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Dissertations / Theses on the topic 'Algebra'

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Published: 4 June 2021

Last updated: 31 July 2024

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1

Alghamdi, Mohamed A. M. A. "Some problems in algebraic topology : polynomial algebras over the Steenrod algebra." Thesis, University of Aberdeen, 1991. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=166808.

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We prove two theorems concerning the action of the Steenrod algebra in cohomology and homology. (i) Let A denote a finitely generated graded F_p polynomial algebra over the Steenrod algebra whose generators have dimensions not divisible by p. The possible sets of dimensions of the generators for such A are known. It was conjectured that if we replaced the polynomial algebra A by a polynomial algebra truncated at some height greater than p over the Steenrod algebras, the sets of all possible dimensions would coincide with the former list. We show that the conjecture is false. For example F₁₁[x₆,x₁₀]¹² truncated at height 12 supports an action of the Steenrod algebra but F₁₁[x₆,x₁₀] does not. (ii) Let V be an elementary abelian 2-group of rank 3. The problem of determining a minimal set of generators for H*(BV,F₂) over the Steenrod algebra was an unresolved problem for many years. (A solution was announced by Kameko in June 1990, but is not yet published.) A dual problem is to determine the subring M of the Pontrjagin ring H*(BV,F₂). We determine this ring completely and in particular give a verification that the minimum number of generators needed in each dimension in cohomology is as announced by Kameko, but by using completely different techniques. Let v ε V - (0) and denote by a_5(v) ε H*(BV,F₂) the image of the non-zero class in H_2s-1(RP^∞,F₂) imeq F₂ under the homomorphism induced by the inclusion of F_{2 → V onto (0,v). We show that M is isomorphic to the ring generated by (a}_s(v),s ≥ 1, v ε V - (0)) except in dimensions of the form 2^r+3 + 2^r+1 + 2^r - 3, r ≥ 0, where we need to adjoin our additional generator.

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2

Rigby, Simon W. "Steinberg algebra and Leavitt path algebras." Master's thesis, University of Cape Town, 2018. http://hdl.handle.net/11427/29433.

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Leavitt path algebras are a new and exciting subject in noncommutative ring theory. To each directed graph E, and unital commutative ring R, is associated an R-algebra called the Leavitt path algebra of E with coefficients in R. It was discovered, when the theory of Leavitt path algebras was already quite advanced, that some of the more difficult questions were susceptible to a new approach using topological groupoids. Taking a special kind of groupoid G, one can construct an R-algebra called the Steinberg algebra of G. Many interesting classes of algebras, including Leavitt path algebras, can be obtained from this process. This dissertation is an exposition of the recent advances achieved by the groupoid approach to Leavitt path algebras. New proofs are presented to show that the boundary path groupoid (which underlies the Steinberg algebra model for Leavitt path algebras) has the necessary topological properties. A new theorem is presented, characterising strongly graded Leavitt path algebras in graphical terms. We show that the main results on the structure theory of Leavitt path algebras, including the simplicity and primitivity theorems, can be recovered using the groupoid approach. We demonstrate how these methods lead to an explicit description of the centre of a Leavitt path algebra.

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Galbucci, Elena. "Confronto tra algebre monounarie e algebra universale." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2011. http://amslaurea.unibo.it/1835/.

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4

Cid, Ruiz Yairon. "Blow-up algebras in Algebra, Geometry and Combinatorics." Doctoral thesis, Universitat de Barcelona, 2019. http://hdl.handle.net/10803/667768.

