If a -module contains a vector of weight annihilated by , there is a unique homomorphism sending the canonical highest-weight vector to . Every highest-weight module of weight is therefore a quotient of .
The sum of all proper submodules of is its unique maximal proper submodule. The quotient is the unique irreducible highest-weight module .
If is a dominant integral weight, the maximal proper submodule of is generated by the simple-root singular vectorsThe generator associated with has weight .
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A Verma module is a type of representation of a highest weight module in the context of the representation theory of Lie algebras, particularly those that are semisimple. Verma modules play an important role in the study of the structure and representation theory of Lie algebras and quantum groups.