Algebras and Representation Theory

, Volume 19, Issue 2, pp 277–313 | Cite as

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras



To each multiquiver Γ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras \(\mathcal {A}({\Gamma })\) carry a canonical representation by differential operators and that \(\mathcal {A}({\Gamma })\) is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of Γ for when this representation is faithful or locally surjective. By forgetting some of the structure of Γ one obtains a Dynkin diagram, D(Γ). We show that the generalized Cartan matrix of \(\mathcal {A}({\Gamma })\) coincides with the one corresponding to D(Γ) and that \(\mathcal {A}({\Gamma })\) contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient U/J of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if J is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types A and C are closely related to \(\mathcal {A}({\Gamma })\) for specific Γ. We also prove one result in the affine case.


Generalized Weyl algebra Enveloping algebra Primitive ring 

Mathematics Subject Classification (2010)

Primary 16S99 Secondary 16W50 16D60 17B35 


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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