181 |
Extension of valuations in skew fieldsKhawaja, T. January 1984 (has links)
The aim of this thesis is to study the extension of valuations in skew field extensions. In Chapter I we look at the following problem. Let K be a field and V a valuation ring of rank 1 in K. Let H be a crossed product division algebra over K. Then we study conditions under which there exists a matrix local ring R in H lying over V and generating H as K-space. We then find that R is a valuation ring in H lying over V iff R is local. Moreover if V is discrete of rank 1, then R is a maximal order in H. In Chapter II we study directly conditions under which a valuation on the centre of a finite dimensional central division algebra can be extended to the whole algebra. In particular if H = (E/K; sigma, a) is a cyclic division algebra and v is a discrete rank 1 valuation on K, then the extension of v to H depends on v(a). We then carry on the study of the extension problem for the tensor product of algebras. In particular if [special characters omitted] and V a rank 1 valuation ring in K and if there exists a valuation ring W in H lying over V with W ∩ H<sub>i</sub> = w<sub>i</sub> (i = 1,...,r), we study conditions under which [special characters omitted]. In Chapter III we look at infinite skew field extensions. We study valuations in skew function fields. The application will include among others, free algebras, universal associative envelopes of Lie algebras and generic crossed product. However our main concern in this chapter is the following question raised by P.M. Cohn. Let K<sub>1</sub>, K<sub>2</sub> be two skew fields with a common subfield K and let v<sub>1</sub>, v<sub>2</sub> be real valued valuations on K<sub> 1</sub> and K<sub>2</sub> respectively such that v<sub>1</sub>|K = v<sub> 2</sub>|K = v. Do v<sub>1</sub>, v<sub>2</sub> have a common extension to H = K<sub> 1</sub> O<sub>K</sub> K<sub>2</sub> (the field coproduct of K<sub>1</sub> and K<sub>2</sub>)? We show that in general the answer is no. Nevertheless we find conditions under which v<sub>1</sub>,v<sup>2</sup> have a common extension to H.
|
182 |
Extremal problems in combinatorial semigroup theoryMitchell, James David January 2002 (has links)
In this thesis we shall consider three types of extremal problems (i.e. problems involving maxima and minima) concerning semigroups. In the first chapter we show how to construct a minimal semigroup presentation that defines a group of non-negative deficiency given a minimal group presentation for that group. This demonstrates that the semigroup deficiency of a group of non-negative deficiency is equal to the group deficiency of that group. Given a finite monoid we find a necessary and sufficient condition for the monoid deficiency to equal the semigroup deficiency. We give a class of infinite monoids for which this equality also holds. The second type of problem we consider concerns infinite semigroups of relations and transformations. We find the relative rank of the full transformation semigroup, over an infinite set, modulo some standard subsets and subsemigroups, including the set of contraction maps and the set of order preserving maps (for some infinite ordered sets). We also find the relative rank of the semigroup of all binary relations (over an infinite set) modulo the partial transformation semigroup, the full transformation semigroup, the symmetric inverse semigroup, the symmetric group and the set of idempotent relations. Analogous results are also proven for the symmetric inverse semigroup. The third, and final, type of problem studied concerns generalising notions of independence from linear algebra to semigroups and groups. We determine the maximum cardinality of an independent set in finite abelian groups, Brandt semigroups, free nilpotent semigroups, and some examples of infinite groups.
|
183 |
The conjugacy problem in groups which are free products with an amalgamated subgroupDalton, Frederick William January 1975 (has links)
No description available.
