A Verma module is a fundamental object in the representation theory of semisimple Lie algebras. It is a specific type of highest weight module that serves as a universal construction for irreducible highest weight modules. Verma modules provide a systematic approach to classifying and understanding a broad class of representations.
Definition
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We choose a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a Borel subalgebra $\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}^+$, where $\mathfrak{n}^+$ is the nilpotent radical (the direct sum of positive root spaces).For any linear functional $\lambda \in \mathfrak{h}^*$ (called a weight), we can define a one-dimensional representation $\mathbb{C}\lambda$ of the Borel subalgebra $\mathfrak{b}$. In this representation, an element $h \in \mathfrak{h}$ acts on a vector $v \in \mathbb{C}\lambda$ by scalar multiplication $\lambda(h)v$, and any element $x \in \mathfrak{n}^+$ acts as zero (i.e., $x \cdot v = 0$). This makes $\mathbb{C}_\lambda$ a left $U(\mathfrak{b})$-module, where $U(\mathfrak{b})$ is the universal enveloping algebra of $\mathfrak{b}$.
The Verma module $M(\lambda)$ associated with the weight $\lambda$ is then defined as the induced module: $M(\lambda) = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$ where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.
Properties
- Highest Weight Module: By construction, $M(\lambda)$ is a highest weight module with highest weight $\lambda$. It is generated by a vector $v_\lambda = 1 \otimes 1$ (where $1 \in \mathbb{C}\lambda$) such that $\mathfrak{n}^+ \cdot v\lambda = 0$ and $h \cdot v_\lambda = \lambda(h)v_\lambda$ for all $h \in \mathfrak{h}$. This vector $v_\lambda$ is called a highest weight vector.
- Universal Property: The Verma module $M(\lambda)$ is the universal highest weight module of highest weight $\lambda$. This means that any $\mathfrak{g}$-module generated by a highest weight vector of weight $\lambda$ is a homomorphic image of $M(\lambda)$.
- Structure: Every Verma module $M(\lambda)$ possesses a unique maximal proper submodule, denoted $K(\lambda)$. This submodule is the sum of all proper submodules of $M(\lambda)$.
- Irreducible Quotient: The quotient module $L(\lambda) = M(\lambda) / K(\lambda)$ is the unique irreducible highest weight module with highest weight $\lambda$. The study of Verma modules is therefore central to classifying all irreducible highest weight modules.
- Dimension: Verma modules are generally infinite-dimensional. They are finite-dimensional only under very specific conditions, for instance, if the Lie algebra is not semisimple or if the definition is adapted for a finite-dimensional context (which is not the standard case for Verma modules). For complex semisimple Lie algebras, $M(\lambda)$ is infinite-dimensional unless $\mathfrak{g}$ is trivial.
- Poincaré-Birkhoff-Witt Basis: Using the Poincaré-Birkhoff-Witt theorem, $M(\lambda)$ can be understood as having a basis indexed by a basis of $U(\mathfrak{n}^-)$, where $\mathfrak{n}^-$ is the nilpotent radical corresponding to negative root spaces. Specifically, $M(\lambda) \cong U(\mathfrak{n}^-) \otimes \mathbb{C}_\lambda$ as a vector space, which makes it isomorphic to $U(\mathfrak{n}^-)$ as a vector space.