Structure and representations of certain classes of infinite-dimensional algebras

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(1)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.

(2) 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).

(3) 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..

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(5) Contents. 1. Introduction. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20 20 23 24 25. 4. Sammanfattning på svenska (Summary in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Populärvetenskaplig introduktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sammanfattning av artiklar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Artikel I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Artikel II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Artikel III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Artikel IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28 28 29 29 29 29 30. 5. Acknowledgements. ..................................................................................... 31. ......................................................................................................... 32. References. ................................................................................................... 7.

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(7) 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 structural 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.

(8) 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.

(9) 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 interested general audience. References and directions for further reading on the various topics are given at the end of each section. All definitions will be given for the context of the complex numbers, C, as this is the predominant case in the papers, but will typically have straightforward analogues for other fields.. 2.1 Associative algebras An associative algebra, A, is a vector space equipped with a bilinear and associative multiplication operation, i.e. a binary operation _·_ (multiplication symbols such as this dot are often omitted in favor of mere juxtaposition) which satisfies for any a, b, c, d ∈ A and k ∈ C the following. (i) (a + b) · (c + d) = a · c + a · d + b · c + b · d, (ii) a · (b · c) = (a · b) · c, (iii) k(a · b) = (ka) · b = a · (kb). An associative algebra A is said to be finite-dimensional if A is finitedimensional as a vector space, and unital if there is a unit element in A, i.e. an element 1 ∈ A such that 1 · a = a = a · 1 for any a ∈ A. Associative algebras are often tacitly assumed to be unital as part of the definition. Paper II studies linear path categories with infinitely many objects, and although these may in fact equivalently be described as (non-unital) associative algebras, we speak of them in the language of category theory in order to emphasize the difference from unital associative algebras. Example 2.1.1. For any integer n ≥ 1, the vector space of n × n-matrices with matrix multiplication as the operation _ · _ is a finite-dimensional unital associative algebra with the n × n identity matrix as unit. The following example generalizes the previous one by the usual identification of matrices with linear maps. 9.

(10) Example 2.1.2. Let V be an arbitrary vector space. Then the vector space End(V ) with function composition as _ · _ is a unital associative algebra with the identity map as unit. For further reading on associative algebras, see [1].. 2.2 Representations and modules of associative algebras Let A be an associative algebra. A representation of A is an algebra homomorphism ϕ : A → End(V ), where End(V ) is the endomorphism algebra of a vector space V , as in Example 2.1.2. The property that there is a homomorphism from A to End(V ) should be intuitively understood to mean that some of the algebraic structure of A is to be found also in End(V ). A vector space V is said to be a (left) A-module if it is equipped with a bilinear A-action _ ∗ _ such that (a · b) ∗ v = a ∗ (b ∗ v) for any a, b ∈ A and v ∈ V and in the case of unital algebras furthermore such that 1 ∗ v = v. Again, the operation symbol ∗ is often omitted in favor of juxtaposition. It can be shown that the notion of a representation ϕ : A → End(V ) of A is in fact equivalent to that of an A-module V by setting a ∗ v = ϕ(a)(v) for a ∈ A and v ∈ V . In the papers to follow, we will most often consider modules, but use the words “representations” and “modules” interchangeably. A subspace W of an A-module V is called a submodule if it is itself a module under the restricted A-action. A non-zero module which has no submodules except for 0 and itself is called simple. Simple modules are of particular interests in representation theory because many classes of modules can be described via some filtration of simple modules. A module which decomposes into a direct sum of simple modules is called semisimple. For further reading on the representation theory of associative algebras, see [1].. 2.3 Categories Category theory is a formal framework for studying structural relationships between algebraic structures (and more general mathematical constructions) of the same kind. Adding additional layers of abstraction, it can even be used to study relationships between seemingly entirely different kinds of constructions. A category C is defined as a collection of the following data. 10.

(11) • A class of objects Ob(C). In this thesis it will typically be the case that each object is an algebraic structure. • For each ordered pair of objects (X, Y ), a class Hom(X, Y ) of morphisms, said to go from X to Y . In this thesis, Hom(X, Y ) will typically be the set of homomorphism from X to Y , where X and Y are algebraic structures. • For each triple of objects (X, Y, Z) a composition map ◦ = ◦X,Y,Z : Hom(Y, Z) × Hom(X, Y ) → Hom(X, Z) satisfying associativity, i.e. that for any f ∈ Hom(X, Y ), g ∈ Hom(Y, Z) and h ∈ Hom(Z, W ), we have h ◦ (g ◦ f ) = (h ◦ g) ◦ f . • For every object X, an identity morphism 1X ∈ Hom(X, X) satisfying for any object Y and any morphisms f ∈ Hom(X, Y ) and g ∈ Hom(Y, X) that f ◦ 1X = f and 1X ◦ g = g. Example 2.3.1. Let the class of all complex vector spaces be the class of objects, and for vector spaces X and Y let Hom(X, Y ) be the set of linear maps from X to Y . Taking ◦ to be function composition and 1X be the identity function, we obtain a category. Example 2.3.2. For an algebra A, we denote by A-Mod the category where objects are A-modules and morphisms are A-module homomorphisms. Again, ◦ and 1X is function composition and the identity function respectively. Similarly to how homomorphisms are structure-preserving maps between algebraic structures, one may define functors, which are structure-preserving maps between categories. Functors that admit inverses are, as is the case for other algebraic structures, called isomorphisms. However, the weaker equivalences of categories are functors that are seen more often in practice and capture the notion of “sameness” of categories that tends to be relevant in category theory. We will in this thesis in particular also consider linear categories, which are categories where every class of morphisms is a vector space such that composition of morphisms is bilinear. A more detailed account of the fundamentals of category theory can be found in [6].. 2.4 Quiver algebras and path categories A quiver is a kind of directed graph so prominent in algebra that it has earned its own name. A quiver Q is a tuple Q = (Q0 , Q1 , s, t), where the constituting data are specified as follows. • Q0 is the set of vertices. • Q1 is the set of arrows. 11.

