Virasoro Algebra and Related Lie Algebras¶
AUTHORS:
- Travis Scrimshaw (2013-05-03): Initial version
-
sage.algebras.lie_algebras.virasoro.
ChargelessRepresentation
¶ A chargeless representation of the Virasoro algebra.
Let \(L\) be the Virasoro algebra over the field \(F\) of characteristic \(0\). For \(\alpha, \beta \in R\), we denote \(V_{a,b}\) as the \((a, b)\)-chargeless representation of \(L\), which is the \(F\)-span of \(\{v_k \mid k \in \ZZ\}\) with \(L\) action
\[\begin{split}\begin{aligned} d_n \cdot v_k & = (a n + b - k) v_{n+k}, \\ c \cdot v_k & = 0, \end{aligned}\end{split}\]This comes from the action of \(d_n = -t^{n+1} \frac{d}{dt}\) on \(F[t, t^{-1}]\) (recall that \(L\) is the central extension of the
algebra of derivations
of \(F[t, t^{-1}]\)), where\[V_{a,b} = F[t, t^{-1}] t^{a-b} (dt)^{-a}\]and \(v_k = t^{a-b+k} (dz)^{-a}\).
The chargeless representations are either irreducible or contains exactly two simple subquotients, one of which is the trivial representation and the other is \(F[t, t^{-1}] / F\). The non-trivial simple subquotients are called the intermediate series.
The module \(V_{a,b}\) is irreducible if and only if \(a \neq 0, -1\) or \(b \notin \ZZ\). When \(a = 0\) and \(b \in \ZZ\), then there exists a subrepresentation isomorphic to the trivial representation. If \(a = -1\) and \(b \in \ZZ\), then there exists a subrepresentation \(V\) such that \(V_{a,b} / V\) is isomorphic to \(K \frac{dt}{t}\) and \(V\) is irreducible.
In characteristic \(p\), the non-trivial simple subquotient is isomorphic to \(F[t, t^{-1}] / F[t^p, t^{-p}]\). For \(p \neq 2,3\), then the action is given as above.
EXAMPLES:
We first construct the irreducible \(V_{1/2, 3/4}\) and do some basic computations:
sage: L = lie_algebras.VirasoroAlgebra(QQ) sage: M = L.chargeless_representation(1/2, 3/4) sage: d = L.basis() sage: v = M.basis() sage: d[3] * v[2] 1/4*v[5] sage: d[3] * v[-1] 13/4*v[2] sage: (d[3] - d[-2]) * (v[-1] + 1/2*v[0] - v[4]) -3/4*v[-3] + 1/8*v[-2] - v[2] + 9/8*v[3] + 7/4*v[7]
We construct the reducible \(V_{0,2}\) and the trivial subrepresentation given by the span of \(v_2\). We verify this for \(\{d_i \mid -10 \leq i < 10\}\):
sage: M = L.chargeless_representation(0, 2) sage: v = M.basis() sage: all(d[i] * v[2] == M.zero() for i in range(-10, 10)) True
REFERENCES:
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sage.algebras.lie_algebras.virasoro.
LieAlgebraRegularVectorFields
¶ 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
\[[d_i, d_j] = (i - j) d_{i+j}.\]This is also known as the Witt (Lie) algebra.
Note
This differs from some conventions (e.g., [Ka1990]), where we have \(d'_i \mapsto -d_i\).
REFERENCES:
See also
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sage.algebras.lie_algebras.virasoro.
VermaModule
¶ A Verma module of the Virasoro algebra.
The Virasoro algebra admits a triangular decomposition
\[V_- \oplus R d_0 \oplus R \hat{c} \oplus V_+,\]where \(V_-\) (resp. \(V_+\)) is the span of \(\{d_i \mid i < 0\}\) (resp. \(\{d_i \mid i > 0\}\)). We can construct the Verma module \(M_{c,h}\) as the induced representation of the \(R d_0 \oplus R \hat{c} \oplus V_+\) representation \(R_{c,H} = Rv\), where
\[V_+ v = 0, \qquad \hat{c} v = c v, \qquad d_0 v = h v.\]Therefore, we have a basis of \(M_{c,h}\)
\[\{ L_{i_1} \cdots L_{i_k} v \mid i_1 \leq \cdots \leq i_k < 0 \}.\]Moreover, the Verma modules are the free objects in the category of highest weight representations of \(V\) and are indecomposable. The Verma module \(M_{c,h}\) is irreducible for generic values of \(c\) and \(h\) and when it is reducible, the quotient by the maximal submodule is the unique irreducible highest weight representation \(V_{c,h}\).
EXAMPLES:
We construct a Verma module and do some basic computations:
sage: L = lie_algebras.VirasoroAlgebra(QQ) sage: M = L.verma_module(3, 0) sage: d = L.basis() sage: v = M.highest_weight_vector() sage: d[3] * v 0 sage: d[-3] * v d[-3]*v sage: d[-1] * (d[-3] * v) 2*d[-4]*v + d[-3]*d[-1]*v sage: d[2] * (d[-1] * (d[-3] * v)) 12*d[-2]*v + 5*d[-1]*d[-1]*v
We verify that \(d_{-1} v\) is a singular vector for \(\{d_i \mid 1 \leq i < 20\}\):
sage: w = M.basis()[-1]; w d[-1]*v sage: all(d[i] * w == M.zero() for i in range(1,20)) True
We also verify a singular vector for \(V_{-2,1}\):
sage: M = L.verma_module(-2, 1) sage: B = M.basis() sage: w = B[-1,-1] - 2 * B[-2] sage: d = L.basis() sage: all(d[i] * w == M.zero() for i in range(1,20)) True
REFERENCES:
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sage.algebras.lie_algebras.virasoro.
VirasoroAlgebra
¶ The Virasoro algebra.
This is the Lie algebra with basis \(\{d_i\}_{i \in \ZZ} \cup \{c\}\) and subject to the relations
\[[d_i, d_j] = (i - j) d_{i+j} + \frac{1}{12}(i^3 - i) \delta_{i,-j} c\]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:
-
sage.algebras.lie_algebras.virasoro.
WittLieAlgebra_charp
¶ 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
\[[d_i, d_j] = (i - j) d_{i+j},\]where the \(i+j\) on the right hand side is identified with its remainder modulo \(p\).
See also