This is an abstract of the PhD thesis On Abelian Ideals in a Borel Subalgebra of a Complex Simple Lie Algebra written by Patrick Browne under the supervision of Dr. John Burns at the Department of Mathematics, National University of Ireland, Galway and submitted in December 2008.
Let g be a complex simple Lie algebra and b a fixed Borel subalgebra of g. We construct maximal (with respect to containment) abelian ideals of b by a variety of methods each having their own merits. We also derive formulas for their dimensions. Then we give a new proof of Kostant's theorem on the dimension of an abelian ideal. Finally we apply our results to give new examples of Einstein solvmanifolds.
We now give a brief historical account of interest in this area. In 1945 A. Malcev [1] determined the commutative subgroups of maximum dimension in the semisimple complex Lie groups. The maximal dimension of these commutative subgroups coincides with the maximal dimension of a commutative subalgebra of g. The next development was Kostant's [2] paper published in 1965, where he gave a connection between Malchev's result and the maximal eigenvalue of the Laplacian acting on the exterior powers ( ∧ g ) of the adjoint representation. In 1998 Kostant reported on the results of Peterson that the number of abelian ideals in the fixed Borel subalgebra of g is 2rank (g), and this paper was the genesis of much of the recent interest in this area. In [3], Panyushev and Röhrle while studying the relationship between spherical nilpotent orbits and abelian ideals of b, constructed all maximal abelian ideals, with the aid of a computer program [4], and observed a bijection between them and the set of long simple roots. Our method does not require the use of computer calculations. Suter in [5] found the maximal dimension of a maximal abelian ideal using the affine Weyl group, in terms of certain Lie theoretic invariants and gave a uniform explanation of the one to one correspondence between the long simple roots and the maximal abelian ideals, something which Panyushev and Röhrle had asked for. In [6] Papi and Cellini gave formulas for the dimension of all maximal abelian ideals in b, similar to that of Suter. Our formulas are simpler and different in nature.We will now give a brief description the structure of this thesis. In chapter one we recall the basic definitions and notations that will be used throughout the thesis. We derive maximality and dimension formulas for abelian ideals in chapter two. Chapter three is concerned with the relationship between abelian ideals and graded Lie algebras. In Chapter four we give an alternative proof of Kostant's theorem, and finally in chapter five we produce new examples of Einstein solvmanifolds
[1] Mal'cev, A. I., Comutative subalgebras of semisimple Lie algebras, Izestiya akademii Nauk SSSR. Seriya Matematiceskaya, vol.9, 1945,pp.291-300.
[2] Kostant, B., Eigenvalues of a Laplacian and commutative Lie subalgebras, Topology, vol.3, 1965,pp.147-159.
[3] Panyushev, D. and Röhrle, G. , Spherical orbits and abelian ideals, Adv. Math., vol.159, 2001,pp.229-246.
[4] Röhrle, G., On normal abelian subgroups in parabolic groups, Annales de l'institut Fourier, vol.5, 1998,pp.1455-1482.
[5] Suter, R., Abelian ideals in a Borel subalgebra of a complex simple Lie algebra, Inventiones Mathematicae, vol.156, 2004,pp.175-221.
[6] Cellini, P. and Papi, P. , Abelian ideals of Borel subalgebras and affine Weyl groups, Adv. Math., vol.187, 2004,pp.320-361.
This is an abstract of the PhD thesis Explicit small classifying spaces for a range of finitely presented infinite groups written by Maura Clancy under the supervision of Dr Graham Ellis at the School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway and submitted in February 2009.
While a classifying space B_G exists for any group G, in reality given a group presentation, finding a productive B_G is by no means trivial. This thesis unearths explicit small classifying spaces for a range of finitely presented infinite groups and uses these spaces to deduce homological information on the groups.