(12) • s, t : Q1 → Q0 are the source and target maps respectively (we say that α ∈ Q1 has source s(α) and target t(α)). A quiver is typically visualized by viewing the vertices as points and the arrows as directed edges pointing from their source to their target. To each quiver we may associate its quiver algebra (or path algebra), which is an associative algebra, CQ, defined as follows. • The underlying vector space of CQ is spanned by paths of one of the following forms. (i) Concatenations α1 α2 . . . αn of arrows in Q1 such that the source of αi is the target of αi+1 for i = 1, 2, . . . , n − 1. (ii) For each vertex v in Q0 a so called empty path εv . • The multiplication ◦ is defined on the first kind of paths by α 1 α 2 . . . α n ◦ β 1 β 2 . . . βm = α 1 α 2 . . . α n β 1 β 2 . . . βm if the source of αn equals the target of β1 ; α 1 α 2 . . . α n ◦ ε v = α1 α 2 . . . αn if αn has source v; εv ◦ α 1 α 2 . . . αn = α1 α 2 . . . αn if α1 has target v; εv ◦ εv = εv , and all other multiplications of paths result in 0. This is then extended to a multiplication on the entire space CQ by bilinearity. As usual in ring theory, we may consider an ideal I of CQ, and then form the quotient CQ/I, which will again be an algebra. The following Theorem, due to Gabriel (see [5]), is arguably the main reason for defining quivers and studying their algebras. Theorem 2.4.1. Let A be a finite-dimensional unital associative algebra. Then there exists a quiver Q and an ideal I of CQ such that A-Mod ∼ = CQ/I-Mod. In the situation of the above theorem, Q is called the Gabriel quiver of A, and I is said to contain the relations corresponding to that quiver. Finding (the infinite-dimensional case analogues of) Gabriel quivers with relations is a main objective in both papers I and III. For references as well as further reading on the role of quivers in the representation theory of associative algebras, see [1]. 12.

(13) 2.5 Groups and group algebras The group is one of the most classical types of algebraic structures, and is often used to model the “symmetries” of some construction (this latter claim is particularly transparent for the dihedral groups described below). A group G is a set endowed with a binary multiplication operation _ · _, with a distinguished element g −1 ∈ G for each g ∈ G and furthermore another distinguished element e ∈ G, subject to the following axioms. (i) e · g = e = g · e, (ii) g · g −1 = e = g −1 · g, (iii) a · (b · c) = (a · b) · c, for all g, a, b, c ∈ G. The complex representation theory of finite groups is a very classical subject, and this is the part of group representation theory that will be of concern in this thesis. The natural notion of a linear representation of a group G turns out to be equivalent to a representation of a certain associative algebra, as it was defined in Section 2.1. This algebra C[G] is called the group algebra of G, and is defined to be the vector space spanned by the elements of G and with algebra multiplication given as the bilinear extension of the group multiplication. This is indeed one of several related reasons why associative algebras have a more distinguished position in representation theory than other algebraic structures. One can prove that every finite-dimensional complex module over a finite group decomposes into a direct sum of modules which are simple, i.e. which have no nontrivial submodules. The representation theory of the finite groups thus largely boils down to the study of the simple modules. Example 2.5.1. (The symmetric groups.) For any positive integer n, the set of permutations on n elements, i.e. bijections from the set {1, 2, . . . , n} to itself, forms a group, Sn , under the multiplication given by composition of functions. This group is called the symmetric group on n elements. The simple modules over Sn are indexed by partitions of n, or equivalently by the Young diagrams of size n. These are typically drawn as top/left adjusted rows (of weakly decreasing length) of boxes. For examples of these drawings, see the vertices of the graph illustrated in Figure 2.8 of Section 2.8. In Paper III, where we study the representation theory of injections between finite sets, the representation theory of Sn finds applications in a natural way since bijections are a special case of injections. Example 2.5.2. (The dihedral groups.) Let n ≥ 3 be an integer. The dihedral group D2n of order 2n is the group of symmetries of the regular n-gon in the plane. Alternatively, it may be defined by the presentation D2n = rn , sn |rnn = 1, sn rn sn = rn−1 , i.e. as the quotient of the free group generated by the symbols rn and sn by the relations rnn = 1 and sn rn sn = rn−1 . Note that we consider dihedral groups 13.