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The primary topic of this thesis lies at the crossroads of Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics. It is mainly focused around the following themes: 1) Defining equations of blow-up algebras; 2) Study of rational maps via blow-up algebras; and 3) Asymptotic properties of the powers of edge ideals of graphs. We are primarily interested in questions that arise in geometrical or combinatorial contexts and try to understand how their possible answers manifest in various algebraic structures or invariants. There is a particular algebraic object, the Rees algebra (or blow-up algebra), that appears in many constructions of Commutative Algebra, Algebraic Geometry, Geometric Modeling, Computer Aided Geometric Design and Combinatorics. The work horse and main topic of this doctoral dissertation has been the study of this algebra under various situations. The Rees algebra was introduced in the field of Commutative Algebra in the famous paper published in 1958. Since then it has become a central and fundamental object with numerous applications. The study of this algebra has been so fruitful that it is difficult to single out particular results or papers. From a geometrical point of view, the Rees algebra corresponds with the bi-homogeneous coordinate ring of two fundamental objects: the blow-up of a projective variety along a subvariety and the graph of a rational map between projective varieties. Therefore, the importance of finding the defining equations of the Rees algebra is probably beyond argument. This is a problem of tall order that has occupied commutative algebraists and algebraic geometers, and despite an extensive effort, it remains open even in the case of polynomial rings in two variables. In Chapter 2 of this dissertation, we use the theory of D-module s to describe the defining ideal of the Rees algebra in the case of a parametrization of a plane curve. In a joint work with Buse and D'Andrea, Chapter 3 of this dissertation , we introduce a new algebra that we call the saturated special fiber ring, which turns out to be an important tool to analyze the degree of a rational map. Later, in Chapter 4 of this dissertation, we compute the multiplicity of this new algebra in the case of perfect ideals of height two, which, in particular, provides an effective method to determine the degree of a rational map having those ideals as base ideal. In a joint work with Simis, Chapte r 5 of this dissertation, we consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The Rees algebra of the edge ideal of a graph is a well studied object, that relates combinatorial properties of a graph with algebraic in variants of the powers of its edge ideal. For the Rees algebra of the edge ideal of a bipartite graph, Chapter 6 of this dissertation, we compute the universal Grobner basis of its defining equations and its total Castelnuovo-Mumford regularity as a bigraded algebra. It is a celeb rated result that the regularity of the powers of a homogeneous ideal is asymptotically a linear function. Considerable efforts have been put forth to understand the form of this asymptotic linear function in the case of edge ideals. In a joint work with Jafari, Picone and Nemati, Chapter 7 of this dissertation, for bicyclic graphs, i.e. graphs containing exactly two cycles, we characterize the regularity of its edge ideal in terms of the induced matching number and determine the previous asymptotic linear function in special cases.

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5

Hadi, Shazia. "Applications of the Pauli algebra and other geometric algebras." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ52560.pdf.

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6

Banjo, Elizabeth. "Representation theory of algebras related to the partition algebra." Thesis, City University London, 2013. http://openaccess.city.ac.uk/2360/.

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The main objective of this thesis is to determine the complex generic representation theory of the Juyumaya algebra. We do this by showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by Martin (the representation theory of which is computable by a combination of classical and category theoretic techniques). We then use this result and general arguments of Cline, Parshall and Scott to prove that the Juyumaya algebra En(x) over the complex field is generically semisimple for all n 2 N. The theoretical background which will facilitate an understanding of the construction process is developed in suitable detail. We also review a result of Martin on the representation theory of the small ramified partition algebra, and fill in some gaps in the proof of this result by providing proofs to results leading to it.

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7

Hmaida, Mufida Mohamed A. "Representation theory of algebras related to the bubble algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15987/.

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In this thesis we study several algebras which are related to the bubble algebra, including the bubble algebra itself. We introduce a new class of multi-parameter algebras, called the multi-colour partition algebra $ P_{n,m} ( \breve{\delta} )$, which is a generalization of both the partition algebra and the bubble algebra. We also define the bubble algebra and the multi-colour symmetric groupoid algebra as sub-algebras of the algebra $ P_{n,m} ( \breve{\delta} ) $. We investigate the representation theory of the multi-colour symmetric groupoid algebra $ \F S_{n,m} $. We show that $ \F S_{n,m} $ is a cellular algebra and it is isomorphic to the generalized symmetric group algebra $ \F \mathbb{Z}_m \wr S_n $ when $ m $ is invertible and $ \F $ is an algebraically closed field. We then prove that the algebra $ P_{n,m} ( \breve{\delta} ) $ is also a cellular algebra and define its cell modules. We are therefore able to classify all the irreducible modules of the algebra $ P_{n,m} ( \breve{\delta} ) $. We also study the semi-simplicity of the algebra $ P_{n,m} ( \breve{\delta} ) $ and the restriction rules of the cell modules to lower rank $ n $ over the complex field. The main objective of this thesis is to solve some open problems in the representation theory of the bubble algebra $ T_{n,m} ( \breve{\delta} ) $. The algebra $ T_{n,m} ( \breve{\delta} ) $ is known to be cellular. We use many results on the representation theory of the Temperley-Lieb algebra to compute bases of the radicals of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $ over an arbitrary field. We then restrict our attention to study representations of $ T_{n,m} ( \breve{\delta} ) $ over the complex field, and we determine the entire Loewy structure of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $. In particular, the main theorem is Theorem 5.41.