In Chapter 2 we derive formulae for the second integral homology of any Artin group for which the K(\pi,1)-conjecture is known to hold, and for the third integral homology of the braid group A_n and the affine braid group \tilde{A}_n. The derivation and proofs are based on the cellular chain complex C_*(\tilde{X}_D), where \tilde{X}_D is the universal cover of a classifying space B_G for the group G \in \{A_n, \tilde{A}_n\}. Chapter 3 defines polytopal groups, actions and classifying spaces. We prove that a group G is polytopal when G is the semi-direct product of two polytopal groups N and Q. We show that \tilde{B}_N \times \tilde{B}_Q is the universal covering space of a polytopal classifying space \tilde{B}_G for G, where \tilde{B}_N (resp. \tilde{B}_Q) is the universal covering space of a polytopal classifying space B_N for N (resp. B_Q for Q). We further show that the cellular chain complex C_*(\tilde{B}_G) can be obtained as the total complex of a double complex with Dim(B_N) rows and Dim(B_Q) columns, a fact alternatively proven by Thomas Brady in his paper ``Free resolutions for semi-direct products". Chapter 4 centres on Bieberbach groups; we realise six of the ten 3-dimensional Bieberbach groups as semi-direct products G = N \rtimes_{\alpha} Q, where N is 2-dimensional Bieberbach and Q = C_\infty. This technique can be extended to determine, inductively, classifying spaces for higher dimensional Bieberbach groups. Chapter 5 introduces twisted Artin groups \mathfrak{A}_{\vec{D}} and shows that in some cases there exists a polytopal classifying space whose t-dimensional cells are indexed by the finite type subsets, of size t, of the generating set S. We show that 3-generator twisted Artin groups of large type admit a two-dimensional classifying space. Using star graph techniques we show that such classifying spaces are non-positively curved for standard Artin groups of large type. In Chapter 6 we conjecture that certain groups are quasi-lattice-ordered and then use a gap routine to experimentally investigate the word-reversing algorithm.
This thesis investigates the practicality of a method for computing group homology described in the paper "Polytopal resolutions for finite groups", by Ellis, Harris and Sköldberg. Using a mix of theoretical and computational results we show that the method is certainly practical for the low-dimensional homology of Mathieu groups, isometry groups of R3, and finite reflection groups and their even subgroups. The main new results in the thesis are:
In this thesis we give new constructions of a number of extremal type II codes. Algebraic proofs are provided that the constructions do in fact yield the codes. The codes are constructed using group rings of which the underlying groups are dihedral. Type II codes are not cyclic, but the constructions here are similar to the constructions of cyclic codes from polynomials.
The first code we construct is the extended binary Golay code. It is constructed from a zero divisor in the group ring of the finite field with two elements and the dihedral group with twenty-four elements. We create a generator matrix of the code that is in standard form and is a reverse circulant generator matrix. The generator matrix generates the code as quasi cyclic of index two.
Algebraic proofs are given of the code's minimum distance, self-duality and doubly evenness. A list of twenty-three other zero divisors that we have found to generate the code is given. Trivial changes adapt the aforementioned algebraic proofs to any of these zero divisors. We also prove that the twenty-four zero divisors are the only ones of their form that will generate the code.
Next we construct the (48,24,12) extremal type II code as a dihedral code. The new construction is similar to that of the extended Golay code. Again, proofs of the self-duality and doubly evenness are given. An algebraic proof of the minimum distance is achieved through the use of two different group ring matrices.
A number of different codes are then constructed, building on the first two constructions. We list some zero divisors that generate type II codes of lengths seventy-two and ninety-six. According to investigations by computer, these codes have minimum distances of twelve and sixteen respectively. No type II codes of each of these lengths are known that have greater respective minimum distances. Some techniques are detailed that vastly reduce the calculations involved in their analysis.
The constructions of the (72,36,12) and (96,48,16) codes are facilitated by the construction of extremal type II codes of all lengths a multiple of eight up to length forty. Type II codes only exist at lengths that are multiples of eight. Overall we have shown that extremal type II codes can be constructed in dihedral group rings at every length a multiple of eight up to and including length forty-eight, and at some lengths beyond forty-eight. We also successfully investigate the possibility to construct some type I codes in the same way.
This is the abstract of the PhD thesis The Hochschild Cohomology Ring of a Quadratic Monomial Algebra, written by David O'Keeffe under the supervision of Dr. Emil Sköldberg at the Department of Mathematics, National University of Ireland, Galway and submitted in January 2009.
Until recently, little was known about the multiplicative structure of the Hochschild cohomology ring for most associative algebras. During the last decade or so, more light has been shed on this topic, with several papers published regarding the structure of the Hochschild cohomology ring for various algebras.
In this thesis, we compute the Hochschild cohomology groups for a special class of associative algebras: the so-called class of Quadratic monomial algebras. These results are then used to compute the Hochschild cohomology ring for the above class of algebras. In doing this, we extend results obtained by Claude Cibils, where he computes the cohomology ring for the class of radical square zero algebras. The latter class of algebras form a subclass of the class of quadratic monomial algebras.
Chapter 1 consists of all the necessary background material that is referred to throughout this work. There are also some new proofs of already known results. However most of the original work is contained in Chapters 2, 3 and 4.
Chapter 2 describes the Hochschild cohomology groups of a quadratic monomial algebra in terms of the cohomology of a graph. An alternative calculation of these groups was performed by Emil Sköldberg. Also in this chapter we describe the generating elements of the cohomology algebra.
Chapter 3 builds on the calculations of the previous chapter by computing the algebra structure on the cohomology algebra. We describe the product structure on the cohomology algebra as a composition of chain maps on a projective resolution.
Finally using the algebra structure calculated in Chapter 3 and the Hilbert series of a vector space, we show in Chapter 4, the cohomology algebra of a quadratic monomial algebra exhibits all possible behaviours as an algebra: It may be
[1] Claude Cibils, Hochschild cohomology algebra of radical square zero algebras, Algebras and modules, II CMS Conf.Proc.,vol.24, Amer.Math.Soc., Providence, RI, 1998,pp.93-101.
[2] Emil Sköldberg, Some Remarks on the Hochschild cohomology of graded algebras, (preprint), available from http://maths.nuigalway.ie/~emil
In this thesis we consider three problems in the broad area of algebraic combinatorics. In the first section we consider the incidence algebras of finite graded posets. We define the incidence algebra of a poset P to be the quotient of the quiver algebra on the Hasse diagram of P with the ideal generated by the differences of paths pi,pj such that source(pi)=source(pj) and destination(pi)=destination(pj). Our main result shows that these algebras have a quadratic Gröbner basis if and only if the poset is pure and lexicographically shellable as defined in [1]. We also generalise our result to pure augmented acyclic categories also defined in [1].
In the second section we consider a symmetric variant of the Orlik-Solomon algebra. The standard Orlik-Solomon algebra [3] can be defined as a quotient of the exterior algebra. It was first introduced as the cohomology algebra of the compliment of a hyperplane arrangement where it was shown to rely only on the dependencies among the hyperplanes. This dependence structure defines a matroid and the Orlik-Solomon algebra has since been studied for arbitrary matroids. Here we introduce a symmetric variant as a quotient of a polynomial ring S=k[x1,...,xn] and discuss a number of its ring theoretical properties and homological invariants. In particular, for uniform matroids U of rank 2, we show that the algebra is Gorenstein and calculate its minimal graded Betti numbers. We also describe the reduced Gröbner basis and Hilbert series of the algebra when the matroid U has even rank. In order to achieve some of these results we also calculate the graded Betti numbers of ideals with linear quotients.
Finally, following results of Sköldberg [2], we use algebraic discrete Morse theory to construct an explicit minimal Hochschild resolution of the commutative algebra A=S/I, where I is a stable ideal; that is, for any minimal generator mxj of I with xj such that j ≥ max{i | xi divides m } then the monomial mxi ∈ I for all i < j. We start with the normalised Hochschild complex HCn:=A ⊗A+⊗n ⊗A, where A+ is the positively graded part of A. This complex is a resolution, however it is far from minimal in general. Sköldberg describes an acyclic partial matching on the basis of this complex which may be used to define a homotopy to a smaller complex. Using this we construct a homotopy equivalent complex and give an explicit description of the differential. From the differential we see that this is a minimal resolution.