(14) for n = 1, 2 undefined, in contrast to going the Coxeter route where it is natural to define dihedral groups also for these n. The dihedral groups are the main objects of study in Paper IV. For a more detailed account of the fundamentals of representations of groups in general and those of the symmetric group in particular, see [10].. 2.6 Lie algebras and universal enveloping algebras Lie algebras arise as the “linearizations” of Lie groups, and have as such often quite direct applications to areas such as quantum physics. A Lie algebra g is a vector space with the Lie bracket binary operation, [_, _]. The Lie bracket is bilinear and furthermore satisfies the following additional axioms. (i) [a, b] = −[b, a], (ii) [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0, for all a, b, c ∈ g. It turns out that the natural notion of representation theory of Lie algebras may be equivalently described in terms of representations of corresponding associative universal enveloping algebras. Because of this, one typically studies representations of universal enveloping algebras in lieu of Lie algebra representations. For a Lie algebra, g, let F (g) be the free associative algebra on a basis of g, and let I be the ideal generated by elements of the form ab − ba − [a, b] for a and b in the chosen basis of g. Then the universal enveloping algebra of g is defined to be U (g) = F (g)/I. The best studied Lie algebras are the finite-dimensional semisimple ones, which may be characterized as finite-dimensional Lie algebras, g, such that every U (g)-module is a direct sum of simple modules. The finite-dimensional semisimple Lie algebras are classified in terms of the celebrated Dynkin diagrams, and their representation theory serves as inspiration also for the representation theory of other Lie algebras, such as the non-semisimple Schrödinger algebra. Example 2.6.1. For a positive integer, n, let sln be the vector space consisting of all n×n-matrices with zero trace, furthermore endowed with the Lie algebra structure given by the commutator sln × sln → sln (a, b) → [a, b] = a · b − b · a, where · is ordinary matrix multiplication. This Lie algebra is arguably the quintessential example of a semisimple Lie algebra. 14.

(15) For a positive integer n and g = sln , there is a vector space decomposition U (g) ∼ = U (n− ) ⊗ U (h) ⊗ U (n+ ), where n− ⊂ g is the Lie subalgebra consisting of the strictly lower triangular matrices, h ⊂ g is the Lie subalgebra consisting of the zero trace diagonal matrices, and n+ ⊂ g is the Lie subalgebra consisting of the strictly upper triangular matrices. This so called triangular decomposition can be generalized to a wide class of Lie algebras (see [9]), including all those that occur in the present thesis, and finds extensive use in the representation theory of such algebras. Example 2.6.2. The (centrally extended) Schrödinger algebra s is the Lie algebra with basis {e, h, f, p, q, z}, where [z, s] = 0 for all s ∈ s, and the rest of the Lie bracket is given by: [h, e] = 2e, [e, f ] = h, [h, f ] = −2f, [e, q] = p, [e, p] = 0, [h, p] = p, [f, p] = q, [f, q] = 0, [h, q] = −q, [p, q] = z. This Lie algebra is not semisimple, and is the main object of study of Paper I. For further details on the theory of Lie algebras and their universal enveloping algebras, see [7] and [4].. 2.7 Homological algebra and Koszulity Since its origin in topology, the language of homological algebra has evolved into one of the main tools of abstract algebra. One may think of homological algebra as a way of clearly expressing how a certain algebraic structure is composed of other (hopefully easier to understand) structures. Example 2.7.1. (Short exact sequences.) Let A be an algebra (say finitedimensional, unital and associative), and let L, M, N be A-modules with an injective homomorphism f : L → M and a surjective homomorphism g : M N such that ker(g) = im(f ). This data is typically called a short exact sequence and written f. g. →M − → N → 0. 0→L− One can see that there is a vector space decomposition M ∼ = L ⊕ N , and that (up to isomorphism) L ⊂ M and M/L = N . 15.

(16) The sense in which the module M in Example 2.7.1 “is composed of” the modules L and N should now be clear. Similar, but more complicated, relationships between modules are captured by the notions of complexes and their homology. A (cochain) complex C • of A-modules C i with i ∈ Z is a collection of homomorphisms di : C i → C i+1 such that di+1 ◦ di = 0 for all i ∈ Z. Such a complex is often written d−2. d−1. 0 1 2 C 1 −→ C 2 −→ .... . . . −−→ C −1 −−→ C 0 −→. d. d. d. The i:th homology1 of C • is defined to be H i (C • ) = ker(di )/ im(di−1 ). The short exact sequence of Example 2.7.1 may be viewed as a complex by setting for instance C −1 = L, C 0 = M , C 1 = N , d−1 = f , d0 = g, and all other modules and maps set to zero. As such, the short exact sequence has i:th homology zero for all i ∈ Z. In general, however, the homologies constitute additional pieces of the “jigsaw puzzle of modules” that complexes describe. Example 2.7.2. (Projective resolutions.) Projective resolutions are complexes of a particular importance, in that they describe modules that can be pieced together using projective modules2. A projective resolution of the A-module M is a complex P • such that the following hold. (i) H 0 (P • ) ∼ = M. i • (ii) H (P ) = 0 for all i < 0. (iii) P i = 0 for all i > 0. (iv) P i is projective for all i ≤ 0. In order to define the Koszul property of unital associative algebras we need to know what Z≥0 -graded algebras and their correspondingly graded modules are. An algebra A is Z≥0 -graded if its underlying vector space decomposes as Ai , A= i∈Z≥0. and Ai · Aj ⊂ Ai+j . A graded A-module, then, is an A-module M whose underlying vector space decomposes as Mi M= i∈Z. and Ai ∗ Mj ⊂ Mi+j . Note that M , in contrast to A, may contain elements of negative degree. A Z≥0 -graded algebra A is Koszul if A0 is semisimple and 1Technically, the correct term is cohomology, whereas “homology” would be used for a complex in the other direction, but let us not be too concerned with this quite inconsequential distinction. 2The reader not familiar with projective modules may think of them as certain generalizations of free modules.. 16.

