## Category O over a deformation of the symplectic oscillator algebra

Venue: | Journal of Pure and Applied Algebra |

Citations: | 25 - 6 self |

### BibTeX

@ARTICLE{Khare_categoryo,

author = {Apoorva Khare},

title = {Category O over a deformation of the symplectic oscillator algebra},

journal = {Journal of Pure and Applied Algebra},

year = {},

pages = {131--166}

}

### OpenURL

### Abstract

Abstract. We discuss the representation theory of Hf, which is a deformation of the symplectic oscillator algebra sp(2n)⋉hn, where hn is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general setup, involving an algebra with a triangular decomposition. Many of the constructions in the classical BGG case go through here as well; in particular, assuming the PBW theorem and finite length of all Vermas, the category O is abelian, finite length, and self-dual. We decompose O as a direct sum of blocks O(λ), and show that enough projectives exist in each block, so that each O(λ) is equivalent to a finite-dimensional algebra. Each block is also a highest weight category, so that we have BGG reciprocity. We focus on the case Hf for n = 1, where we show all these assumptions to hold,

### Citations

222 | Koszul duality patterns in representation theory
- Beilinson, Ginzburg, et al.
- 1996
(Show Context)
Citation Context ...(λ), then P(λ0) = Z(λ0) is the projective cover of V (λ0), and I(λ0) = A(λ0) is the injective hull. Proof. We only have to show that enough projectives exist in our abelian category O(λ). We refer to =-=[BGS]-=-, §3.2§. Following Remark (3) there, we only need to verify five things (here) about O(λ), to conclude that enough projectives exist. We do so now. (1) A = O(λ) is a finite length abelian k-category. ... |

113 |
Finite dimensional algebras and highest weight categories
- Cline, Parshall, et al.
- 1988
(Show Context)
Citation Context ...bjects (for M ∈ O), and induced homomorphisms for its morphisms. More generally, we can define F : H → H in the same way. Our analysis in the next few sections is in the spirit of [BGG1], [GGOR], and =-=[CPS1]-=-.6 APOORVA KHARE Theorem 2. Let M ∈ H. (1) dim(F(M))µ = dim(Mµ) for each weight µ. (2) F(F(M)) is canonically isomorphic to M. (3) HomA(M, N) = HomA(F(N), F(M)) if M, N ∈ H. The proof is standard, gi... |

67 | On the category O for rational Cherednik algebras
- Ginzburg, Guay, et al.
(Show Context)
Citation Context ...M) for its objects (for M ∈ O), and induced homomorphisms for its morphisms. More generally, we can define F : H → H in the same way. Our analysis in the next few sections is in the spirit of [BGG1], =-=[GGOR]-=-, and [CPS1].6 APOORVA KHARE Theorem 2. Let M ∈ H. (1) dim(F(M))µ = dim(Mµ) for each weight µ. (2) F(F(M)) is canonically isomorphic to M. (3) HomA(M, N) = HomA(F(N), F(M)) if M, N ∈ H. The proof is ... |

62 | Abelian Categories, an Introduction to the Theory of Functors - Freyd - 1964 |

49 |
Enveloping algebras, Graduate Studies
- Dixmier
- 1996
(Show Context)
Citation Context ...on of blocks O(χ), where χ ∈ HomC−alg(Z(U(g)), C). Thus, a g-module V is in O(χ) iff for each z in the center Z one can find an n so that (z − χ(z)) n kills V . Furthermore, (as in [H], Ex.(23.9), or =-=[Dix]-=-(7.4.8),) every algebra map from the center to C is of the form χµ for some µ ∈ h ∗ . Thus, the irreducible module V = V (λ) is in O(χ) iff χλ = χ = χµ, iff λ + δ and µ + δ are W-conjugate (by Harish-... |

49 |
The q-Schur algebra
- Donkin
(Show Context)
Citation Context ...If λ is minimal in S, then ∃M ′ ∈ F(∆) so that 0 → M ′ → M → Z(λ) → 0 is exact. (3) Suppose M1, M2 ∈ ON. Then M1 ⊕ M2 ∈ F(∆) iff M1, M2 ∈ F(∆). □ Proof. (1) and (3) follow from [BGG1], and (2) is cf. =-=[Don]-=-(A3.1)(i). □ The next result comes from [GGOR], and involves h-diagonalizable modules M. Proposition 6. Suppose M is h-diagonalizable. Then the following are equivalent : (1) M ∈ O. (2) M is a quotien... |

16 | Projective modules in the category O for the Cherednik algebra
- Guay
(Show Context)
Citation Context ... − t − 1, E). (2) The above theorem holds for any Z(r) → V → 0. In any such V , any maximal vector of a given weight r ′ (if it exists)is unique upto scalar. (3) Further, we have Verma’s Theorem (cf. =-=[Ver]-=-, [Dix](7.6.6)) : HomHf (Z(r′ ), Z(r)) = 0 or k for general r, r ′ ∈ k. All nonzero homomorphisms are injective. Theorem 13. We work again in the Verma module Z(r) for any r ∈ k. (1) ∆0 acts on F m Y ... |

8 |
A category of -modules, Funct
- Bernstein, Gelfand, et al.
- 1976
(Show Context)
Citation Context ...genvector for B+. A standard cyclic module is an A-module generated by exactly one maximal vector. Certain universal standard cyclic modules are called Verma modules, just as in the classical case of =-=[BGG1]-=- or [H]. Then there exist maximal vectors (i.e. eigenvectors for B+) in any object of O, and all finite dimensional modules are in O. We now look at standard cyclic modules, namely V = A ·vλ, where vλ... |