Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations higher Frobenius kernels reduces to first kernel. In preceding paper in this series, deforming distribution algebra a kernel yielded family deformations called reduced enveloping algebras. we prove Steinberg decomposition can be similarly deformed, allowing us reduce representa...

In this paper, by using of Frobenius theorem a relation between Lie subalgebras of the Lie algebra of a top space T and Lie subgroups of T(as a Lie group) is determined. As a result we can consider these spaces by their Lie algebras. We show that a top space with the finite number of identity elements is a C^{∞} principal fiber bundle, by this method we can characterize top spaces.

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain func...

This paper consists of three results on Frobenius categories: (1)we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen–Macaulaymodules over some additive category; this is an analogue of Gabriel–Quillen’...

In this thesis we present the formalization of three principal results that are the Jordan normal form of a matrices, the Bolzano-Weierstraß theorem, and the Perron-Frobenius theorem. To formalize the Jordan normal form, we introduce many concepts of linear algebra like block diagonal matrices, companion matrices, invariant factors, ... The formalization of Bolzano-Weierstraß theorem needs to d...

We prove a representation theorem of projections of sets of integer points by an integer matrix W . Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of non-linear integer optimization problems in variable dimension. Concretely, we aim to optimize f(Wx) over a set F = P ∩ Z, where f ...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

in this paper, the concept of intuitionistic fuzzy mapping as a generalization of fuzzy mapping is presented, and its' relationship with intuitionistic fuzzy relations is derived. moreover, some basicoperations of intuitionistic fuzzy mappings are defined, hence we can conclude that all of intuitionistic fuzzy mappings constitute a soft algebrawith respect to these operations. afterwards, the a...

The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. It includes semisimple Frobenius manifolds as a particular case. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singulari...