Alternating forms on free modules

Given a free module \(M\) of finite rank over a commutative ring \(R\) and a positive integer \(p\), an alternating form of degree \(p\) on \(M\) is a map

\[a:\ \underbrace{M\times\cdots\times M}_{p\ \; \mbox{times}} \longrightarrow R\]

that (i) is multilinear and (ii) vanishes whenever any of two of its arguments are equal. An alternating form of degree \(p\) is a tensor on \(M\) of type \((0,p)\).

Alternating forms are implemented via the class FreeModuleAltForm, which is a subclass of the generic tensor class FreeModuleTensor.

AUTHORS:

  • Eric Gourgoulhon, Michal Bejger (2014-2015): initial version

REFERENCES:

  • Chap. 23 of R. Godement: Algebra, Hermann (Paris) / Houghton Mifflin (Boston) (1968)
  • Chap. 15 of S. Lang: Algebra, 3rd ed., Springer (New York) (2002)
class sage.tensor.modules.free_module_alt_form.FreeModuleAltForm(fmodule, degree, name=None, latex_name=None)

Bases: sage.tensor.modules.free_module_tensor.FreeModuleTensor

Alternating form on a free module of finite rank over a commutative ring.

This is a Sage element class, the corresponding parent class being ExtPowerFreeModule.

INPUT:

  • fmodule – free module \(M\) of finite rank over a commutative ring \(R\), as an instance of FiniteRankFreeModule
  • degree – positive integer; the degree \(p\) of the alternating form (i.e. its tensor rank)
  • name – (default: None) string; name given to the alternating form
  • latex_name – (default: None) string; LaTeX symbol to denote the alternating form; if none is provided, name is used

EXAMPLES:

Alternating form of degree 2 on a rank-3 free module:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e')
sage: a = M.alternating_form(2, name='a') ; a
Alternating form a of degree 2 on the
 Rank-3 free module M over the Integer Ring
sage: type(a)
<class 'sage.tensor.modules.free_module_alt_form.ExtPowerFreeModule_with_category.element_class'>
sage: a.parent()
2nd exterior power of the dual of the Rank-3 free module M over the Integer Ring
sage: a[1,2], a[2,3] = 4, -3
sage: a.display(e)
a = 4 e^1/\e^2 - 3 e^2/\e^3

The alternating form acting on the basis elements:

sage: a(e[1],e[2])
4
sage: a(e[1],e[3])
0
sage: a(e[2],e[3])
-3
sage: a(e[2],e[1])
-4

An alternating form of degree 1 is a linear form:

sage: b = M.linear_form('b') ; b
Linear form b on the Rank-3 free module M over the Integer Ring
sage: b[:] = [2,-1,3]  # components w.r.t. the module's default basis (e)

A linear form is a tensor of type (0,1):

sage: b.tensor_type()
(0, 1)

It is an element of the dual module:

sage: b.parent()
Dual of the Rank-3 free module M over the Integer Ring
sage: b.parent() is M.dual()
True

The members of a dual basis are linear forms:

sage: e.dual_basis()[1]
Linear form e^1 on the Rank-3 free module M over the Integer Ring
sage: e.dual_basis()[2]
Linear form e^2 on the Rank-3 free module M over the Integer Ring
sage: e.dual_basis()[3]
Linear form e^3 on the Rank-3 free module M over the Integer Ring

Any linear form is expanded onto them:

sage: b.display(e)
b = 2 e^1 - e^2 + 3 e^3

In the above example, an equivalent writing would have been b.display(), since the basis e is the module’s default basis. A linear form maps module elements to ring elements:

sage: v = M([1,1,1])
sage: b(v)
4
sage: b(v) in M.base_ring()
True

Test of linearity:

sage: u = M([-5,-2,7])
sage: b(3*u - 4*v) == 3*b(u) - 4*b(v)
True

The standard tensor operations apply to alternating forms, like the extraction of components with respect to a given basis:

sage: a[e,1,2]
4
sage: a[1,2]  # since e is the module's default basis
4
sage: all( a[i,j] == - a[j,i] for i in {1,2,3} for j in {1,2,3} )
True

