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.

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...

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...

In this paper, we give a notation on the Singleton bounds for linear codes over a finite commutative quasi-Frobenius ring in the work of Shiromoto [5]. We show that there exists a class of finite commutative quasi-Frobenius rings. The Singleton bounds for linear codes over such rings satisfy d(C) − 1 A ≤ n− log|R| |C|. keywords : Linear codes, General weight, Quasi-Frobenius ring, Singleton bounds