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8

Bertram, Wolfgang. "The geometry of Jordan and Lie structures /." Berlin [u.a.] : Springer, 2000. http://www.loc.gov/catdir/enhancements/fy0816/00066150-d.html.

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9

Pawloski, Robert Michael. "Computing the Cohomology Ring and Ext-Algebra of Group Algebras." Diss., The University of Arizona, 2006. http://hdl.handle.net/10150/194298.

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This dissertation describes an algorithm and its implementation in the computer algebra system GAP for constructing the cohomology ring and Ext-algebra for certain group algebras kG. We compute in the Morita equivalent basic algebra B of kG and obtain the cohomology ring and Ext-algebra for the group algebra kG up to isomorphism. As this work is from a computational point of view, we consider the cohomology ring and Ext-algebra via projective resolutions.There are two main methods for computing projective resolutions. One method uses linear algebra and the other method uses noncommutative Grobner basis theory. Both methods are implemented in GAP and results in terms of timings are given. To use the noncommutative Grobner basis theory, we have implemented and designed an alternative algorithm to the Buchberger algorithm when given a finite dimensional algebra in terms of a basis consisting of monomials in the generators of the algebra and action of generators on the basis.The group algebras we are mainly concerned with here are for simple groups in characteristic dividing the order of the group. We have computed the Ext-algebra and cohomology ring for a variety of simple groups to a given degree and have thus added many more examples to the few that have thus far been computed.

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10

Linckelmann, Markus. "Variations sur les blocs a groupes de defaut cycliques." Paris 7, 1988. http://www.theses.fr/1988PA077209.

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Le present travail a pour but d'une part de donner une presentatgion complete des resultats connus sur les blocs a groupes de defaut cycliques, en traitant d'abord les proprietes qui ne dependent que de l'algebre du bloc en tant qu'algebre abstraite et ensuite les proprietes qui proviennent de la presence d'un groupe, et d'autre part d'exposer quelques resultats nouveaux concernant la structure de l'algebre de source et une description de l'algebre du bloc en tant que algebre commutante des suites exactes de green associes au bloc

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11

Razafimanantsoa, Gérard. "La k-nilpotence minimale dans les boucles de moufang commutatives : classification partielle des applications trilineaires alternees d'un espace vectoriel." Toulouse 3, 1988. http://www.theses.fr/1988TOU30161.

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Cette these concerne les boucles de mougang commutatives (bmc) et la classification des applications trilineaires alternees (ata). On etudie d'abord les bmc "k-nilpotentes minimales" i. E. Les bmc k-nilpotentes dont tout quotient propre est de classe (k-1) au plus. Ce sont des 3-boucles et celles qui sont d'ordre minimum s::(k) sont d'exposant 3 ou 9. On montre que s::(5)=3**(77). Les trois bmc d'exposant 3, de classe 3 et d'ordre s::(3)=3**(8) sont construites explicitement. D'autre part, soit v et w deux espaces vectoriels sur un corps commutatif k avec dim(v)=n. L'ensemble at(n,m,k) des ata des v**(3) dans w dont l'image est de rang m est muni de la double action naturelle des groupes lineaires gl(v) et gl(w). On considere le probleme du decompte des orbites. Un theoreme de correspondance est etabli, montrant que, pour m=1,2,3, le nombre o(n,m) des orbites de at(n,m,gf(3)) peut etre vu comme le nombre total de bmc d'exposant 3 et de classe 2 deux a deux non isomorphes dont l'ordre et le rang sont respectivement n et 3**((n+m)). On utilise un programme pascal pour etablir que o(5,2)=6. On en deduit qu'il existe 13 bmc d'exposant 3 d'ordre 3**(7)

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12

Barrat, Pierre. "Sur la dimension de la cohomologie parabolique de sous-groupes arithmetiques de sl(3)." Paris 6, 1987. http://www.theses.fr/1987PA066250.

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On utilise la correspondance de gelbart-jacquet entre representations de gl(2) et de gl(3) pour minorer la dimension de la cohomologie parabolique de certains sous-groupes arithmetiques de sl(3) explicitement decrits, en examinant les conducteurs des representations admissibles irreductibles generiques de gl(n)

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13

Angermund, Sebastian. "Algebra och linjär algebra tillämpat i kombinatorik." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210834.

