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Virasoro Algebra and Related Lie Algebras¶

AUTHORS:

  • Travis Scrimshaw (2013-05-03): Initial version
class sage.algebras.lie_algebras.virasoro.LieAlgebraRegularVectorFields(R)¶

Bases: sage.algebras.lie_algebras.lie_algebra.InfinitelyGeneratedLieAlgebra, sage.structure.indexed_generators.IndexedGenerators

The Lie algebra of regular vector fields on \(\CC^{\times}\).

This is the Lie algebra with basis \(\{d_i\}_{i \in \ZZ}\) and subject to the relations

\[[d_i, d_j] = (j - i) d_{i+j}.\]

This is also known as the Witt (Lie) algebra.

REFERENCES:

  • Wikipedia article Witt_algebra

See also

WittLieAlgebra_charp

Element¶

alias of LieAlgebraElement

bracket_on_basis(i, j)¶

Return the bracket of basis elements indexed by x and y where x < y.

(This particular implementation actually does not require x < y.)

EXAMPLES:

sage: L = lie_algebras.regular_vector_fields(QQ)
sage: L.bracket_on_basis(2, -2)
-4*d[0]
sage: L.bracket_on_basis(2, 4)
2*d[6]
sage: L.bracket_on_basis(4, 4)
0
lie_algebra_generators()¶

Return the generators of self as a Lie algebra.

EXAMPLES:

sage: L = lie_algebras.regular_vector_fields(QQ)
sage: L.lie_algebra_generators()
Lazy family (generator map(i))_{i in Integer Ring}
some_elements()¶

Return some elements of self.

EXAMPLES:

sage: L = lie_algebras.regular_vector_fields(QQ)
sage: L.some_elements()
[d[0], d[2], d[-2], d[-1] + d[0] - 3*d[1]]
class sage.algebras.lie_algebras.virasoro.VirasoroAlgebra(R)¶

Bases: sage.algebras.lie_algebras.lie_algebra.InfinitelyGeneratedLieAlgebra, sage.structure.indexed_generators.IndexedGenerators

The Virasoro algebra.

This is the Lie algebra with basis \(\{d_i\}_{i \in \ZZ} \cup \{c\}\) and subject to the relations

\[[d_i, d_j] = (j - i) d_{i+j} + \frac{1}{12}(j^3 - j) \delta_{i,-j} c\]

and

\[[d_i, c] = 0.\]

(Here, it is assumed that the base ring \(R\) has \(2\) invertible.)

This is the universal central extension \(\widetilde{\mathfrak{d}}\) of the Lie algebra \(\mathfrak{d}\) of regular vector fields on \(\CC^{\times}\).

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)

REFERENCES:

  • Wikipedia article Virasoro_algebra
Element¶

alias of LieAlgebraElement

basis()¶

Return a basis of self.

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)
sage: B = d.basis(); B
Lazy family (basis map(i))_{i in Disjoint union of
                            Family ({'c'}, Integer Ring)}
sage: B['c']
c
sage: B[3]
d[3]
sage: B[-15]
d[-15]
bracket_on_basis(i, j)¶

Return the bracket of basis elements indexed by x and y where x < y.

(This particular implementation actually does not require x < y.)

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)
sage: d.bracket_on_basis('c', 2)
0
sage: d.bracket_on_basis(2, -2)
-4*d[0] - 1/2*c
c()¶

The central element \(c\) in self.

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)
sage: d.c()
c
d(i)¶

Return the element \(d_i\) in self.

EXAMPLES:

sage: L = lie_algebras.VirasoroAlgebra(QQ)
sage: L.d(2)
d[2]
lie_algebra_generators()¶

Return the generators of self as a Lie algebra.

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)
sage: d.lie_algebra_generators()
Lazy family (generator map(i))_{i in Integer Ring}
some_elements()¶

Return some elements of self.

EXAMPLES:

sage: d = lie_algebras.VirasoroAlgebra(QQ)
sage: d.some_elements()
[d[0], d[2], d[-2], c, d[-1] + d[0] - 1/2*d[1] + c]
class sage.algebras.lie_algebras.virasoro.WittLieAlgebra_charp(R, p)¶

Bases: sage.algebras.lie_algebras.lie_algebra.FinitelyGeneratedLieAlgebra, sage.structure.indexed_generators.IndexedGenerators

The \(p\)-Witt Lie algebra over a ring \(R\) in which \(p \cdot 1_R = 0\).

Let \(R\) be a ring and \(p\) be a positive integer such that \(p \cdot 1_R = 0\). The \(p\)-Witt Lie algebra over \(R\) is the Lie algebra with basis \(\{d_0, d_1, \ldots, d_{p-1}\}\) and subject to the relations

\[[d_i, d_j] = (j - i) d_{i+j},\]

where the \(i+j\) on the right hand side is identified with its remainder modulo \(p\).

See also

LieAlgebraRegularVectorFields

Element¶

alias of LieAlgebraElement

bracket_on_basis(i, j)¶

Return the bracket of basis elements indexed by x and y where x < y.

(This particular implementation actually does not require x < y.)

EXAMPLES:

sage: L = lie_algebras.pwitt(Zmod(5), 5)
sage: L.bracket_on_basis(2, 3)
d[0]
sage: L.bracket_on_basis(3, 2)
4*d[0]
sage: L.bracket_on_basis(2, 2)
0
sage: L.bracket_on_basis(1, 3)
2*d[4]
lie_algebra_generators()¶

Return the generators of self as a Lie algebra.

EXAMPLES:

sage: L = lie_algebras.pwitt(Zmod(5), 5)
sage: L.lie_algebra_generators()
Finite family {0: d[0], 1: d[1], 2: d[2], 3: d[3], 4: d[4]}
some_elements()¶

Return some elements of self.

EXAMPLES:

sage: L = lie_algebras.pwitt(Zmod(5), 5)
sage: L.some_elements()
[d[0], d[2], d[3], d[0] + 2*d[1] + d[4]]

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