|
184 |
Deformations of Cayley submanifoldsOhst, Matthias January 2016 (has links)
Cayley submanifolds of R^8 were introduced by Harvey and Lawson as an instance of calibrated submanifolds, extending the volume-minimising properties of complex submanifolds in Kähler manifolds. More generally, Cayley submanifolds are 4-dimensional submanifolds which may be defined in an 8-manifold M equipped with a certain differential 4-form Phi invariant at each point under the spin representation of Spin(7). If this 4-form Phi is closed, then the holonomy of M is contained in Spin(7) and Cayley submanifolds are calibrated minimal submanifolds. McLean studied the deformations of closed Cayley submanifolds. The deformation problem is elliptic but in general obstructed. We show that for a generic choice of Spin(7)-structure, there are no obstructions, and hence the moduli space is a finite-dimensional smooth manifold. Then we study the deformations of compact, connected Cayley submanifolds with non-empty boundary contained in a given submanifold W of M, where we require that the Cayley submanifolds meet the submanifold W orthogonally. We show that for a generic choice of Spin(7)-structure, the Cayley submanifolds are rigid. Moreover, we show that also for a generic choice of the submanifold W, the Cayley submanifolds are rigid. We further discuss some examples for this deformation theory. In addition, we study the deformations of asymptotically cylindrical Cayley submanifolds inside asymptotically cylindrical Spin(7)-manifolds. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that for a generic choice of Spin(7)-structure, there are no obstructions, and hence the moduli space is a finite-dimensional smooth manifold whose dimension is equal to the index of the operator of Dirac type. We further construct examples of asymptotically cylindrical Cayley submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds with holonomy Spin(7) constructed by Kovalev.
|
185 |
The finite dual of crossed productsJahn, Astrid January 2015 (has links)
In finite dimensions, Hopf algebras have a very nice duality theory, as the vector space dual of a finite-dimensional Hopf algebra is also a Hopf algebra in a canonical way. This breaks down in the infinite-dimensional setting, as here the dual need not be a Hopf algebra. Instead, one chooses a subalgebra of the vector space dual called the finite dual. This subalgebra is always canonically a Hopf algebra. In this thesis, we aim to better understand the finite dual by trying to understand how the finite dual of a crossed product relates to the finite duals of its components. We start by investigating what the assignment sending a Hopf algebra to its finite dual does to functions. Unlike in the finite-dimensional case, this is no longer a contravariant exact monoidal functor and might not even be a functor at all. However, many of the results true thanks to this in finite dimensions still always hold, while we can find necessary and sufficient conditions for others to hold as well as specific situations in which they are always true. Crossed products generalise the notion of a smash product, which can be viewed as the Hopf algebra equivalent of the semidirect product. Many Hopf algebras of interest can be written as crossed products. We study the finite dual of such a product and find numerous results when assuming conditions such as one of the components being finite-dimensional or the crossed product being a smash product. These can be combined for strong statements about the finite dual under certain assumptions. Finally, we consider Noetherian Hopf algebras which are finite modules over central Hopf subalgebras. Many of these algebras decompose as crossed products, so that we can use our previous results to study them. However, we also find results that are true without assuming such a decomposition. This allows us to calculate the finite duals of numerous examples, including a quantised enveloping algebra at a root of unity and all the prime affine regular Hopf algebras of Gelfand-Kirillov dimension one with prime PI degree.
|
186 |
Topics regarding close operator algebrasDickson, Liam January 2014 (has links)
In this thesis we focus on two topics. For the first we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of B(H). By showing C*-algebras have row length (in the sense of Pisier) of at most two we show that the row metric is equivalent to the original Kadison- Kastler metric. We then use this result to obtain universal constants for a recent perturbation result of Ino and Watatani, which states that succiently close intermediate subalgebras must occur as small unitary perturbations, by removing the dependence on the structure of inclusion. Roydor has recently proved that injective von Neumann algebras are Kadison-Kastler stable in a non-self adjoint sense, extending seminal results of Christensen. We prove a one-sided version, showing that an injective von Neumann algebra which is nearly contained in a weak*-closed non-self adjoint algebra can be embedded by a similarity close to the natural inclusion map. This theorem can then be used to extend results of Cameron et al. by demonstrating Kadison-Kastler stability of certain crossed products in the non self-adjoint setting. These crossed products can be chosen to be non-amenable.
