7 Verma modules
7.1 A generalized notion of Verma modules
Let \(A\) be an algebra over a commutative ring \(\mathbb {K}\). Let \(\eta = (a_i, r_i)_{i \in I} \in (A \times \mathbb {K})^I\) be a family of pairs of algebra elements \(a_i \in A\) and scalars \(r_i \in \mathbb {K}\). The generalized Verma module associated to the data \(\eta \) is the quotient
where \(J^{\eta } \subset A\) is the left ideal (an \(A\)-submodule) of \(A\) generated by the elements \(a_i - r_i \, 1 \in A\). Note that \(\mathscr {V}^{\eta }\) is a (left) \(A\)-module.
The vector \(\mathbb {v}^{\eta } := 1 + J^{\eta } \in \mathscr {V}^{\eta }\) (the equivalence class of the unit element of \(A\)) is called the highest weight vector of \(\mathscr {V}^{\eta }\). It is cyclic, i.e., it generates the whole Verma module \(\mathbb {v}^{\eta }\) as an \(A\)-module, and it satisfies
for all \(i \in I\), where \(a_i \in A\) and \(r_i \in \mathbb {K}\) are the algebra elements and scalars in the collection \(\eta = (a_i, r_i)_{i \in I}\).
Clear from the construction.
Suppose that \(M\) is an \(A\)-module and \(v \in M\) is a vector such that
for all \(i \in I\), where \(a_i \in A\) and \(r_i \in \mathbb {K}\) are the algebra elements and scalars in the collection \(\eta = (a_i, r_i)_{i \in I}\). Then there exists a (unique) \(A\)-module homomorphism \(\phi \colon \mathscr {V}^{\eta } \to M\) such that \(\phi (\mathbb {v}^{\eta }) = v\). The range of the map \(\phi \) is the submodule generated by \(v\) in \(M\).
By cyclicity of the highest weight vector \(\mathbb {v}^{\eta }\), any element of \(\mathscr {V}^{\eta }\) can be written as \(a \cdot \mathbb {v}^{\eta }\) for some \(a \in A\). The map \(A \to M\) given by \(a \mapsto a \cdot v\) factors through the quotient by the ideal \(J^{\eta }\) by our assumption on \(v\), and since \(\mathscr {V}^{\eta } = A / J^{\eta }\), we thus get a well-defined map \(\phi \colon \mathscr {V}^{\eta } \to M\) with the asserted properties.
7.2 Verma modules for Lie algebras with a triangular decomposition
Let \(\mathfrak {g}\) be a Lie algebra over \(\mathbb {K}\). A triangular decomposition of \(\mathfrak {g}\) is a decomposition
of \(\mathfrak {g}\) into a vector space direct sum of three vector subspaces: Cartan part (or Cartan subalgebra) \(\mathfrak {g}^0\subseteq \mathfrak {g}\), the upper part \(\mathfrak {g}^+ \subseteq \mathfrak {g}\), and the lower part \(\mathfrak {g}^- \subseteq \mathfrak {g}\).
(Note that in this definition we do not yet require \(\mathfrak {g}^0, \mathfrak {g}^+, \mathfrak {g}^- \subseteq \mathfrak {g}\) to be Lie subalgebras, with the Cartan subalgebra being abelian and the upper and lower parts being nilpotent. In intended use cases, we typically have these properties, however.)
Let \(\mathfrak {g}\) be a Lie algebra over \(\mathbb {K}\), with a triangular decomposition \(\mathfrak {g}= \mathfrak {g}^0 \oplus \mathfrak {g}^+ \oplus \mathfrak {g}^-\). Let \(\eta \colon \mathfrak {g}^0 \to \mathbb {K}\) be a linear functional on the Cartan part. Then the Verma module associated to the "highest weight" \(\eta \) is the generalized Verma module for the universal enveloping algebra \(\mathscr {U}(\mathfrak {g})\), associated with the data consisting of the pairs \((H, \eta (H)) \in \mathscr {U}(\mathfrak {g}) \times \mathbb {K}\) for \(H \in \mathfrak {g}^0\) and pairs \((E,0) \in \mathscr {U}(\mathfrak {g}) \times \mathbb {K}\) for \(E \in \mathfrak {g}^+\). The Verma module is (by a mild abuse of notation) still denoted by \(\mathscr {V}^{\eta }\).
7.3 Virasoro Verma modules
A triangular decomposition
of \(\mathfrak {vir}\) is defined so that \(\mathfrak {vir}^0\) is spanned by \(L_0, C \in \mathfrak {vir}\), \(\mathfrak {vir}^+\) is spanned by \(L_n\) for \(n {\gt} 0\), and \(\mathfrak {vir}^-\) is spanned by \(L_n\) for \(n {\lt} 0\).
(Without further comment, for the Virasoro algebra we always use this triangular decomposition.)
Let \(c, h \in \mathbb {K}\). The Virasoro Verma module with central charge \(c\) and conformal weight \(h\) is the Verma module \(\mathscr {V}^{\eta }\) associated to the linear functional \(\eta \colon \mathfrak {vir}^0 \to \mathbb {K}\) with
We denote the Virasoro Verma module by \(\mathscr {V}^{c,h}\).