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We continue our exploration of various approaches to integration of representations from a Lie algebra $\mbox{Lie} (G)$ to an algebraic group $G$ in positive characteristic. In the present paper we concentrate on an approach exploiting exponentials. This approach works well for over-restricted representations, introduced in this paper, and takes no note of $G$-stability., Accepted by Transactions of the AMS. This paper is split off the earlier versions (1, 2 and 3) of arXiv:1708.06620. Some of the statements in these versions of arXiv:1708.06620 contain mistakes corrected here. Version 2 of this paper: close to the accepted version by the journal, minor improvements, compared to Version 1

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_\chi({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic $p>0$ and $\chi\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result for ${\mathfrak g}$ of type $G_2$ in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic., Comment: Main paper: 8 pages; Appendix: 8 pages. Version 2: Table added

We develop the theory of a category ${\mathscr C}_A$ which is a generalisation to non-restricted ${\mathfrak g}$-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted ${\mathfrak g}$-modules, where ${\mathfrak g}$ is the Lie algebra of a reductive group $G$ over an algebraically closed field ${\mathbb K}$ of characteristic $p>0$. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra $U_\chi$ associated to ${\mathfrak g}$ and to $\chi\in{\mathfrak g}^{*}$ in standard Levi form. On the right, they are modules over a commutative Noetherian $S({\mathfrak h})$-algebra $A$, where ${\mathfrak h}$ is the Lie algebra of a maximal torus of $G$. We develop here certain important modules $Z_{A,\chi}(\lambda)$, $Q_{A,\chi}^I(\lambda)$ and $Q_{A,\chi}(\lambda)$ in ${\mathscr C}_A$ which generalise familiar objects when $A={\mathbb K}$, and we prove some key structural results regarding them., Comment: 49 pages. v2: Index of notation added and minor changes made

We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples., Version 2: many minor improvements, one minor correction. Version 3: further minor improvements, close to the final published version. To appear in Canadian Mathematical Bulletin

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper we prove that Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity., 22 pages. Version 2: Minor corrections and clarifications. Version 3: Corrections and clarifications. Final version, published by Nagoya Mathematical Journal

In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group., Comment: 30 pages. Version 2: Minor corrections. Version 3: Changes to Sections 4 and 7 and corrections throughout. Version 4: Changes to Section 7 and other edits throughout. Accepted for publication by the Journal of the Mathematical Society of Japan

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other., 9 pages. Version 2: historical review added. Version 3 (14 pages): historical review revised, minor corrections elsewhere, some of the results did not make it into the final version. Version 4: final journal version with a slightly different (to Version 3) approach

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group., Comment: Version 2: Some changes in terminology, examples of over-restricted modules are added. Version 3: mistakes in tables are corrected. Version 4: major revision, the paper is split into two parts, the exponential part is published separately (see arXiv:1807.08698). Version 5: minor corrections. Version 6: final, minor corrections

Ulf Rehmann and Jun Morita, in their 1989 paper "A Matsumoto-type theorem for Kac-Moody groups", gave a presentation of $K_2(A,F)$ for any generalised Cartan matrix $A$ and field $F$. The purpose of this paper is to use this presentation to compute $K_2(A,F)$ more explicitly in the case when $A$ is hyperbolic. In particular, we shall show that these $K_2(A,F)$ can always be expressed as a product of quotients of $K_2(F)$ and $K_2(2,F)$. Along the way, we shall also prove a similar result in the case when $A$ has an odd entry in each column., 25 pages