|
187 |
The outer automorphism groups of three classes of groupsLogan, Alan D. January 2014 (has links)
The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the first direction we start with a class of groups and then classify their outer automorphism groups. In the other direction we start with a broad class of groups, for example finitely generated groups, and for each group Q in this class we construct a group G_Q such that Q is related, in a suitable sense, to the outer automorphism group of G_Q. We give a list of 14 groups which precisely classifies the outer automorphism groups of one-ended two-generator, one-relator groups with torsion. We also describe the outer automorphism groups of such groups which have more than one end. Combined with recent algorithmic results of Dahmani–Guirardel, this work yields an algorithm to compute the outer automorphism group of a two-generator, one-relator group with torsion. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a particular kind of HNN-extension. We apply this to prove two main results. These results demonstrate a universal property of triangle groups and are as follows. Fix an arbitrary hyperbolic triangle group H. If Q is a finitely generated group then there exists an HNN-extension G_Q of H such that Q embeds with finite index into the outer automorphism group of G_Q. Moreover, if Q is residually finite then G_Q can be taken to be residually finite. Secondly, fix an equilateral triangle group H = ⟨a, b; a^i, bi, (ab)^i⟩ with i > 9 arbitrary. If Q is a countable group then there exists an HNN-extension G_Q of H such that Q is isomorphic to the outer automorphism group of G_Q. The proof of this second main result applies a theory of Wise underlying his recent work leading to the resolution of the virtually fibering and virtually Haken conjectures. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a semi-direct product H.
|
188 |
Cayley automaton semigroupsMcLeman, Alexander Lewis Andrew January 2015 (has links)
Let S be a semigroup, C(S) the automaton constructed from the right Cayley graph of S with respect to all of S as the generating set and ∑(C(S)) the automaton semigroup constructed from C(S). Such semigroups are termed Cayley automaton semigroups. For a given semigroup S we aim to establish connections between S and ∑(C(S)). For a finite monogenic semigroup S with a non-trivial cyclic subgroup C[sub]n we show that ∑(C(S)) is a small extension of a free semigroup of rank n, and that in the case of a trivial subgroup ∑(C(S)) is finite. The notion of invariance is considered and we examine those semigroups S satisfying S ≅ ∑(C(S)). We classify which bands satisfy this, showing that they are those bands with faithful left-regular representations, but exhibit examples outwith this classification. In doing so we answer an open problem of Cain. Following this, we consider iterations of the construction and show that for any n there exists a semigroup where we can iterate the construction n times before reaching a semigroup satisfying S ≅ ∑(C(S)). We also give an example of a semigroup where repeated iteration never produces a semigroup satisfying S ≅ ∑(C(S)). Cayley automaton semigroups of infinite semigroups are also considered and we generalise and extend a result of Silva and Steinberg to cancellative semigroups. We also construct the Cayley automaton semigroup of the bicyclic monoid, showing in particular that it is not finitely generated.
|
189 |
Some new classes of division algebras and potential applications to space-time block codingSteele, Andrew January 2014 (has links)
In this thesis we study some new classes of nonassociative division algebras. First we introduce a generalisation of both associative cyclic algebras and of Waterhouse's nonassociative quaternions. An important aspect of these algebras is the simplicity of their construction, which is a modification of the classical definition of associative cyclic algebras. By taking the parameter used in the classical definition from a larger field, we lose the property of associativity but gain many new examples of division algebras. This idea is also applied to obtain a generalisation of the first Tits construction. We go on to study constructions of Menichetti, Knuth, and Hughes and Kleinfeld, which have previously only been considered over finite fields. We extend these definitions to infinite fields and get new examples of division algebras, including some over the real numbers. Recently, both associative and nonassociative division algebras have been applied to the theory of space-time block coding. We explore this connection and show how the algebras studied in this thesis can be used to construct space-time block codes.
|
190 |
Aspects of abstract and profinite group theoryHardy, Philip David January 2002 (has links)
This thesis is in two parts. The first is a study of soluble profinite groups with the maximal condition (max) and mirrors the development of the fundamental properties of polycyclic groups. Two of the main results are a profinite version of Hall's criterion for nilpotence and the existence of nilpotent almost-supplements for the Fitting subgroup. It is shown that the automorphism group of a soluble profinite group with max itself has max and that a soluble profinite group has max if and only if each of its subnormal subgroups of defect at most 2 has max. Quantitative versions of these results are also obtained. The second part derives a new characterization of abstract and profinite branch groups. This is achieved by examining the subnormal subgroup structure of just non-(virtually abelian) groups and a partial classification of this larger class is obtained. Weak branch groups are shown to satisfy no abstract group laws and to contain many abstract free subgroups in the profinite case. The notion of a branch group is weakened to that of a generalized branch group. Associated with each generalized branch group is its structure graph, and the circumstances under which this graph is a tree are characterized.
|
Page generated in 0.0441 seconds