the tensor product:

sage: c = b*b ; c
Symmetric bilinear form  b*b on the Rank-3 free module M over the
 Integer Ring
sage: c.parent()
Free module of type-(0,2) tensors on the Rank-3 free module M over the
 Integer Ring
sage: c.display(e)
b*b = 4 e^1*e^1 - 2 e^1*e^2 + 6 e^1*e^3 - 2 e^2*e^1 + e^2*e^2
 - 3 e^2*e^3 + 6 e^3*e^1 - 3 e^3*e^2 + 9 e^3*e^3

the contractions:

sage: s = a.contract(v) ; s
Linear form on the Rank-3 free module M over the Integer Ring
sage: s.parent()
Dual of the Rank-3 free module M over the Integer Ring
sage: s.display(e)
4 e^1 - 7 e^2 + 3 e^3

or tensor arithmetics:

sage: s = 3*a + c ; s
Type-(0,2) tensor on the Rank-3 free module M over the Integer Ring
sage: s.parent()
Free module of type-(0,2) tensors on the Rank-3 free module M over the
 Integer Ring
sage: s.display(e)
4 e^1*e^1 + 10 e^1*e^2 + 6 e^1*e^3 - 14 e^2*e^1 + e^2*e^2
 - 12 e^2*e^3 + 6 e^3*e^1 + 6 e^3*e^2 + 9 e^3*e^3

Note that tensor arithmetics preserves the alternating character if both operands are alternating:

sage: s = a - 2*a ; s
Alternating form of degree 2 on the Rank-3 free module M over the
 Integer Ring
sage: s.parent() # note the difference with s = 3*a + c above
2nd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring
sage: s == -a
True

An operation specific to alternating forms is of course the exterior product:

sage: s = a.wedge(b) ; s
Alternating form a/\b of degree 3 on the Rank-3 free module M over the
 Integer Ring
sage: s.parent()
3rd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring
sage: s.display(e)
a/\b = 6 e^1/\e^2/\e^3
sage: s[1,2,3] == a[1,2]*b[3] + a[2,3]*b[1] + a[3,1]*b[2]
True

The exterior product is nilpotent on linear forms:

sage: s = b.wedge(b) ; s
Alternating form b/\b of degree 2 on the Rank-3 free module M over the
 Integer Ring
sage: s.display(e)
b/\b = 0
degree()

Return the degree of self.

EXAMPLES:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: a = M.alternating_form(2, name='a')
sage: a.degree()
2
disp(basis=None, format_spec=None)

Display the alternating form self in terms of its expansion w.r.t. a given module basis.

The expansion is actually performed onto exterior products of elements of the cobasis (dual basis) associated with basis (see examples below). The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • basis – (default: None) basis of the free module with respect to which the alternating form is expanded; if none is provided, the module’s default basis is assumed
  • format_spec – (default: None) format specification passed to self._fmodule._output_formatter to format the output

EXAMPLES:

Display of an alternating form of degree 1 (linear form) on a rank-3 free module:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: e.dual_basis()
Dual basis (e^0,e^1,e^2) on the Rank-3 free module M over the Integer Ring
sage: a = M.linear_form('a', latex_name=r'\alpha')
sage: a[:] = [1,-3,4]
sage: a.display(e)
a = e^0 - 3 e^1 + 4 e^2
sage: a.display()  # a shortcut since e is M's default basis
a = e^0 - 3 e^1 + 4 e^2
sage: latex(a.display())  # display in the notebook
\alpha = e^0 -3 e^1 + 4 e^2

A shortcut is disp():

sage: a.disp()
a = e^0 - 3 e^1 + 4 e^2

Display of an alternating form of degree 2 on a rank-3 free module:

sage: b = M.alternating_form(2, 'b', latex_name=r'\beta')
sage: b[0,1], b[0,2], b[1,2] = 3, 2, -1
sage: b.display()
b = 3 e^0/\e^1 + 2 e^0/\e^2 - e^1/\e^2
sage: latex(b.display())  # display in the notebook
\beta = 3 e^0\wedge e^1 + 2 e^0\wedge e^2 -e^1\wedge e^2

