arXiv subjects
Publications and source records attributed to Wee Liang Gan.
We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.
We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo-Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FI-module which is generated and related in finite degree, if it has a nonzero higher homology, then its homological degrees are strictly increasing (starting from the first homological degree).
It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.
We determine explicitly the center of the twisted graded Hecke algebras associated to homocyclic groups. Our results are a generalization of formulas by M. Douglas and B. Fiol in [J. High Energy Phys. 2005 (2005), no. 9, 053, 22 pages, hep-th/9903031].
We give a direct proof of (a slight generalization of) the recent result of A. Premet related to generalized Gelfand-Graev representations and of an equivalence due to Skryabin.
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology.
Let M be a smooth, simply-connected, closed oriented manifold, and LM the free loop space of M. Using a Poincare duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's renormalization Hopf algebra, as well as to pre-Lie versions. These results are direct analogues of Turaev's results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then explain how all of the Hopf algebras are uniquely determined by the pre-Lie structures, and discuss noncommutative versions of the Hopf algebras.
A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with U_q(sl_2). We study its representation theory, and in particular, its category O.
We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA is the algebra defined by Sahi in q-alg/9710032, which is a generalization of the Cherednik algebra of type BCn. We prove the formal PBW theorem for GDAHA, and parametrize its irreducible representations in the case when D is affine (i.e. extended D4, E6, E7, E8) and q=1. We formulate a series of conjectures regarding algebraic properties of GDAHA. We expect that, similarly to how GDAHA of rank 1 provide quantizations of del Pezzo surfaces (as shown in math.QA/0406480), GDAHA of higher rank provide quantizations of deformations of Hilbert schemes of these surfaces. The proofs are based on the study of the rational version of GDAHA (which is closely related to the algebras studied in math.QA/0401038), and differential equations of Knizhnik-Zamolodchikov type.
We study a scheme M closely related to the set of pairs of n by n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n+1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. Simple objects of that category are analogous to Lusztig's character sheaves. We construct a functor of Quantum Hamiltonian reduction from category C to the category O for type A rational Cherednik algebra.
The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.
We study a generalization of graded Hecke algebras introduced by Drinfeld in 1986, in which the role of a finite group G is played by a reductive algebraic group. This includes continuous generalizations of symplectic reflection algebras and of rational Cherednik algebras. We prove direct and converse PBW theorems, and also study versions of our algebras in which the group G is replaced with its Lie algebra. We also begin to develop the representation theory of continuous Hecke algebras, which unifies representation theories of real reductive groups, Drinfeld-Lusztig degenerate affine Hecke algebras, and symplectic reflection (in particular, Cherednik) algebras. The paper is dedicated to V. Drinfeld's 50-th birthday.
We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.
We prove that the graph complex is a strong homotopy Lie (super) bialgebra.
We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.
We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver.