Frobenius algebras play an important role in the representation theory of finite groups. In the present work, we investigate the (quasi) Frobenius property of n-group algebras. Using the (quasi-) Frobenius property of ring, we can obtain some information about constructions of module category over this ring ([2], p. 66–67).

We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. prove Frobenius Theorem with sharp regularity estimate when subbundle is log-Lipschitz: if $${\mathcal {V}}$$ a log-Lipschitz involutive rank r, then for any $$\varepsilon >0$$ , locally there homeomorphism $$\Phi (u,v)$$ such that ,\frac{\partial \Phi }{\partial u^1},\dots u^r}\in C^{0,1-\...

In 1969 Kunz [Ku] proved a fundamental result, connecting the regularity of a local ring of positive characteristic with the flatness of its Frobenius endomorphism φ. This was a first indication of the important role that φ would play in homological commutative algebra, especially in reflecting basic homological properties of the ring. Some results in Peskine and Szpiro’s groundbreaking thesis,...

In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhäuser, and Zhu. We find a sequence of gauge invariant central elements of H such that...

If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R). We associate a directed graph to any homogeneous, monotone function, f : (R) → (R), and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R). Several results in the literature emerge as c...

This paper provides a simple proof for the Perron-Frobenius theorem concerned with positive matrices using a homotopy technique. By analyzing the behaviour of the eigenvalues of a family of positive matrices, we observe that the conclusions of Perron-Frobenius theorem will hold if it holds for the starting matrix of this family. Based on our observations, we develop a simple numerical technique...