1. p-centers 1 1.1. Notation 1 1.2. Frobenius twist for symplectic varieties 1 1.3. Differential operators 2 1.4. Universal enveloping algebras 2 2. Classical Hamiltonian reduction 2 2.1. GIT in positive characteristic 2 2.2. Frobenius kernel 3 2.3. Hamiltonian actions and reduction 4 3. Frobenius constant quantizations 4 3.1. Definition 4 3.2. Quantum Hamiltonian reduction 5 4. Quantization of...

Given a primitive integer vector a ∈ Z>0, the largest integer b such that the knapsack polytope P = {x ∈ R≥0 : 〈a,x〉 = b} contains no integer point is called the Frobenius number of a. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed ||a|| 1+...

This paper sets a foundation for the study of linear codes over nite rings. The nite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from nite elds to nite Frobenius rings. It is over Frobenius rings that certain key identi cations can be made bet...

A notion of a Frobenius manifold with a nice real structure was introduced by Hertling. It is called CDV structure (Cecotti-Dubrovin-Vafa structure). In this paper, we introduce a “positivity condition” on CDV structures and show that any Frobenius manifold of rank two with real spectrum can be equipped with a positive CDV structure. We extend naturally the symmetries of Frobenius structures gi...

In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to constru...

From any given Frobenius manifold one may construct a so-called ‘dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotent...

A finite dimensional Lie algebra f is Frobenius if there is a linear Frobenius functional F : f→ C such that the skew bilinear form BF defined by BF (x, y) = F ([x, y]) is non-degenerate. The principal element of f is then the unique element F̂ such that F (x) = F ([F̂ , x]); it depends on the choice of functional. However, if f is a subalgebra of a simple Lie algebra g and not an ideal of any la...