements:

>>> repl.update({F(-i0, -i1): [
... [0, Ex/c, Ey/c, Ez/c],
... [-Ex/c, 0, -Bz, By],
... [-Ey/c, Bz, 0, -Bx],
... [-Ez/c, -By, Bx, 0]]})

Now it is possible to retrieve the contravariant form of the Electromagnetic tensor:

>>> F(i0, i1).replace_with_arrays(repl, [i0, i1])
[[0, -E_x/c, -E_y/c, -E_z/c], [E_x/c, 0, -B_z, B_y], [E_y/c, B_z, 0, -B_x], [E_z/c, -B_y, B_x, 0]]

and the mixed contravariant-covariant form:

>>> F(i0, -i1).replace_with_arrays(repl, [i0, -i1])
[[0, E_x/c, E_y/c, E_z/c], [E_x/c, 0, B_z, -B_y], [E_y/c, -B_z, 0, B_x], [E_z/c, B_y, -B_x, 0]]

Energy-momentum of a particle may be represented as:

>>> from sympy import symbols
>>> P = tensorhead('P', [Lorentz], [[1]])
>>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True)
>>> repl.update({P(i0): [E, px, py, pz]})

The contravariant and covariant components are, respectively:

>>> P(i0).replace_with_arrays(repl, [i0])
[E, p_x, p_y, p_z]
>>> P(-i0).replace_with_arrays(repl, [-i0])
[E, -p_x, -p_y, -p_z]

The contraction of a 1-index tensor by itself:

>>> expr = P(i0)*P(-i0)
>>> expr.replace_with_arrays(repl, [])
E**2 - p_x**2 - p_y**2 - p_z**2

Attributes

name
index_types
rank
types	( equal to typ.types)
symmetry	(equal to typ.symmetry)
comm	(commutation group)

commutes_with(other)

Returns 0 if self and other commute, 1 if they anticommute.

Returns None if self and other neither commute nor anticommute.

class sympy.tensor.tensor.TensExpr

Abstract base class for tensor expressions

Notes

A tensor expression is an expression formed by tensors; currently the sums of tensors are distributed.

A TensExpr can be a TensAdd or a TensMul.

TensAdd objects are put in canonical form using the Butler-Portugal algorithm for canonicalization under monoterm symmetries.

TensMul objects are formed by products of component tensors, and include a coefficient, which is a SymPy expression.

In the internal representation contracted indices are represented by (ipos1, ipos2, icomp1, icomp2), where icomp1 is the position of the component tensor with contravariant index, ipos1 is the slot which the index occupies in that component tensor.

Contracted indices are therefore nameless in the internal representation.

fun_eval(*index_tuples)

Return a tensor with free indices substituted according to index_tuples

index_types list of tuples (old_index, new_index)

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l)

get_matrix()

DEPRECATED: do not use.

Returns ndarray components data as a matrix, if components data are available and ndarray dimension does not exceed 2.

replace_with_arrays(replacement_dict, indices)

Replace the tensorial expressions with arrays. The final array will correspond to the N-dimensional array with indices arranged according to indices.

Parameters

replacement_dict

dictionary containing the replacement rules for tensors.

indices

the index order with respect to which the array is read.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> from sympy.tensor.tensor import tensorhead
>>> from sympy import symbols, diag

>>> L = TensorIndexType("L")
>>> i, j = tensor_indices("i j", L)
>>> A = tensorhead("A", [L], [[1]])
>>> A(i).replace_with_arrays({A(i): [1, 2]}, [i])
[1, 2]
>>> expr = A(i)*A(j)
>>> expr.replace_with_arrays({A(i): [1, 2]}, [i, j])
[[1, 2], [2, 4]]

For contractions, specify the metric of the TensorIndexType, which in this case is L, in its covariant form:

>>> expr = A(i)*A(-i)
>>> expr.replace_with_arrays({A(i): [1, 2], L: diag(1, -1)}, [])
-3

Symmetrization of an array:

>>> H = tensorhead("H", [L, L], [[1], [1]])
>>> a, b, c, d = symbols("a b c d")
>>> expr = H(i, j)/2 + H(j, i)/2
>>> expr.replace_with_arrays({H(i, j): [[a, b], [c, d]]}, [i, j])
[[a, b/2 + c/2], [b/2 + c/2, d]]

Anti-symmetrization of an array:

>>> expr = H(i, j)/2 - H(j, i)/2
>>> repl = {H(i, j): [[a, b], [c, d]]}
>>> expr.replace_with_arrays(repl, [i, j])
[[0, b/2 - c/2], [-b/2 + c/2, 0]]

The same expression can be read as the transpose by inverting i and j:

>>> expr.replace_with_arrays(repl, [j, i])
[[0, -b/2 + c/2], [b/2 - c/2, 0]]

class sympy.tensor.tensor.TensAdd

Sum of tensors

Parameters

free_args : list of the free indices

Notes

Sum of more than one tensor are put automatically in canonical form.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(a) + q(a); t
p(a) + q(a)
>>> t(b)
p(b) + q(b)