Display of an alternating form of degree 3 on a rank-3 free module:

sage: c = M.alternating_form(3, 'c')
sage: c[0,1,2] = 4
sage: c.display()
c = 4 e^0/\e^1/\e^2
sage: latex(c.display())
c = 4 e^0\wedge e^1\wedge e^2

Display of a vanishing alternating form:

sage: c[0,1,2] = 0  # the only independent component set to zero
sage: c.is_zero()
True
sage: c.display()
c = 0
sage: latex(c.display())
c = 0
sage: c[0,1,2] = 4  # value restored for what follows

Display in a basis which is not the default one:

sage: aut = M.automorphism(matrix=[[0,1,0], [0,0,-1], [1,0,0]],
....:                      basis=e)
sage: f = e.new_basis(aut, 'f')
sage: a.display(f)
a = 4 f^0 + f^1 + 3 f^2
sage: a.disp(f)     # shortcut notation
a = 4 f^0 + f^1 + 3 f^2
sage: b.display(f)
b = -2 f^0/\f^1 - f^0/\f^2 - 3 f^1/\f^2
sage: c.display(f)
c = -4 f^0/\f^1/\f^2

The output format can be set via the argument output_formatter passed at the module construction:

sage: N = FiniteRankFreeModule(QQ, 3, name='N', start_index=1,
....:                   output_formatter=Rational.numerical_approx)
sage: e = N.basis('e')
sage: b = N.alternating_form(2, 'b')
sage: b[1,2], b[1,3], b[2,3] = 1/3, 5/2, 4
sage: b.display()  # default format (53 bits of precision)
b = 0.333333333333333 e^1/\e^2 + 2.50000000000000 e^1/\e^3
 + 4.00000000000000 e^2/\e^3

The output format is then controlled by the argument format_spec of the method display():

sage: b.display(format_spec=10)  # 10 bits of precision
b = 0.33 e^1/\e^2 + 2.5 e^1/\e^3 + 4.0 e^2/\e^3

Check that the bug reported in trac ticket #22520 is fixed:

sage: M = FiniteRankFreeModule(SR, 2, name='M')
sage: e = M.basis('e')
sage: a = M.alternating_form(2)
sage: a[0,1] = SR.var('t', domain='real')
sage: a.display()
t e^0/\e^1
display(basis=None, format_spec=None)

Display the alternating form self in terms of its expansion w.r.t. a given module basis.

The expansion is actually performed onto exterior products of elements of the cobasis (dual basis) associated with basis (see examples below). The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • basis – (default: None) basis of the free module with respect to which the alternating form is expanded; if none is provided, the module’s default basis is assumed
  • format_spec – (default: None) format specification passed to self._fmodule._output_formatter to format the output

EXAMPLES:

Display of an alternating form of degree 1 (linear form) on a rank-3 free module:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: e.dual_basis()
Dual basis (e^0,e^1,e^2) on the Rank-3 free module M over the Integer Ring
sage: a = M.linear_form('a', latex_name=r'\alpha')
sage: a[:] = [1,-3,4]
sage: a.display(e)
a = e^0 - 3 e^1 + 4 e^2
sage: a.display()  # a shortcut since e is M's default basis
a = e^0 - 3 e^1 + 4 e^2
sage: latex(a.display())  # display in the notebook
\alpha = e^0 -3 e^1 + 4 e^2

A shortcut is disp():

sage: a.disp()
a = e^0 - 3 e^1 + 4 e^2

Display of an alternating form of degree 2 on a rank-3 free module:

sage: b = M.alternating_form(2, 'b', latex_name=r'\beta')
sage: b[0,1], b[0,2], b[1,2] = 3, 2, -1
sage: b.display()
b = 3 e^0/\e^1 + 2 e^0/\e^2 - e^1/\e^2
sage: latex(b.display())  # display in the notebook
\beta = 3 e^0\wedge e^1 + 2 e^0\wedge e^2 -e^1\wedge e^2

