This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

Ferrets are amusing, flexible creatures that have been under represented in computer models. Because their bodies can assume almost any curved shape, splines are the natural tool for modelling ferrets. Surface pasting is a hierarchical method of modelling with spline surfaces, where features are added onto a base surface. Existing surface pasting techniques are limited to modelling rectilinear shapes. Using the task of modelling a ferret as a driving force, I propose a method of pasting cylinders in world space; I looked at methods for reducing distortion of pasted features; and I created a method for pasting trimmed features to allow for features that do not have the rectilinear shape of standard pasting. With my methods, modelling ferrets with surface pasting is easier, and the resulting models are closer to a real ferret.

Ferrets are amusing, flexible creatures that have been under represented in computer models. Because their bodies can assume almost any curved shape, splines are the natural tool for modelling ferrets. Surface pasting is a hierarchical method of modelling with spline surfaces, where features are added onto a base surface. Existing surface pasting techniques are limited to modelling rectilinear shapes. Using the task of modelling a ferret as a driving force, I propose a method of pasting cylinders in world space; I looked at methods for reducing distortion of pasted features; and I created a method for pasting trimmed features to allow for features that do not have the rectilinear shape of standard pasting. With my methods, modelling ferrets with surface pasting is easier, and the resulting models are closer to a real ferret.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.

The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

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An n × n matrix factorization of a polynomial f is a pair of n × n matrices (P, Q) such that PQ = f In, where In is the n × n identity matrix. In this dissertation, we study matrix factorizations of an arbitrary element in a given unital ring. This study is motivated on the one hand by the construction of the unit object in the bicategory LGK of Landau-Ginzburg models (of great utility in quantum physics) whose 1−cells are matrix factorizations of polynomials over a commutative ring K, and on the other hand by the existing tensor product of matrix factorizations b⊗. We observe that the pair of n × n matrices that appear in the matrix factorization of an element in a unital ring is not unique. Next, we propose a new operation on matrix factorizations denoted e⊗ which is such that if X is a matrix factorization of an element f in a unital ring (e.g. the power series ring K[[x1, ..., xr]] f) and Y is a matrix factorization of an element g in a unital ring (e.g. g ∈ K[[y1, ..., ys]]), then Xe⊗Y is a matrix factorization of f g in a certain unital ring (e.g. in case f ∈ K[[x1, ..., xr]] and g ∈ K[[y1, ..., ys]], then f g ∈ K[[x1, ..., xr , y1, ..., ys]]). e⊗ is called the multiplicative tensor product of X and Y. After proving that this product is bifunctorial, many of its properties are also stated and proved. Furthermore, if MF(1) denotes the category of matrix factorizations of the constant power series 1, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of (MF(1),e⊗) which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that (MF(1),e⊗) is an example of this concept. Furthermore, we define a summand-reducible polynomial to be one that can be written in the form f = t1 + · · · + ts + g11 · · · g1m1 + · · · + gl1 · · · glml under some specified conditions where each tk is a monomial and each gji is a sum of monomials. We then use b⊗ and e⊗ to improve the standard method for matrix factorization of polynomials on this class and we prove that if pji is the number of monomials in gji, then there is an improved version of the standard method for factoring f which produces factorizations of size 2 Qm1 i=1 p1i+···+ Qml i=1 pli−( Pm1 i=1 p1i+···+ Pml i=1 pli) times smaller than the size one would normally obtain with the standard method. Moreover, details are given to elucidate the intricate construction of the unit object of LGK. Thereafter, a proof of the naturality of the right and left unit maps of LGK with respect to 2−morphisms is presented. We also prove that there is no direct inverse for these (right and left) unit maps, thereby justifying the fact that their inverses are found only up to homotopy. Finally, some properties of matrix factorizations are exploited to state and prove a necessary condition to obtain a Morita context in LGK.

On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis. The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product. We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ãn and the double loop quiver with relations βα=αβ=αn=βn=0.