We establish a new universal relation between the Lie bracket and • –multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of " weak Frobenius manifold " which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary,...

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics....

This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in ε0 play a prominent role. The ...

New family of flat potential (Darboux–Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius structure on Hurwitz spaces found by B. Dubrovin more than 10 years ago. The branch points of a ramifie...

We show that the theory of Frobenius fields is decidable. This is conjectured in [4], [8] and [13], and we prove it by solving a group theoretic problem to which this question is reduced there. To do this we present and develop the notion of embedding covers of finite and pro-finite groups. We also answer two other problems from [8], again by solving a corresponding group theoretic problem: A f...

Quantum systems in which the position and momentum take values in the ring Zd and which are described with d-dimensional Hilbert space, are considered. When d is the power of a prime, the position and momentum take values in the Galois field GF (p), the position-momentum phase space is a finite geometry and the corresponding ‘Galois quantum systems’ have stronger properties. The study of these ...

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the represent...

the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.

Building on the results of Deligne and Illusie liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only dimension certain cohomology groups vector bundles arising from Frobenius pushforward de Rham complex. Using bundle methods, apply this show exceptional Enriques surfaces, class introduced by Ekedahl Shepherd-Barron, do not lift yet base miniversal formal...

We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras ar...