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Detta arbete i kombinatorik undersöker i första hand storlekar av delmängder till vektorrum Fn, därF är en ändlig mängd av q element. Arbetet börjar med att definiera begreppen kropp och linjärarum och visar speciellt att Fn är ett vektorrum. Sedan visas hur speciella mängder som behandlasi frågeställningarna kan ses både som delmängder till någon ändlig mängd f1; 2; 3; :::; ng och somvektorer i Fn.Resultaten som erhålls är delvis begränsningar och delvis explicita storlekar av vektormängderi Fn. Det visas att delmängder till Fn2bestående av mängder med udda storlek och parvis jämnasnitt är linjärt oberoende och har därmed storlek n. Sedan undersöks även variationen av dettaproblem då endast mängder av jämn storlek tillåts, vilka visas kunna anta storleksordningar avåtminstone 2bn=2c. Till slut visas en generalisering från Fn2till Fnq, speciellt generaliseras arbetetshuvudsakliga frågeställning. Slutligen gås ett arbete [3] av Z. Dvir igenom, i vilket han bestämmeren undre gräns till Kakeyamängder K Fnq.i

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14

Schoenecker, Kevin J. "An infinite family of anticommutative algebras with a cubic form." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1187185559.

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15

Ronagh, Pooya. "The inertia operator and Hall algebra of algebraic stacks." Thesis, University of British Columbia, 2016. http://hdl.handle.net/2429/58120.

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We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line q = [A¹]. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks. Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting polynomials in q) and their corresponding eigenvalue decompositions are used to define a graded structure on the Grothendieck ring. We then define the structure of a Hall algebra on the space of stack functions. The commutative and non-commutative products of the Hall algebra respect the graded structure defined above. Moreover, the two multiplications coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. Finally, for any algebroid, an ε-element is defined and shown to be contained in the space of virtually indecomposables. This is a new approach to the theory of generalized Donaldson-Thomas invariants.
Science, Faculty of
Mathematics, Department of
Graduate

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16

Tam, King-lam, and 譚敬霖. "Using computer algebra system in teaching secondary algebra." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B29954472.

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17

Jöllenbeck, Michael. "Algebraic discrete Morse theory and applications to commutative algebra (Algebraische diskrete Morse-Theorie und Anwendungen in der kommutativen Algebra) /." [S.l. : s.n.], 2005. http://archiv.ub.uni-marburg.de/diss/z2005/0108/.

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18

Phan, Christopher Lee 1980. "Koszul and generalized Koszul properties for noncommutative graded algebras." Thesis, University of Oregon, 2009. http://hdl.handle.net/1794/10367.

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xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and co-authored materials.
Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics

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19

Rife, Susan A. "Matrix algebra." Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1996. http://handle.dtic.mil/100.2/ADA316035.

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Delatorre, Anthony R., and William K. Cooke. "Matrix algebra." Thesis, Monterey, California. Naval Postgraduate School, 1998. http://hdl.handle.net/10945/8658.

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Approved for public release; Distribution is unlimited
This thesis is designed to act as an instructor's supplement for refresher matrix algebra courses at the Naval Postgraduate School (NPS). The need for a beginning matrix algebra supplement is driven by the unique circumstances of most NPS students. Most military students attend XPS several years after receiving their undergraduate degrees. This supplement, unlike most college textbooks, bridges the gap between the student's educational lay-off and the rigors of mathematically oriented degrees such as applied math, operations research and engineering. By reviewing the fundamental concepts of vectors and matrices, and performing basic operations with them, the student quickly develops the background needed in NPS's demanding curriculums. This supplement focuses on matrix and vector operations, linear transformations, systems of linear equations, and computational techniques for solving systems of linear equations. The goal is to enhance current matrix algebra textbooks and help the beginning student build a foundation for higher level engineering and mathematics based courses.

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21

Lobo, Francisco. "Multiversal algebra." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/multiversal-algebra(9d3f8fa5-a4e7-4ea9-abb4-9688a5812e0a).html.

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This thesis discusses two ideas, multiversal algebra and algebraic enrichment, and one potential application for the latter, sequential scheduling. Multiversal algebra is a proposal for the reconsideration of semigroupoid and category theory within a framework that extends the approach of universal algebra. The idea is to introduce the notion of algebraic operation relative to a given binary relation, as an alternative to the notion of operation on a carrier class. It is shown that for a particular class of relations the derived notion of category coincides with that of standard category theory. Algebraic enrichment is the name given to a series of similar constructions translating between external and internal algebraic structure, which are studied as a ﬁrst step towards generalizing the seminal results of Eckmann and Hilton, and for the application to sequential scheduling. This well-known combinatorial engine of game semantics is shown to form part of a double semigroupoid, and this new algebraic perspective on scheduling oﬀers a new direction for the study of game models and their innocence condition.

