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:
See also
-
Element
¶ alias of
LieAlgebraElement
-
bracket_on_basis
(i, j)¶ Return the bracket of basis elements indexed by
x
andy
wherex < 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:
-
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
andy
wherex < 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
-
Element
¶ alias of
LieAlgebraElement
-
bracket_on_basis
(i, j)¶ Return the bracket of basis elements indexed by
x
andy
wherex < 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]]
-