(17) furthermore has a projective resolution P • of graded modules such that P i is generated by Pii for every i ∈ Z. Koszul algebras have been described as “as close to semisimple as a Z-graded algebra can be”. The Koszul property plays a main role in Papers II and III, albeit in the slightly generalized context of path categories rather than unital associative algebras. For a thorough treatment of the fundamentals of homological algebra, see [6], and for an introduction to the theory of Koszul rings in general, see [2].. 2.8 Induction and restriction of representations Consider two unital and associative algebras B ⊂ A with the same unit. For an A-module V , there is a natural way of obtaining a B-module structure on the vector space V : simply let every b ∈ B act like b ∈ A. This restriction procedure gives rise to a functor Res : A-Mod → B-Mod. It turns out that there is a “universal” way also of assigning to every B-module V an A-module. This induction procedure gives rise to a functor Ind : B-Mod → A-Mod, V → A ⊗B V, which is left adjoint to the restriction functor. Example 2.8.1. (The case for the symmetric group algebras, and the Young lattice.) Let n be a positive integer and consider the symmetric groups as in Example 2.5.1. Then we have a natural inclusion of group algebras C[Sn ] ⊂ C[Sn+1 ], which gives rise to a restriction Res : C[Sn+1 ]-Mod → C[Sn ]-Mod, and a corresponding induction functor in the other direction. The so called Young lattice is the graph obtained by taking all Young diagrams as vertices, and using the branching rule to draw an edge between two Young diagrams if the simple module corresponding to one of them is a direct summand of the result of applying an induction or a restriction functor to the simple module corresponding to the other. This way, the Young lattice arises as the main object of study in paper III. The (truncated) Young lattice is illustrated in Figure 2.8. Example 2.8.2. (The case for the dihedral group algebras.) For integers n ≥ 3, consider the dihedral groups Dn as in example 2.5.2. For any integer p, there 17.

(18) Figure 2.1. The Young lattice truncated after diagrams of size four. is a natural inclusion D2n → D2pn p rn → rpn sn → spn . With the geometrical interpretation of dihedral groups as the groups of symmetries of regular n-gons, the above inclusion corresponds to inscribing the regular n-gon into the regular pn-gon. This inclusion induces a corresponding inclusion of group algebras C[D2n ] ⊂ C[D2pn ]. The restriction and induction functors obtained from these inclusions form the main objects of study of paper IV, and certain graphs with vertices indexed by simple modules similar to the Young lattice of the symmetric groups play a prominent role in this paper. An introduction to the induction and restriction procedure for general algebras can be found in [12], and a proof of the branching rule for the symmetric groups can be found in [10].. 2.9 Category O For nearly all Lie algebras, it is an overwhelming task to classify its entire category of modules. As is often done in such situations in algebra, mathematicians restrict their efforts to certain more tractable subcategories which 18.

(19) nevertheless have interesting properties. In the case of semisimple Lie algebras, one classical such category is the category O due to Bernstein, Gelfand and Gelfand, see [3]. Let g be a Lie algebra with a triangular decomposition U (g) ∼ = U (n− ) ⊗ ∗ U (h) ⊗ U (n+ ) (see Section 2.6). Let also h be the dual vector space of h. Then the category O ⊂ U (g)-Mod associated to this triangular decomposition is the subcategory consisting of all U (g)-modules3 M satisfying the following conditions. (i) M is finitely generated. (ii) M decomposes into a sum of weight spaces, i.e. M ∼ = λ∈h∗ Mλ such that for every x ∈ h ⊂ U (g) and v ∈ Mλ we have xv = λ(x)v. (iii) For every v ∈ M , the vector space U (n+ )v is finite-dimensional. Any category O is abelian and contains important classes of modules. Example 2.9.1. (Verma modules.) One important class of modules in O is the Verma modules. Let λ ∈ h∗ and let V be a one-dimensional U (h⊕n+ )-module V on which U (n+ ) acts by 0 and each x ∈ U (h) acts by the scalar λ(x). Then the module Δ(λ) = U (g) ⊗U (h⊕n+ ) V is a Verma module. Note that we in effect obtain the Verma modules by applying an induction functor as in Section 2.8. A category O may be defined for the non-semisimple Schrödinger algebra (see Example 2.6.2), and this category is studied in Paper I. A modern treatment of the category O for semisimple Lie algebras can be found in [8].. 3Consistent with our habit (and sometimes abuse of notation) of studying the representation theory of g via that of U (g), we define O to be a category of U (g)-modules, even though the essentially equivalent definition using g is more common.. 19.

(20) 3. Summary of papers. 3.1 Paper I In this paper we examine the blocks of the category O of the Schrödinger algebra, s, and prove some consequences of the results thus obtained. Consider the standard triangular decomposition s∼ = n+ ⊕ h ⊕ n− of the Schrödinger algebra. In order to state the main results, we need to list a few definitions. • Let h∗ be the dual vector space of h, and for the basis {h, z} in h, let {h , z } be the dual basis. • Let R = {±2h , ±h } be the set of roots • For ξ ∈ h∗ /ZR, let O[ξ] ⊂ O be the full subcategory consisting of modules M such that there is a nonzero v ∈ M satisfying that for any x ∈ h we have xv = λ(x)v for some λ ∈ ξ. • For λ ∈ h∗ , let O[ξ]λ be the Serre subcategory of O[ξ] generated by Δ(λ), i.e. which contains Δ(λ) and is closed under taking short exact sequences with at least one module of the sequence in the subcategory. • Let c be the Casimir element of s, and for λ ∈ h∗ , let ϑλ be the scalar with which c acts on Δ(λ). It is easily seen that we have the decomposition O[ξ]. O∼ = ξ∈h∗ /ZR. This decomposition may be refined by further decomposing each O[ξ], in ways that depend on ξ. In particular, it is crucial whether h acts by a (half-)integer or not, and whether z acts by zero or not. We have the following results, where the central charge refers to the value λ(z) for any λ ∈ ξ ⊂ h∗ . Proposition 3.1.1. Let ξ ∈ h∗ /ZR be of nonzero central charge. Assume that λ(h) ∈ 12 Z for any λ ∈ ξ. Then the following hold. (i) The module Δ(λ) is simple for any λ ∈ ξ. (ii) We have the decomposition O[ξ]λ O[ξ] ∼ = λ∈ξ. 20.