Examples with components data added to the tensor expression:

>>> from sympy import symbols, diag
>>> x, y, z, t = symbols("x y z t")
>>> repl = {}
>>> repl[Lorentz] = diag(1, -1, -1, -1)
>>> repl[p(a)] = [1, 2, 3, 4]
>>> repl[q(a)] = [x, y, z, t]

The following are: 2**2 - 3**2 - 2**2 - 7**2 ==> -58

>>> expr = p(a) + q(a)
>>> expr.replace_with_arrays(repl, [a])
[x + 1, y + 2, z + 3, t + 4]

Attributes

args	(tuple of addends)
rank	(rank of the tensor)
free_args	(list of the free indices in sorted order)

canon_bp()

canonicalize using the Butler-Portugal algorithm for canonicalization under monoterm symmetries.

contract_metric(g)

Raise or lower indices with the metric g

Parameters

g : metric

contract_all : if True, eliminate all g which are contracted

Notes

see the TensorIndexType docstring for the contraction conventions

fun_eval(*index_tuples)

Return a tensor with free indices substituted according to index_tuples

Parameters

index_types : list of tuples (old_index, new_index)

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j) + A(i, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l) + A(k, l)

substitute_indices(*index_tuples)

Return a tensor with free indices substituted according to index_tuples

Parameters

index_types : list of tuples (old_index, new_index)

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)

class sympy.tensor.tensor.TensMul

Product of tensors

Parameters

coeff : SymPy coefficient of the tensor

args

Notes

args[0] list of TensorHead of the component tensors.

args[1] list of (ind, ipos, icomp) where ind is a free index, ipos is the slot position of ind in the icomp-th component tensor.

args[2] list of tuples representing dummy indices. (ipos1, ipos2, icomp1, icomp2) indicates that the contravariant dummy index is the ipos1-th slot position in the icomp1-th component tensor; the corresponding covariant index is in the ipos2 slot position in the icomp2-th component tensor.

Attributes

components	(list of TensorHead of the component tensors)
types	(list of nonrepeated TensorIndexType)
free	(list of (ind, ipos, icomp), see Notes)
dum	(list of (ipos1, ipos2, icomp1, icomp2), see Notes)
ext_rank	(rank of the tensor counting the dummy indices)
rank	(rank of the tensor)
coeff	(SymPy coefficient of the tensor)
free_args	(list of the free indices in sorted order)
is_canon_bp	(True if the tensor in in canonical form)

canon_bp()

Canonicalize using the Butler-Portugal algorithm for canonicalization under monoterm symmetries.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[2]])
>>> t = A(m0,-m1)*A(m1,-m0)
>>> t.canon_bp()
-A(L_0, L_1)*A(-L_0, -L_1)
>>> t = A(m0,-m1)*A(m1,-m2)*A(m2,-m0)
>>> t.canon_bp()
0

contract_metric(g)

Raise or lower indices with the metric g

Parameters

g : metric

Notes

see the TensorIndexType docstring for the contraction conventions

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m0)*q(m1)*g(-m0, -m1)
>>> t.canon_bp()
metric(L_0, L_1)*p(-L_0)*q(-L_1)
>>> t.contract_metric(g).canon_bp()
p(L_0)*q(-L_0)

get_free_indices()

Returns the list of free indices of the tensor

The indices are listed in the order in which they appear in the component tensors.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_free_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_free_indices()
[m2]

get_indices()

Returns the list of indices of the tensor

The indices are listed in the order in which they appear in the component tensors. The dummy indices are given a name which does not collide with the names of the free indices.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_indices()
[L_0, -L_0, m2]

perm2tensor(g, is_canon_bp=False)

Returns the tensor corresponding to the permutation g

For further details, see the method in TIDS with the same name.

sorted_components()

Returns a tensor product with sorted components.

split()

Returns a list of tensors, whose product is self

Dummy indices contracted among different tensor components become free indices with the same name as the one used to represent the dummy indices.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(a,b)*B(-b,c)
>>> t
A(a, L_0)*B(-L_0, c)
>>> t.split()
[A(a, L_0), B(-L_0, c)]

sympy.tensor.tensor.canon_bp(p)

Butler-Portugal canonicalization

sympy.tensor.tensor.tensor_mul(*a)

product of tensors

sympy.tensor.tensor.riemann_cyclic_replace(t_r)

replace Riemann tensor with an equivalent expression

R(m,n,p,q) -> 2/3*R(m,n,p,q) - 1/3*R(m,q,n,p) + 1/3*R(m,p,n,q)

sympy.tensor.tensor.riemann_cyclic(t2)

replace each Riemann tensor with an equivalent expression satisfying the cyclic identity.

This trick is discussed in the reference guide to Cadabra.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, riemann_cyclic
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> R = tensorhead('R', [Lorentz]*4, [[2, 2]])
>>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
>>> riemann_cyclic(t)
0