In this thesis, we present a detailed exposition of Koszul algebras and Koszul duality. We begin with an overview of the required concepts of graded algebras and homological algebra. We then give a precise treatment of Koszul and quadratic algebras, together with their dualities. We fill in some arguments that are omitted in the literature and work out a number of examples in full detail to illustrate the abstract concepts.

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In computations over many-sorted algebras, one typically encounters error cases, caused by attempting to evaluate an operation outside its domain (e.g. division by the integer 0; taking the square root of a negative integer; popping an empty stack). We present a method for systematically dealing with such error cases, namely the construction of an "error algebra" based on the original algebra. As an application of this method, we show that it provides a good semantics for (possibly improper) function tables. / Thesis / Master of Science (MSc)

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Let A be a Banach algebra over a field IF that is either the real field IR or the complex field ℂ, and let A' be its first dual space and A" its second dual space. R. Arens in 1950, gave a way of defining two Banach algebra products on A" , such that each of these products is an extension of the original product of A when A is naturally embedded in A" . These two products mayor may not coincide. Arens calls the multiplication in A regular provided these two products in A" coincide. Perhaps the first important result on the Arens second dual, due essentially to Shermann and Takeda, is that any C*-algebra is Arens regular and the second dual is again a C*-algebra. Indeed if A is identified with its universal representation then A" may be identified with the weak operator closure of A-hat. In a significant paper Civin and Yood, obtain a variety of results. They show in particular that for a locally compact Abelian group G ,Ll(G) is Arens regular if and only if G is finite. (Young showed that this last result holds for arbitrary locally compact groups.) Civin and Yood also identify certain quotient algebras of [Ll(G)]". Pak-Ken Wong proves that A-hat is an ideal in A" when A is a semi-simple annihilator algebra, and this topic has been taken up by S. Watanabe to show that [L 1 (G)]-hat is ideal in [L 1 (G) ]" if and only if G is compact and [M (G)]-hat is an ideal in [M(G)]" if and only if G is finite. One shoulu also note in this context the well known fact that if E is a reflexive Banach space with the approximation property and A is the algebra of compact operators on E, (in particular A is semi-simple annihilator algebra) then A" may be identified with BL(E). S.J. Pym [The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc., (3) 15 (1965)] has proved that A is Arens regular if and only if every linear functional on A is weakly almost periodic. A general study of those Banach algebras which are Arens regular has been done by N.J. Young and Craw and Young. But in general, results and theorems about the representations of A" are rather few. In Chapter One we investigate some relationships between the Banach algebra A and its second dual space. We also show that if A" is a C*-algebra, then * is invariant on A. In Chapter Two we analyse the relations between certain weakly compact and compact linear operators on a Banach algebra A, associated with the two Arens products defined on A". We clarify and extend some known results and give various illustrative examples. Chapter Three is concerned with the second dual of annihilator algebras. We prove in particular that the second dual of a semi-simple annihilator algebra is an annihilator algebra if and only if A is reflexive. We also describe in detail the second dual of various classes of semi-simple annihilator algebras. In Chapter Four, we particularize some of the problems in Chapters Two and Three to the Banach algebra ℓ1 (S) when S is a semigroup. We also investigate some examples of ℓ1(S) in relation to Arens regularity. Throughout we shall assume familiarity with standard Banach algebra ideas; where no definition is given in the thesis we intend the definition to be as in Bonsall and Duncan. Whenever possible we also use their notation.

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B₂. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.

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