Abstract
Let 
be a toroidal Lie algebra corresponding to a semisimple Lie algebra 
We describe all Borel subalgebras of 
which contain the Cartan subalgebra 
where 
is a fixed Cartan subalgebra of We show that each such Borel subalgebra 
determines a parabolic decomposition 
where 
is a proper toroidal subalgebra of and 
Our first main result is that, for any weight λ which does not vanish on 
, an arbitrary subquotient of the Verma module 
is induced from its 
submodule of 
invariant vectors. This reduces the study of subquotients of to the study of subquotients of Verma modules over . We then introduce categories 
and 
and their respective blocks 
and 
corresponding to a central charge ν which is nonzero on . Our second main result is that the functors of induction and invariants are mutually inverse equivalences of the category and the full subcategory of whose objects are generated by their invariants.
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Communicated by L. Takhtajan
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Dimitrov, I., Futorny, V. & Penkov, I. A Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras. Commun. Math. Phys. 250, 47–63 (2004). https://doi.org/10.1007/s00220-004-1142-3
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DOI: https://doi.org/10.1007/s00220-004-1142-3