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In this work we examine C*-algebras of Toeplitz operators over the unit ball in ℂⁿ and the unit polydisc in ℂ². Toeplitz operators are interesting examples of non-normal operators that generate non-commutative C*-algebras. Moreover, in the nice cases (depending on the geometry of the domain) of algebras of Toeplitz operators we can recover some analogues of the spectral theorem up to compact operators. In this setting, we can capture the index of a Fredholm operator which is a fundamental numerical invariant in Operator Theory. / Master of Science

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A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is completely determined by a C*-correspondence, which consists of a right Hilbert A- module, E, and a *-homomorphism from the C*-algebra A into L(E), the adjointable operators on E. Some familiar examples of C*-algebras which can be recognized as Cuntz-Pimsner algebras include the Cuntz algebras, Cuntz-Krieger algebras, and crossed products of a C*-algebra by an action of the integers by automorphisms. In this dissertation, we construct a Cuntz-Pimsner Algebra associated to a dynam- ical system of a substitution tiling, which provides an alternate construction to the groupoid approach found in [3], and has the advantage of yielding a method for com- puting the K-Theory. / Graduate

In this thesis we present the connection between vertex operator algebras and modular forms which lies at the heart of Borcherds’ proof of the Monstrous Moonshine conjecture. In order to do so we introduce modular forms, vertex algebras, vertex operator algebras and their partition functions. Each notion is illustrated with examples.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).

This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.

Estudamos a estrutura de álgebras de potências associativas que são álgebras train. Primeiramente, mostramos a existência de idempotentes, que são todos principais e absolutamente primitivos. Em seguida, vemos as equações train envolvendo a decom- posição de Peirce. Quando a álgebra é de dimensão finita, resulta que a dimensão das componentes de Peirce são invariantes e o limite superior para seus nilndices são es- tudados para alguns idempotentes. Além disso, mostramos que as álgebras localmente train são álgebras train. Damos então uma descrição completa para o conjunto dos idempotentes para obter suas fórmulas explcitas. É voltada uma atenção para o caso de álgebras de Jordan, onde discutimos condições para que álgebras train de potências as- sociativas sejam álgebras de Jordan. Também mostramos que álgebras train de Jordan são de dimensão finita. Para álgebras de Bernstein de ordem n e perodo p, provamos que para termos associatividade nas potências necessitamos p = 1. Neste caso, existem 2 n1 possibilidades de equações train, que são explicitamente descritas. / We study the structure of power associative algebras which are train algebras. First we show the existence of idempotents, which are all principal and absolutely primitive. Then consider the train equations involving the Peirce decomposition. When the alge- bra is finite dimensional, it follows that the size of the Pierce components are invariant and the upper limit for its nil-indexes are studied for some idempotent. Furthermore, we show that locally train algebras are train algebras. Then we get a complete de- scription for the set of idempotents to obtain their explicit formulas. We give attention to the case of Jordan algebras, where we discuss conditions for train power associa- tive algebras be Jordan algebras. We also show that Jordan train algebras are finite dimensional. For Bernstein algebras of order n and period p, we prove that to have associativity in the powers we need p = 1. In this case, there are 2 n1 possibilities of train equations, which are explicitly described.

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.