(21) (iii) The functor defined on objects as N → Nλ and on morphisms in the obvious way provides an equivalence between O[ξ]λ and the category of finite dimensional complex vector spaces, so that O[ξ]λ ∼ = C-mod, with the latter being the category of finite-dimensional vector spaces. Proposition 3.1.2. Let ξ ∈ h∗ /ZR be of nonzero central charge and assume that λ(h) ∈ Z + 12 for any λ ∈ ξ. For i ∈ Z+ denote by λi the element in ξ such that λi (h) = − 32 + i. Then we have the following: (i) For λ ∈ ξ the module Δ(λ) is simple if and only if λ(h) ≤ − 32 . (ii) For each i ∈ N we have a non-split short exact sequence 0 → Δ(−λi − 3h ) → Δ(λi ) → L(λi ) → 0. (iii) We have the decomposition O[ξ] ∼ =. . O[ξ]λi .. i∈Z+. (iv) We have O[ξ]λ0 ∼ = C-mod, more precisely, the functor defined on objects as N → Nλ and on morphisms in the obvious way provides an equivalence between O[ξ]λ0 and the category of finite dimensional complex vector spaces. (v) For i ∈ N the category O[ξ]λi is equivalent to the category of finite dimensional representations over C of the following quiver with relations: •j. a. *•. ab = 0.. b. Proposition 3.1.3. Let ξ ∈ h∗ /ZR be of nonzero central charge and assume that λ(h) ∈ Z for any λ ∈ ξ. For i ∈ Z+ denote by λi the element in ξ such that λi (h) = −1 + i. Then we have the following: (i) The module Δ(λ) is simple for each λ ∈ ξ. (ii) We have the decomposition O[ξ]λi . O[ξ] ∼ = i∈Z+. (iii) We have O[ξ]λi ∼ = C⊕C-mod for all i ∈ Z+ , where C⊕C-mod denotes the category of finite-dimensional C ⊕ C-modules. For the next results, consider the following two quivers: ∞ Q∞. a. :. ... k b. +. a. -1 k b. *0j. a b. *1j. a. +. . .. b. 21.

(22) and Q∞ :. a. 0j. a. *1j. b. a. *2j. b. +. . .. b. with imposed commutativity relation ab = ba (which includes the relation ab = 0 for the vertex 0 in Q∞ ). We denote by ∞ Q+ ∞ -lfmod the category of locally finite dimensional ∞ Q∞ modules (in which ab = ba) that are bounded from the right, that is modules in which i is represented by the zero vector space for all i 0. We also denote by Q∞ -fmod the category of finite dimensional Q∞ -modules (in which again ab = ba), that is modules in which each i is represented by a finite dimensional vector space and these vector spaces are zero for all but finitely many i. Theorem 3.1.1. Let ξ ∈ h∗ /ZR be of zero central charge and assume λ(h) ∈ Z for any λ ∈ ξ. Then the category O[ξ] is equivalent to ∞ Q+ ∞ -lfmod. Denote by Of the full subcategory of O consisting of all finite-dimensional modules in O. Theorem 3.1.2. The categories Of and Q∞ -fmod are equivalent. Consider next the following quiver which we call Γ: a. 0J j. b. -1 k a. . b s. t. + k -2. + k -3. a. a. +. . .. b s. t b. a. *2k J. b s. t b. a. *1j J. + k -4. b. +. . .. a. Proposition 3.1.4. Let ξ ∈ h∗ /ZR be of zero central charge and assume λ(h) ∈ Z for any λ ∈ ξ. The quiver Γ is the Gabriel quiver for the category O[ξ]. As a consequence of the above results, one may arrive at a description of the center, Z(s), of the Schrödinger algebra. Theorem 3.1.3. We have Z(s) = C[z, c]. Let mλ be the ideal of Z(s) that is generated by z − λ(z) and c − ϑλ . Theorem 3.1.4. The annihilator in U (s) of Δ(λ) is the ideal U (s)mλ . 22.

