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Twisted generalized Weyl algebras and pr...
Hartwig, Jonas T....

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Twisted generalized Weyl algebras and primitive quotients of enveloping algebras by Hartwig, Jonas T. ( Author )
Lemonjar Open Access
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21-04-2015
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 A(Γ) carry a canonical representation by differential operators and that A(Γ) 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 A(Γ) coincides with the one corresponding to D(Γ) and that A(Γ) 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 A(Γ) for specific Γ. We also prove one result in the affine case.
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MYR 0.00
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https://arxiv.org/abs/1504.05361
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