1,721,002 research outputs found
A new linear quotient of C 4 admitting a symplectic resolution
Full text linkC2 C2 ∼= C4.We show that the quotient C[superscript 4]/G admits a symplectic resolution for G = Q[subscript 8] x [subscript Z/2]D[subscript 8] < Sp[subscript 4](C). Here Q[subscript 8] is the quaternionic group of order eight and D[subscript 8] is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation C[superscript 2] ⊠ C[superscript 2] ≅ C[superscript 4]. This group is also naturally a subgroup of the wreath product group Q[superscript 8][subscript 2] ⋊ S[subscript 2] < Sp[subscript 4](C). We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C[superscript 4]/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions
Multiplicative preprojective algebras are 2-Calabi-Yau
Full text linkWe prove that multiplicative preprojective algebras, defined by
Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers
containing unoriented cycles. If the quiver is not itself a cycle, we show that
the center is trivial, and hence the Calabi-Yau structure is unique. If the
quiver is a cycle, we show that the algebra is a non-commutative crepant
resolution of its center, the ring of functions on the corresponding
multiplicative quiver variety with a type A surface singularity. We also prove
that the dg versions of these algebras (arising as certain Fukaya categories)
are formal. We conjecture that the same properties hold for all non-Dynkin
quivers, with respect to any extended Dynkin subquiver (note that the cycle is
the type A case). Finally, we prove that multiplicative quiver varieties-for
all quivers-are formally locally isomorphic to ordinary quiver varieties. In
particular, they are all symplectic singularities (which implies they are
normal and have rational Gorenstein singularities). This includes character
varieties of Riemann surfaces with punctures and monodromy conditions. We
deduce this from a more general statement about 2-Calabi--Yau algebras
(following Bocklandt, Galluzzi, and Vaccarino).Comment: 48 pages, Remarks 1.6 and 4.7 adde
Co-invariants of Lie algebras of vector fields on algebraic varieties
Full text linkWe prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us. Keywords: Lie algebras; D-modules; Poisson homology; Poisson varieties; Calabi–Yau varieties; Jacobi varietie
Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities
Full text linkOriginal manuscript July 10 2009Let X ⊂ ℂ[superscript 3] be a surface with an isolated singularity at the origin, given by the equation Q(x, y, z) = 0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = ℂ[superscipt 2]/G for G < SL[subscript 2](ℂ) finite. Let Y ≔ S[superscript n]X be the n-th symmetric power of X. We compute the zeroth Poisson homology HP[subscript 0]([subscript Y]), as a graded vector space with respect to the weight grading, where [subscript Y] is the ring of polynomial functions on Y. In the Kleinian case, this confirms a conjecture of Alev, that HP[subscript 0] ( [G [superscipt n]⋊ S[subscript n]over ℂ[2n]) ≃ HH [subscript 0] (Weyl ( [G [superscipt n]⋊ S[subscript n]over ℂ[2n]), where Weyl[subscript 2n] is the Weyl algebra on 2n generators. That is, the Brylinski spectral sequence degenerates in degree zero in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, A[subscript γ], for all but countably many parameters γ in the elliptic curve. As a consequence, we deduce a bound on the number of irreducible finite-dimensional representations of all quantizations of Y. This includes the noncommutative spherical symplectic reflection algebras associated to G[superscript n] ⋊ S[subscript n].National Science Foundation (U.S.) (Grant DMS-0504847)American Institute of Mathematics (Fellowship)Massachusetts Institute of Technology. Undergraduate Research Opportunities Progra
POISSON TRACES AND D-MODULES ON POISSON VARIETIES
No full textTo every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism [phi]: X → Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M [subscript phi] (X,N)M(XN) on X, and prove that it is holonomic if X has finitely many symplectic leaves, [phi] is finite, and N is coherent.
As an application, we deduce that noncommutative filtered algebras, for which the associated graded algebra is finite over its center whose spectrum has finitely many symplectic leaves, have finitely many irreducible finite-dimensional representations. The appendix, by Ivan Losev, strengthens this to show that, in such algebras, there are finitely many prime ideals, and they are all primitive. This includes symplectic reflection algebras.
Furthermore, we describe explicitly (in the settings of affine varieties and compact C ∞-manifolds [C superscript infinity symbol -manifolds]) the finite-dimensional space of Poisson traces on X when X = V/G, where V is symplectic and G is a finite group acting faithfully on V
Traces on finite W-algebras
No full textWe compute the space of Poisson traces on a classical W-algebra, i.e., linear functionals invariant under Hamiltonian derivations. Modulo any central character, this space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology of the corresponding quantum W-algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology, which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the centrally reduced classical W-algebra.National Science Foundation (U.S.) (grant DMS-0504847)National Science Foundation (U.S.) (grant DMS-0900233
Computational Approaches to Poisson Traces Associated to Finite Subgroups of Sp[subscript 2n](C)
No full textOriginal manuscript January 26, 2011We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one by proving several results that bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions that are invariant under invariant Hamiltonian flow. We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of GL[subscript 2](C) < Sp[subscript 4](C), Coxeter groups of rank <3 and types A 4, B 4=C 4, and D 4, and subgroups of SL[subscript 2](C).National Science Foundation (U.S.) (Grant DMS-1000113)American Institute of Mathematics (Fellowship)National Science Foundation (U.S.) (Grant DMS-0900233
Combinatorial aspects of bow varieties
Full text linkFirst introduced by Cherkis in theoretical physics, bow varieties form a rich family of symplectic varieties generalizing Nakajima quiver varieties. An algebro-geometric definition was later given by Nakajima and Takayama via moduli spaces of quiver representations. The main goal of this thesis is to study the torus equivariant cohomology of bow varieties. Our study is motivated by classical Schubert calculus and lays the foundation for a Schubert calculus for bow varieties where the underlying quiver is of finite type A. The crucial main mathematical tool we use is the theory of stable envelopes of Maulik and Okounkov. We show that this theory applies to bow varieties and study it with the main focus on explicit calculations. As a main result of this thesis we generalize a fundamental ingredient of classical Schubert calculus to the world of bow varieties: The Chevalley--Monk formula. Our generalization of this formula characterizes the multiplication of tautological divisors with respect to the stable envelope basis
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