(23) Finally one may prove. Theorem 3.1.5. Primitive ideals in U (s) with nonzero central charge are exactly the annihilators of simple highest weight modules with nonzero central charge. Going into the proofs of these results is outside of the scope of this summary.. 3.2 Paper II Let k be an algebraically closed field, let G be some simple connected graph which is orientable so that it contains an infinite directed walk, and construct a k-linear category as follows: First form a quiver by replacing the edges of G with one arrow in each direction. Then consider the k-linear category C generated by the path algebra of this quiver. Finally, obtain another k-linear category C by taking the quotient by the two following kinds of relations. 1. For arrows b : i → j and a : j → k such that i = j, j = k and k = i i. /j. b. a. /k. set ab = 0. 2. For arrows a1 : i → j, b1 : j → i, b2 : j → k and a2 : k → j such that i = j, j = k and k = i io. a1. /. jo. b1. a2 b2. /k. set a1 b1 = a2 b2 . The following is the main result of this paper. Theorem 3.2.1. The category C is Koszul. To prove this result, we construct for each simple C-module a projective resolution that is linear up to arbitrary degree. The idea is to start with taking the projective cover of a simple module, and considering the corresponding short exact sequence as a (non-projective) resolution. We proceed to replace the kernel in this sequence with a resolution (also in the form of a short exact sequence), and then keep doing the same thing with any non-projective module that appears. It turns out that with the proper choices, only three kinds of short exact sequences appear in this way, and the end result is a resolution that is both projective and linear up to arbitrary degree. The idea is formalized by letting the replacement procedure be handled by mapping cones, and the iteration by induction. 23.

(24) 3.3 Paper III The finite sets and injections between them form a category, which we denote by I. Let Q be the quiver given by (the Hasse diagram of) the Young lattice, and let C be the quotient of the path category of Q by the relations that composition of two arrows corresponding to addition of two boxes to the same column is zero. The following theorem was recently shown to hold by Sam and Snowden, see [11]. Theorem 3.3.1. The Gabriel quiver of CI is Q, and CI is Morita equivalent to CC, i.e. CI-Mod ∼ = CC-Mod. In this paper, we give a new and more direct proof of this theorem. This is done by computing subquotients of indecomposable projective CI-modules, which is in turn done by translating the problem into the language of the representation theory of the symmetric group and then applying the LittlewoodRichardson rule. Next, we study Koszulity properties of C. The following technical lemma is key. Lemma 3.3.1. Each arrow, say from λ to μ of the Young quiver Q may be assigned a “sign” sλμ = ±1 such that for any set {λ1 , λ2 , λ3 , λ4 } of Young diagrams such that λ2 and λ3 are obtained by adding a node to λ1 , and λ4 is obtained by adding a node to λ2 and λ3 , we have sλλ24 sλλ12 = −sλλ34 sλλ13 . While C was previously known to be Koszul, we here construct explicit linear resolutions of simple C-modules. We do this as follows. Let Pλ i denote the projective C-module generated at the object λ and in degree −i. Fix some (Young) diagram ξ. Let Ii be the set of diagrams that can be obtained by adding −i nodes, no two of which to the same row, to ξ. Define i λ∈Ii Pλ i , for i ≤ 0 P = 0, for i > 0. Fix non-zero elements vλλ,i ∈ Pλ i (λ). Consider a diagram μ. If μ is a subquotient of Pλ i , we have a uniquely determined x ∈ C(λ, μ). Otherwise set x = 0. Finally define vμλ,i = x · vλλ,i , which is a basis element of the subquotient μ in Pλ i , provided that μ is a subquotient of Pλ , and 0 otherwise. Define the maps πi,λ : P i → P i+1 vμμ,i → sλμ vμλ,i+1 , 24.

(25) where the sign sλμ is the one defined in Lemma 3.3.1, and also λ∈Ii+1 πi,λ , for i < 0 δi = 0, for i ≥ 0. Let Lξ denote the simple C-module at ξ and concentrated in degree zero. Theorem 3.3.2. The modules P i and maps δ i form a linear resolution, P • , of Lξ . Along the way, we obtain the following theorem, where CC ! denotes the quadratic dual of CC. Theorem 3.3.3. There is an isomorphism CC ! ∼ = (CC)op .. 3.4 Paper IV Inspired by an influential similar construction for the symmetric groups, we consider the dihedral groups D2n and construct complex algebras AP,M as follows. For P being any set of primes, define AP to be the free algebra generated by the symbols Resp and Indp with p ∈ P . The complexified Grothendieck group G = C ⊗Z Groth[ D2n -Mod] n≥3. of dihedral groups becomes an AP -module with action induced by the actions of Resp and Indp on the Grothendieck group. For any submodule M ⊂ G, let AnnAP (M) be the ideal of elements of AP that annihilate each element of M, and finally let AP,M = AP / AnnAP (M). Our main results hold when no p ∈ P or n with a D2n -module in M is even, and furthermore for every n either all D2n -modules lie in M or none does. Fix some arbitrary p ∈ P and define the algebras TP,M = AP,M /Resp Indp −1 . In this setting, our results are summarized in the following theorems. The first theorem gives a decomposition of AP,M into indecomposable algebras. Theorem 3.4.1. There is an isomorphism of algebras AP,M ∼ = TP,M ⊕ TP,M , 25.

