Higher Deformations of Lie Algebra Representations I

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 Japa

Higher Deformations of Lie Algebra Representations II

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.Comment: 21 pages. Version 2: Minor corrections and clarification

K2K_2 of Kac-Moody Groups

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.Comment: 25 page

Integration of Modules II: Exponentials

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.Comment: 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

Integration of Modules I: Stability

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 correction

Covering Groups of Nonconnected Topological Groups and 2-Groups

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.Comment: 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) approac

An empirical assessment of real-time progressive stereo reconstruction

3D reconstruction from images, the problem of reconstructing depth from images, is one of the most well-studied problems within computer vision. In part because it is academically interesting, but also because of the significant growth in the use of 3D models. This growth can be attributed to the development of augmented reality, 3D printing and indoor mapping. Progressive stereo reconstruction is the sequential application of stereo reconstructions to reconstruct a scene. To achieve a reliable progressive stereo reconstruction a combination of best practice algorithms needs to be used. The purpose of this research is to determine the combinat ion of best practice algorithms that lead to the most accurate and efficient progressive stereo reconstruction i.e the best practice combination. In order to obtain a similarity reconstruction the in t rinsic parameters of the camera need to be known. If they are not known they are determined by capturing ten images of a checkerboard with a known calibration pattern from different angles and using the moving plane algori thm. Thereafter in order to perform a near real-time reconstruction frames are acquired and reconstructed simultaneously. For the first pair of frames keypoints are detected and matched using a best practice keypoint detection and matching algorithm. The motion of the camera between the frames is then determined by decomposing the essential matrix which is determined from the fundamental matrix, which is determined using a best practice ego-motion estimation algorithm. Finally the keypoints are reconstructed using a best practice reconstruction algorithm. For sequential frames each frame is paired with t he previous frame and keypoints are therefore only detected in the sequential frame. They are detected , matched and reconstructed in the same fashion as the first pair of frames, however to ensure that the reconstructed points are in the same scale as the points reconstructed from the first pair of frames the motion of the camera between t he frames is estimated from 3D-2D correspondences using a best practice algorithm. If the purpose of progressive reconstruction is for visualization the best practice combination algorithm for keypoint detection was found to be Speeded Up Robust Features (SURF) as it results in more reconstructed points than Scale-Invariant Feature Transform (SIFT). SIFT is however more computationally efficient and thus better suited if the number of reconstructed points does not matter, for example if the purpose of progressive reconstruction is for camera tracking. For all purposes the best practice combination algorithm for matching was found to be optical flow as it is the most efficient and for ego-motion estimation the best practice combination algorithm was found to be the 5-point algorithm as it is robust to points located on planes. This research is significant as the effects of the key steps of progressive reconstruction and the choices made at each step on the accuracy and efficiency of the reconstruction as a whole have never been studied. As a result progressive stereo reconstruction can now be performed in near real-time on a mobile device without compromising the accuracy of reconstruction

Tilting modules and highest weight theory for reduced enveloping algebras

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, and let ${\mathfrak g}$ be its Lie algebra. Given $\chi\in{\mathfrak g}^{*}$ in standard Levi form, we study a category ${\mathscr C}_\chi$ of graded representations of the reduced enveloping algebra $U_\chi({\mathfrak g})$. Specifically, we study the effect of translation functors and wall-crossing functors on various highest-weight-theoretic objects in ${\mathscr C}_\chi$, including tilting modules. We also develop the theory of canonical $\Delta$-flags and $\overline{\nabla}$-sections of $\Delta$-flags, in analogy with similar concepts for algebraic groups studied by Riche and Williamson.Comment: 45 page

On graded representations of modular Lie algebras over commutative algebras

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 mad