[1] D. Kozlov. Combinatorial Algebraic Topology. Springer, 2008.
[2] E. Sköldberg, Morse theory from an algebraic viewpoint. Trans. Amer. Math. Soc., 358:115-129, 2006.
[3] S. A. Yuzvinsky. Orlik-Solomon algebras in algebra and topology., Russ. Math. Surv., 56:293-364, 2001.
This thesis is concerned with the design and analysis of numerical methods for some classes of singularly perturbed differential equations.
We first consider a parameterised reaction-diffusion problem, based on the Rayleigh and Orr-Sommerfeld models of hydrodynamic stability, where it is necessary to determine a function u and a parameter κ. Since κ depends on the derivative of u at a boundary, it is important to accurately estimate u in that region, and so we must resolve the layer. We develop an iterative technique that uses a finite difference method on a piecewise uniform, Shishkin mesh. An analysis of the method shows that the errors are parameter uniform and are almost second order convergent, which is supported with numerical results.
In the main body of the thesis we investigate the use of iterative overlapping Schwarz methods for coupled systems of reaction diffusion problems. We first consider the case when all the differential equations in the system have identical singular perturbation parameters ε1=ε2=...=εM=&epsilon. By appropriately choosing the overlaps of the subdomains we can ensure parameter uniform results and furthermore, when $epsilon; is sufficiently small we only require one iteration.
When the problem is generalised so that each equation may have distinct perturbation parameters the solutions have overlapping layers of different widths. Consequently we require more subdomains and, although the results are parameter uniform, the fast convergence of the iterates is lost. We overcome this problem by developing a semi-iterative algorithm.
Finally we extend the Schwarz methods to two dimensional systems and demonstrate the benefits of the domain decomposition algorithms, in particular the reduction in computational effort.
The first test to be developed for comparing non-nested models was that of Cox, introduced in 1962. Since then, few, if any such tests have been presented in mainstream statistical literature; in the econometrics literature Vuong has proposed a test that is generally applicable.
We present a detailed outline of Cox's test for non-nested models, both in its original analytic form and its simulation-based form, as proposed by Williams and Hinde. Vuong's test is also outlined, and compared to Cox's test, and shown to have no advantages over it other than speed and ease of use. The analytic and simulation-based versions of Cox's test are shown to produce similar results. We also show that whilst the estimation of p-values by both the analytic and simulation based versions of Cox's test is biased, the level of such bias is not great. We investigate the extension of simulation based Cox tests to nested models, and show that whilst this is possible, the consequent level of bias is enormous, and may not be feasibly reduced by multiple bootstrap techniques. The bias present in simulation-based Cox tests is a consequence of the composite nature of the null hypotheses: model parameters are refitted at each stage of the resampling procedure. It may be eliminated by adopting the null hypotheses to be simple by fixing the model parameters, and using these fixed parameters throughout. We present two tests based upon such simple null hypotheses: the Hybrid and Dragnet tests. The former test is essentially a simulation based Cox test where the model parameters are fixed at their maximum likelihood estimates. We show that, in relation to non-nested models, the hybrid test is more conservative than Cox's test, but only slightly so in any situation where it is likely to be used; and that as well as being non-biased, it also possess the practical advantage of being much quicker to complete. The Dragnet tests extends the concept of the hybrid test: the models are tested not only at the maximum likelihood estimates of the model parameters, but at a cross section of possible parameters, hence we obtain multiple test results, from which conclusions concerning the relative merits of the models may be drawn.
Also presented is a discussion of zero-modified models. Current statistical practice is to assume that zero-modification does not involve zero-deflation. It is shown that this practice may, in certain circumstances be unwise. It is also shown that whilst the interpretation of zero-modified and hurdle models is quite different, their fits tend to be extremely similar.