(26) and TP,M is an indecomposable algebra. Furthermore, the center of AP,M is generated by 1 and Resp Indp , where p ∈ P is arbitrary. The next theorems describe the algebra TP,M explicitly in terms of a basis and relations. We get two cases depending on whether the prime factors of the involved dihedral group orders lie in P or not. Theorem 3.4.2. Assume that there is no D2n -module in M with all prime factors of n belonging to P . Then TP,M has a basis consisting of the monomials of the forms l | k|P | Reskp11 . . . Resp|P Indlp11 . . . Indp|P |P | | with ki , li ∈ N, and relations generated by the ones of the following forms. (i) Resp Resq = Resq Resp . (ii) Indp Indq = Indq Indp . (iii) Indq Resp = Resp Indq , for p = q. (iv) Resp Indp = 1. In order to state the next theorem, we need to define one of the key concepts of the paper. We call a terminal subsequence (i.e. a right monomial factor) z of z a nadir in z with respect to p if the number of Indp minus the number of Resp in z is minimal over all terminal subsequences of z. If z is a nadir in z with respect to all p ∈ P simultaneously, then we call z a total nadir in z. Theorem 3.4.3. Assume that there is some D2n -module in M with all prime factors of n belonging to P . Then TP,M has a basis consisting of the monomials of the forms l | k|P | Reskp11 . . . Resp|P (i) Indlp11 . . . Indp|P |P | | with ki , li ∈ N, l | k|P | Resp|P (ii) Indlp11 Reskp11 . . . Indp|P |P | | with ki , li ∈ N such that ki = 0 = li for at least two i, (iii) Reskpi Indlpj with i = j, and k, l ∈ Z>0 , (iv) Respi (mod |P |)+1 Indlpi Reskpi Indpi (mod |P |)+1 with k, l ∈ Z>0 , l | (v) Indlpj Reskpj Indlp11 . . . Indp|P |P | with j ∈ {1, 2, . . . , |P |}, with k, l ∈ Z>0 , and li ∈ N such that lj = 0 but li = 0 for at least one i, k|P | Indlpj Reskpj (vi) Reskp11 . . . Resp|P | with j ∈ {1, 2, . . . , |P |}, with k, l ∈ Z>0 , and ki ∈ N such that kj = 0 but ki = 0 for at least one i, and relations generated by the ones of the following forms. (i) Resp Resq = Resq Resp . 26.

(27) (ii) Indp Indq = Indq Indp . (iii) z1 = z1 , where z2 is the result of reordering the factors of z1 in a way such that the relative order of factors Resp and Indp for each fixed p ∈ P is unchanged, and where either both or none of z1 and z2 has a total nadir. (iv) Resp Indp = 1. Two features of the actions of Resp and Indp on M are heavily used in the proofs of our results. The first is that Resp acts locally nilpotently on M; more concretely, applying Resp to a D2n -module where n/p is not an integer ≥ 3 yields zero. It is to capture the significance of this that we define the concept of the nadir of a monomial in AP , which was used in Theorem 3.4.3. The second feature is that the pattern with which Resp and Indp act on modules is, in a certain sense, invariant under shifts of the order of the underlying dihedral group, so that their actions on M are determined by their action on a finite-dimensional subspace of M. The paper is concluded by some more modest results and conjectures for more general P and M.. 27.

(28) 4. Sammanfattning på svenska (Summary in Swedish). Denna sammanfattning utgör en något komprimerad och till svenska översatt version av föregående kapitel 1 och 3. Den första delen ger en kort populärvetenskaplig introduktion till abstrakt algebra och delområdet representationsteori. Den andra delen sammanfattar resultaten hos avhandlingens fyra ingående vetenskapliga artiklar.. 4.1 Populärvetenskaplig introduktion För de flesta av oss (och högst sannolikt för mänskligheten som sådan) började resan in i matematikens värld med en samling konkreta föremål – låt säga en handfull stenar – och operationen där specifika antal av dem adderas. I en första, trevande abstraktion insåg vi att addition av naturliga tal modellerar addition av antal föremål oberoende av de fysiska egenskaperna hos föremålen; endast antalen räknas. De naturliga talen utgör tillsammans med additionsoperationen ett grundläggande exempel på en algebraisk struktur. Vi började därefter studera ytterligare operationer (subtraktion, multiplikation och division) och ytterligare abstraktioner av den fysiska världen som vi kan tillämpa dessa på (negativa tal, rationella tal, reella tal, komplexa tal och senare matriser av tal). Den tidiga algebran kretsade till stor del kring studier av ekvationer med detta begränsade antal operationer. Under de senaste två århundradena så har utvecklingen såväl inom dessa studier som på fysikens område motiverat helt nya abstraktioner av fysiska företeelser, och den moderna algebran omfattar en uppsjö av algebraiska strukturer. Inom delområdet representationsteori studerar algebraiker särskilda algebraiska strukturer – så kallade representationer – som var och en införlivar strukturella egenskaper hos en annan algebraisk struktur av intresse. Detta görs i regel antingen eftersom den senare algebraiska strukturen uppträder genom sina representationer i direkta tillämpningar, eller eftersom representationerna delar några av den senare algebraiska strukturens inressanta egenskaper samtidigt som de är lättare att studera. Den mest klassiska och välanvända sortens representationer är den som utgörs av särskilda vektorrum av matriser försedda med operationen matrismultiplikation (eller mer allmänt linjära transformationer försedda med operationen funktionssammansättning). Denna sorts representationer kan användas för att studera många olika algebraiska strukturer, däribland kogeralgebror, grupper och Liealgebror, och representationerna kan i samtliga av dessa fall betraktas som representationer av associativa algebror. 28.