Display of an alternating form of degree 3 on a rank-3 free module:

sage: c = M.alternating_form(3, 'c')
sage: c[0,1,2] = 4
sage: c.display()
c = 4 e^0/\e^1/\e^2
sage: latex(c.display())
c = 4 e^0\wedge e^1\wedge e^2

Display of a vanishing alternating form:

sage: c[0,1,2] = 0  # the only independent component set to zero
sage: c.is_zero()
True
sage: c.display()
c = 0
sage: latex(c.display())
c = 0
sage: c[0,1,2] = 4  # value restored for what follows

Display in a basis which is not the default one:

sage: aut = M.automorphism(matrix=[[0,1,0], [0,0,-1], [1,0,0]],
....:                      basis=e)
sage: f = e.new_basis(aut, 'f')
sage: a.display(f)
a = 4 f^0 + f^1 + 3 f^2
sage: a.disp(f)     # shortcut notation
a = 4 f^0 + f^1 + 3 f^2
sage: b.display(f)
b = -2 f^0/\f^1 - f^0/\f^2 - 3 f^1/\f^2
sage: c.display(f)
c = -4 f^0/\f^1/\f^2

The output format can be set via the argument output_formatter passed at the module construction:

sage: N = FiniteRankFreeModule(QQ, 3, name='N', start_index=1,
....:                   output_formatter=Rational.numerical_approx)
sage: e = N.basis('e')
sage: b = N.alternating_form(2, 'b')
sage: b[1,2], b[1,3], b[2,3] = 1/3, 5/2, 4
sage: b.display()  # default format (53 bits of precision)
b = 0.333333333333333 e^1/\e^2 + 2.50000000000000 e^1/\e^3
 + 4.00000000000000 e^2/\e^3

The output format is then controlled by the argument format_spec of the method display():

sage: b.display(format_spec=10)  # 10 bits of precision
b = 0.33 e^1/\e^2 + 2.5 e^1/\e^3 + 4.0 e^2/\e^3

Check that the bug reported in trac ticket #22520 is fixed:

sage: M = FiniteRankFreeModule(SR, 2, name='M')
sage: e = M.basis('e')
sage: a = M.alternating_form(2)
sage: a[0,1] = SR.var('t', domain='real')
sage: a.display()
t e^0/\e^1
wedge(other)

Exterior product of self with the alternating form other.

INPUT:

  • other – an alternating form

OUTPUT:

EXAMPLES:

Exterior product of two linear forms:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: a = M.linear_form('A')
sage: a[:] = [1,-3,4]
sage: b = M.linear_form('B')
sage: b[:] = [2,-1,2]
sage: c = a.wedge(b) ; c
Alternating form A/\B of degree 2 on the Rank-3 free module M
 over the Integer Ring
sage: c.display()
A/\B = 5 e^0/\e^1 - 6 e^0/\e^2 - 2 e^1/\e^2
sage: latex(c)
A\wedge B
sage: latex(c.display())
A\wedge B = 5 e^0\wedge e^1 -6 e^0\wedge e^2 -2 e^1\wedge e^2

Test of the computation:

sage: a.wedge(b) == a*b - b*a
True

Exterior product of a linear form and an alternating form of degree 2:

sage: d = M.linear_form('D')
sage: d[:] = [-1,2,4]
sage: s = d.wedge(c) ; s
Alternating form D/\A/\B of degree 3 on the Rank-3 free module M
 over the Integer Ring
sage: s.display()
D/\A/\B = 34 e^0/\e^1/\e^2

Test of the computation:

sage: s[0,1,2] == d[0]*c[1,2] + d[1]*c[2,0] + d[2]*c[0,1]
True

Let us check that the exterior product is associative:

sage: d.wedge(a.wedge(b)) == (d.wedge(a)).wedge(b)
True

and that it is graded anticommutative:

sage: a.wedge(b) == - b.wedge(a)
True
sage: d.wedge(c) == c.wedge(d)
True