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Structure and Representations of Certain Classes of Infinite-Dimensional Algebras UPPSALA DISSERTATIONS IN MATHEMATICS 110 Structure and representations of certain classes of infinite-dimensional algebras Brendan Frisk Dubsky Department of Mathematics Uppsala University UPPSALA 2018 Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Lägerhyddsvägen 1, Uppsala, Wednesday, 5 December 2018 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Rolf Farnsteiner (Christian-Albrechts-Universität zu Kiel). Abstract Frisk Dubsky, B. 2018. Structure and representations of certain classes of infinite-dimensional algebras. Uppsala Dissertations in Mathematics 110. 32 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2728-2. We study several infinite-dimensional algebras and their representation theory. In Paper I, we study the category O for the (centrally extended) Schrödinger Lie algebra, which is an analogue of the classical BGG category O. We decompose the category into a direct sum of "blocks", and describe Gabriel quivers of these blocks. For the case of non-zero central charge, we in addition find the relations of these quivers. Also for the finite-dimensional part of O do we find the Gabriel quiver with relations. These results are then used to determine the center of the universal enveloping algebra, the annihilators of Verma modules, and primitive ideals of the universal enveloping algebra which intersect the center of the Schrödinger algebra trivially. In Paper II, we construct a family of path categories which may be viewed as locally quadratic dual to preprojective algebras. We prove that these path categories are Koszul. This is done by constructing resolutions of simple modules, that are projective and linear up to arbitrary position. This is done by using the mapping cone to piece together certain short exact sequences which are chosen so as to fall into three managable families. In Paper III, we consider the category of injections between finite sets, and also the path category of the Young lattice subject to the relations that two boxes added to the same column in a Young diagram yields zero. We construct a new and direct proof of the Morita equivalence of the linearizations of these categories. We also construct linear resolutions of simple modules of the latter category, and show that it is quadratic dual to its opposite. In Paper IV, we define a family of algebras using the induction and restriction functors on modules over the dihedral groups. For a wide subfamily, we decompose the algebras into indecomposable subalgebras, find a basis and relations for each algebra, as well as explicitly describe each center. Keywords: infinite-dimensional algebras, representation theory, category O, koszul, koszulity, injections, dihedral, preprojective, young lattice Brendan Frisk Dubsky, Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden. © Brendan Frisk Dubsky 2018 ISSN 1401-2049 ISBN 978-91-506-2728-2 urn:nbn:se:uu:diva-363403 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-363403) List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Brendan Dubsky, Rencai Lü, Volodymyr Mazorchuk and Kaiming Zhao. Category O for the Schrödinger algebra. Linear Algebra and its Applications, 460:17-50, 2014. II Brendan Dubsky. Koszulity of some path categories. Communications in Algebra, 45(9):4084-4092, 2017. III Brendan Dubsky. Incidence category of the Young lattice, injections between finite sets, and Koszulity. Manuscript, 2018. arXiv:1607.00426 IV Brendan Dubsky. Induction and restriction on representations of dihedral groups. Manuscript, 2018. arXiv:1805.02567 Reprints were made with permission from the publishers. Contents 1 Introduction .................................................................................................. 7 2 Preliminaries ................................................................................................ 9 2.1 Associative algebras ......................................................................... 9 2.2 Representations and modules of associative algebras .................. 10 2.3 Categories ....................................................................................... 10 2.4 Quiver algebras and path categories ............................................. 11 2.5 Groups and group algebras ............................................................ 13 2.6 Lie algebras and universal enveloping algebras ........................... 14 2.7 Homological algebra and Koszulity .............................................. 15 2.8 Induction and restriction of representations ................................. 17 2.9 Category O ..................................................................................... 18 3 Summary of papers .................................................................................... 20 3.1 Paper I ............................................................................................. 20 3.2 Paper II ............................................................................................ 23 3.3 Paper III .......................................................................................... 24 3.4 Paper IV .......................................................................................... 25 4 Sammanfattning på svenska (Summary in Swedish) .............................. 28 4.1 Populärvetenskaplig introduktion ................................................. 28 4.2 Sammanfattning av artiklar ........................................................... 29 4.2.1 Artikel I ............................................................................ 29 4.2.2 Artikel II ........................................................................... 29 4.2.3 Artikel III ......................................................................... 29 4.2.4 Artikel IV ......................................................................... 30 5 Acknowledgements .................................................................................... 31 References ........................................................................................................ 32 1. Introduction For most of us (and indeed most probably for humanity itself), the journey into the world of mathematics began with a collection of concrete object – say a handful of pebbles – and the operation of adding numbers of them together. In a first, tentative leap of abstraction, we realized that the rules of addition of natural numbers model addition of numbers of objects irrespective of any physical properties those objects might have; only the numbers count1. Natural numbers together with the addition operation form a basic example of an algebraic structure. We soon proceeded to consider more operations (subtraction, multiplication, and division) and more abstractions of the physical world to which we can apply them (negative numbers, rational numbers, real numbers, complex numbers, and later matrices of numbers). Early algebra to a large extend revolved around the study of equations involving this quite limited number of operations. Over the past two centuries or so, developments both in the study of these equations and in physics have motivated radically new kinds of operations and abstractions of physical features, and the modern field of algebra comprises the study of a plethora of algebraic structures. In the subdiscipline of representation theory, algebraists consider certain algebraic structures – so-called representations – each of which subsume struc- tural properties of another algebraic structure of interest. This is typically done either because the representations is how that algebraic structure arises in some application, or because the representations embody interesting properties of the original structure while being easier to study. The most classical and widespread kind of representation is the one consisting of certain collections of complex matrices equipped with the structure of a vector space and the operation of matrix multiplication (or more generally linear transformations equipped with function composition). This kind of representation may be used to study many different algebraic structures, including quiver algebras, groups and Lie algebras, and every such representation may be viewed as a representation of some associative algebra. The present thesis is a collection of results on the complex representation theory of various associative algebras (in paper II viewed as path categories). In paper I, we study the category O of the Schrödinger Lie algebra. In paper II, we consider the representation theoretic property of Koszulity, and prove that path algebras of a certain class are Koszul. In paper III, we derive a more 1No pun intended. 7 elementary and explicit proof of Koszulity as well as a description of the quiver of the algebra of injections of finite sets and a proof of its Koszul self-duality. Finally, in paper IV, we study certain algebras generated by the induction and restriction functors on representations of dihedral groups. 8 2. Preliminaries Here we introduce some of the main concepts used in the papers to follow. While a solid background in mathematics will be necessary to understand the content of the papers, the present chapter is intended to provide an accessible reference for the mathematician whose algebra is a bit rusty, as well as a relatively self-contained overview of the studied areas for the mathematically
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