(29) 4.2 Sammanfattning av artiklar Denna avhandling innehåller fyra vetenskapliga artiklar där olika associativa algebror och deras representationer studeras. Artikel I studerar representationsteorin för en Liealgebra, artiklarna II och III behandlar egenskapen koszulitet hos två algebror, och artikel IV studerar en familj av algebror som uppstår ur induktion och restriktion av representationer av dihedralgrupperna.. 4.2.1 Artikel I I denna artikel studerar vi kategorin O för Schrödinger-algebran. Närmare bestämt delar vi upp kategorin som en summa av olika "block", och beskriver Gabriel-kogren för blocken. För fallet med "nollskild central laddning" hittar vi även kogrens relationer. Också för den ändligtdimensionella delen av O hittar vi såväl Gabriel-kogret som dess relationer. Ovanstående resultat används sedan för att hitta centret till den universella omslutande algebran till Schrödinger-algebran, annihilatorerna till Vermamodulerna, samt de primitiva ideal till den universella omslutande algebran som har triviellt snitt med Schrödinger-algebrans center.. 4.2.2 Artikel II I denna artikel konstruerar vi en familj, C, av vägkategorier som kan betraktas som lokalt kvadratiskt duala till preprojektiva algebror. Vi bevisar att dessa kategorier har Koszul-egenskapen. I beviset konstruerar vi resolutioner av de enkla C-modulerna som är projektiva och linjära till godtyckligt hög position. Detta görs genom att med hjälp av kon-konstruktionen sätta samman korta exakta följer, som kan väljas så att de alla återfinns i tre hanterbara familjer.. 4.2.3 Artikel III Låt I vara kategorin av ändliga mängder och injektionerna mellan dem. Låt även Q vara kogret som ges av Young-gittret, och C vara vägkategorin av Q modulo relationerna där de vägar som motsvarar addition av två lådor i samma kolumn i ett Young-diagram sätts till noll. I denna artikel hittar vi ett nytt och mer direkt bevis för att Q är Gabriel-kogret till I, och att CI och CC är Morita-ekvivalenta. Vi hittar därefter ett nytt bevis för det kända faktum att C har Koszulegenskapen genom att konstruera explicita linjära resolutioner till de enkla C-modulerna, och bevisar på vägen att C är kvadratiskt självdual. 29.

(30) 4.2.4 Artikel IV I denna artikel studerar vi en familj av algebror AP,M som definieras med hjälp av aktionen av restriktionsoch induktionsfunktorer på moduler över dihedralgrupperna D2n och där P är en mängd primtal och M är (en delmodul av grothendieckgruppen av) en samling moduler över dihedralgrupperna. Närmare bestämt låter vi AP,M vara den fria algebra som genereras av restriktionsoch induktionsfunktorerna Resp och Indp där p ∈ P modulo annihilatorn vid den naturliga aktion av dessa på M. Våra huvudsakliga resultat gäller givet att inget p ∈ P eller n med någon D2n -modul i M är jämnt, samt att det för varje n gäller att antingen ingen eller samtliga D2n -moduler finns i M. Dessa resultat inkluderar en uppdelning av AP,M i odelbara delalgebror, en explicit beskrivning av en bas och relationer för algebran, samt en beskrivning av dess center.. 30.

(31) 5. Acknowledgements. It is with great delight that I’m presented this opportunity to express my appreciation for some of the people who are the most important in my life, or to whom I’m otherwise indebted. You all deserve to hear this more often really. Walter, from the first course (in combinatorics!) where you taught me, I recognized your extraordinary ability as a teacher. Whatever was your way of thinking about mathematics, I knew I wanted to partake in it. You have since impressed and inspired me also with you energy, your patience, and your generosity, and have indeed shaped my way of thinking. Thank you for being the best advisor one could wish for. Thank you also to my coadvisor, Martin, who has always been eager to help, and whose frequent interjections during seminars have reliably taught me more than the talks themselves. Thank you to my other colleagues at the math department for all the help and the relaxed and positive environment, not least to the administration staff whose heroics keep the place running. A special thanks to Jakob, Sam, Andrea, Erik, Marta, Anya, Sebastian, and the rest of my PhD student colleagues. I regret that I haven’t been in a position to fully appreciate the warm camaraderie offered by your company. Lastly, a loving thank you to all my friends and family, especially to my mother Liselotte, my late father Frantisek, my sister Susanna, and my brother Dennis. All and any faith I have in myself, I owe to the faith that you have put in me.. 31.

(32) References. [1] Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory. Cambridge University Press, 2006. [2] Alexander Beilinson, Viktor Ginzburg, and Wolfgang Soergel. Koszul duality patterns in representation theory. Journal of the American Mathematical Society, 9(2):473–527, 1996. [3] Joseph Bernstein, Izrail Gel’fand, and Sergei Gel’fand. A category of g-modules. Functional Analysis and its Applications, 10(2):87–92, 1976. [4] Jacques Dixmier. Enveloping Algebras. North-Holland Publishing Company, 1977. [5] Peter Gabriel. Indecomposable representations II. Symposia Mathematica Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971). Academic Press, 1973. [6] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer-Verlag, 2nd edition, 2003. [7] James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer-Verlag, 1972. [8] James E. Humphreys. Representations of Semisimple Lie Algebras in the BGG Category O. American Mathematical Society, 2008. [9] Robert Moody and Arturo Pianzola. Lie algebras with triangular decompositions. Canadian Mathematical Society series of monographs and advanced texts. J. Wiley, 1995. [10] Bruce E. Sagan. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer-Verlag, 2nd edition, 2001. [11] Steven Sam and Andrew Snowden. Gl-equivariant modules over polynomial rings in infinitely many variables. Transactions of the American Mathematical Society, 368(2):1097–1158, 2016. [12] Alexander Zimmermann. Representation Theory: A Homological Algebra Point of View. Algebra and Applications. Springer International Publishing, 2014.. 32.

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