Travis Schedler

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

1.1. Classical r-matrices. In the early eighties, Belavin and Drinfeld [BD] classified nonskewsymmetric classical r-matrices for simple Lie algebras. It turned out that such r-matrices, up to isomorphism and twisting by elements from the exterior square of the Cartan subalgebra, are classified by combinatorial objects which are now called Belavin-Drinfeld triples. By definition, a Belavin-Drinfeld triple for a simple Lie algebra g is a triple (Γ1,Γ2, T ), where Γ1,Γ2 are subsets of the Dynkin diagram Γ of g, and T : Γ1 → Γ2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition: if α ∈ Γ1, then there exists k such that T k−1(α) ∈ Γ1 but T (α) / ∈ Γ1. The r-matrix corresponding to such a triple is given by a certain explicit formula. These results generalize in a straightforward way to semisimple Lie algebras. In [S], the third author generalized the work of Belavin and Drinfeld and classified classical nonskewsymmetric dynamical r-matrices for simple Lie algebras. It turns out that they have an even simpler classification: up to gauge transformations, they are classified by generalized Belavin-Drinfeld triples, which are defined as the usual Belavin-Drinfeld triples but without any nilpotency condition. The dynamical rmatrix corresponding to such a triple is given by a certain explicit formula. As before, these results can be generalized to semisimple Lie algebras.

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Abstract Let X ⊂ ℂ3 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 = ℂ2/G for G < SL2(ℂ) finite. Let Y ≔ SnX be the n-th symmetric power of X. We compute the zeroth Poisson homology HP0(𝒪Y), as a graded vector space with respect to the weight grading, where 𝒪Y is the ring of polynomial functions on Y. In the Kleinian case, this confirms a conjecture of Alev, that , where Weyl2n 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, Aγ, 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 Gn ⋊ Sn.

Computational Approaches to Poisson Traces Associated to Finite Subgroups of

We 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 , Coxeter groups of rank ⩽3 and types A 4, B 4=C 4, and D 4, and subgroups of .

The necklace Lie coalgebra and renormalization algebras

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 quivertheoretic 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.

Invariants of Hamiltonian flow on locally complete intersections

We consider the Hamiltonian flow on complex complete intersection surfaces with isolated singularities, equipped with the Jacobian Poisson structure. More generally we consider complete intersections of arbitrary dimension equipped with Hamiltonian flow with respect to the natural top polyvector field, which one should view as a degenerate Calabi–Yau structure. Our main result computes the coinvariants of functions under the Hamiltonian flow. In the surface case this is the zeroth Poisson homology, and our result generalizes those of Greuel, Alev and Lambre, and the authors in the quasihomogeneous and formal cases. Its dimension is the sum of the dimension of the top cohomology and the sum of the Milnor numbers of the singularities. In other words, this equals the dimension of the top cohomology of a smoothing of the variety. More generally, we compute the derived coinvariants, which replaces the top cohomology by all of the cohomology. Still more generally we compute the

NORMALITY AND QUADRATICITY FOR SPECIAL AMPLE LINE BUNDLES ON TORIC VARIETIES ARISING FROM ROOT SYSTEMS

{\mathcal{D}}

Superpotentials and higher order derivations

D-module which represents all invariants under Hamiltonian flow, which is a nontrivial extension (on both sides) of the intersection cohomology