On primitive ideals

By V. Ginzburg. 2002

88 citations recorded by the source corpus.

AbstractWe extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive ideals, to much wider classes of algebras. Our general version of the Irreducibility Theorem says that if A is a positively filtered associative algebra such that gr A is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of the Duflo theorem says that if A is an algebra with a “triangular structure”, see § 2, then any primitive ideal in A is the annihilator of a simple highest weight module. Applications to symplectic reflection algebras and Cherednik algebras are discussed.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c8e7a3c27927b1252210126963fac93e8fade438

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In Algebra and Number Theory

1. I. Losev
2. V. Ginzburg
3. I. Gordon
4. E. Gorsky
5. Pavel Etingof
6. A. Premet
7. R. Bezrukavnikov
8. J. T. Stafford

In Mathematical Physics

1. I. Losev
2. V. Ginzburg
3. I. Gordon
4. Pavel Etingof
5. E. Gorsky
6. J. T. Stafford
7. A. Premet
8. R. Bezrukavnikov

Citations in the corpus

• Enveloping algebras of Slodowy slices and the Joseph ideal (2005, unknown): “By the Irreducibility Theorem, the associated variety of the annihilator in U(g) of any simple g-module coincides with the closure of a nilpotent orbit in g; see [2, 18, 21, 35, 12].”
• Rational Cherednik algebras and Hilbert schemes (2004, background): “a primitive ideal, and hence, by [ Gi , Generalized Duflo Theorem], annihilate an object from category Od.”
• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, background): “Z,M) (an isomorphism of R-modules). The right hand side is definitely an H-module. Now we can finish the proof of the theorem. We may assume that Mis irreducible. In this case, it is shown by Ginzburg, [Gi1], that supp(M)/Wis an irreducible subvariety of h/W. Hence, supp(M) is an equidimensional varie”
• Commutative quotients of finite W-algebras (2008, methodology): “As in [34], we assume that e = ek for some k ≤ t and O(ei) ⊂ O(e) for i ≤ k.”
• Parabolic induction and restriction functors for rational Cherednik algebras (2008, background): “M). The functor J is called the Jacquet functor ([ Gi ]).”
• On category O for cyclotomic rational Cherednik algebras (2011, result): “The algebra Hp is known to be simple for p ∈ css 1 this follows, for instance, from results of [30].”
• Primitive ideals, non-restricted representations and finite W-algebras (2006, background): “By the Irreducibility Theorem, the associated variety VA(I) of any primitive ideal I of U(g) is the closure of a nilpotent orbit in g∗; see [12] and references therein.”
• Quantized symplectic actions and W -algebras (2007, result): “Set A∧ f := lim ← −k→∞ A/Af k . There is a natural topological algebra structure on A∧ f (where the kernels of the natural epimorphisms A∧ f →A /Af k form a fundamental set of neighborhoods of 0); compare with [ Gi1 ], Section 5. Clearly, A∧ f is complete w.r.t.”
• Cherednik algebras and differential operators on quasi-invariants (2001, background): “ebra D(hreg)#Wis known to be simple [Mo]. Hence, I| 6= 0 implies Hc| hreg= I| . Thus, (Hc/I)| = 0. Now, Corollary 3.6 yields: 0 = V = Hc/I, and Hc = I. From Proposition 3.5 one also obtains (see also [Gi]) the following important Corollary 3.7. If c∈C[R]W reg , then every Harish-Chandra module has fi”
• Lecture notes on Cherednik algebras (2010, background): “lent to the category of finite dimensional representations of G. We can now define the composition functor J: Oc(G,h) →Oc(G,h)0, by the formula J(M) = E(Mc). The functor Jis called the Jacquet functor ([Gi2]). 5.3. The duality functor. Recall that in Section 3.11, for any H1,c(G,h)-module M, the full ”
• Completions of symplectic reflection algebras (2010, methodology): “ve ideals in IdL(H1,c) and any fiber of the map is a single Ξ-orbit. Recall that a two-sided ideal Iin an associative unital algebra Ais called primitive if it is the annihilator of a simple module. In[G2]Ginzburgproved thatanyprimitive ideals ofH1,c iscontained insomeset IdL(H1,c). This is an analog”