Igor Mencattini

Insertion and Elimination Lie Algebra: The Ladder Case

We prove that the insertion-elimination Lie algebra of Feynman graphs in the ladder case has a natural interpretation in terms of a certain algebra of infinite dimensional matrices. We study some aspects of its representation theory and we will discuss some relations with the representation of the Heisenberg algebra.

The Structure of the Ladder Insertion-Elimination Lie Algebra

We continue our investigation into the insertion-elimination Lie algebra of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson–Schwinger equations. We work out the relation to the classical infinite dimensional Lie algebra and we determine the cohomology of .

Hyper-symplectic structures on integrable systems

Abstract We prove that an integrable system over a symplectic manifold whose symplectic form is covariantly constant carries a natural hyper-symplectic structure. Moreover, a special Kahler structure is induced on the base manifold.

Post-lie algebras and isospectral flows

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.

Post-Lie algebras and factorization theorems

Abstract In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang–Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

G-systems and deformation of G-actions on Rd

Given a (smooth) action of a Lie group G on Rd we construct a DGA whose Maurer-Cartan elements are in one to one correspondence with some class of defomations of the (induced) G-action on the ring of formal power series with coefficients in the ring of smooth functions on Rd. In the final part of this note we discuss the cohomological obstructions to the existence and to the uniqueness (in a sense to be clarified) of such deformations.

A Note on the Automorphism Group of the Bielawski-Pidstrygach Quiver

We show that there exists a morphism between a group alg introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase spaceCn;2 of the Gibbons{Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image of alg together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus ofCn;2, the subgroup contains an element sending the first point to the second.

Deformations of momentum maps and G-systems

In this note we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of deformations is parametrized by the formal G-systems introduced in B. Dherin and I. Mencattini [“G-systems and deformations of G-actions on Rd,” J. Math. Phys. 55, 011702 (2014)] (see also B. Dherin and I. Mencattini [“Quantization of (volume-preserving) actions on Rd,” e-print arXiv:1202.0886]) and allows us to obtain classical invariant Hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal G-systems.