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Lindberg, Lars. "Doubled quadratic division algebras /." Uppsala, 2004. http://www.math.uu.se/research/pub/Lindberg3lic.pdf.

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23

Pierce, Robyn U. "An exploration of algebraic insight and effective use of computer algebra systems /." Connect to thesis, 2001. http://eprints.unimelb.edu.au/archive/00000739.

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24

Kronewitter, Frank Dell. "Noncommutative computer algebra in linear algebra and control theory /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2000. http://wwwlib.umi.com/cr/ucsd/fullcit?p9963663.

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Levandovskyy, Viktor. "Non-commutative computer algebra for polynomial algebras Gröbner bases, applications and implementation /." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=976594358.

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Yang, Qunfeng. "Some graded Lie algebra structures associated with Lie algebras and Lie algebroids." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0007/NQ41350.pdf.

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27

Yoshii, Yoji. "The coordinate algebra of extended affine Lie algebras of type A(1)." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0015/NQ46556.pdf.

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Ahmed, Chwas Abas. "Representation theory of diagram algebras : subalgebras and generalisations of the partition algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15997/.

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This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.

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Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368142.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

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Burton, Cynthia L. "Hopf algebras and Dieudonné modules /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/5808.

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Wilcox, Stewart. "Cellularity of Twisted Semigroup Algebras of Regular Semigroups." Thesis, The University of Sydney, 2005. http://hdl.handle.net/2123/720.

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There has been much interest in algebras which have a basis consisting of diagrams, which are multiplied in some natural diagrammatic way. Examples of these so-called diagram algebras include the partition, Brauer and Temperley-Lieb algebras. These three examples all have the property that the product of two diagram basis elements is always a scalar multiple of another basis element. Motivated by this observation, we find that these algebras are examples of twisted semigroup algebras. Such algebras are an obvious extension of twisted group algebras, which arise naturally in various contexts; examples include the complex numbers and the quaternions, considered as algebras over the real numbers. The concept of a cellular algebra was introduced in a famous paper of Graham and Lehrer; an algebra is called cellular if it has a basis of a certain form, in which case the general theory of cellular algebras allows us to easily derive information about the semisimplicity of the algebra and about its representation theory, even in the non-semisimple case. Many diagram algebras (including the above three examples) are known to be cellular. The aim of this thesis is to deduce the cellularity of these examples (and others) by proving a general result about the cellularity of twisted semigroup algebras. This will extend a recent result of East. In Chapters 2 and 3 we discuss semigroup theory and twisted semigroup algebras, and realise the above three examples as twisted semigroup algebras. Chapters 4 to 7 detail and extend slightly the theory of cellular algebras. In Chapter 8 we state and prove the main theorem, which shows that certain twisted semigroup algebras are cellular. Under the assumptions of the main theorem, we explore the cell representations of twisted semigroup algebras in Chapter 9. Finally in Chapter 10, we apply the theorem to various examples, including the three diagram algebras mentioned above.

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Bengtsson, Svetlana. "Vad algebra är bra till? : Några gymnasieelevers förståelse av algebra." Thesis, Högskolan Kristianstad, Sektionen för lärande och miljö, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:hkr:diva-16833.

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Enligt mina erfarenheter upplever många elever i skolan algebra som svårt. De uppfattar ofta algebraisk räkning som abstrakt och obegriplig och ser ingen mening med att lära sig det. Det är vanligt att eleverna så sent som på gymnasiet uppvisar svårigheter med variabelbegreppet och hur det används. Detta har gjort mig nyfiken. Vad är det som eleverna inte förstår och varför? Med min undersökning ville jag få en uppfattning om varför algebra är så svårt för många elever och vad man kan göra för att underlätta algebrainlärningen för dem. Under min tid som gymnasielärare har jag mött flest elever i matematiksvårigheter på de olika yrkesförberedande programmen. Jag antog att de eleverna har sämre algebrakunskaper än de övriga eleverna på gymnasiet. Därför fokuserade jag mig i min studie på en grupp elever som går på ett av gymnasieskolans yrkesförberedande program. Syftet med mitt examensarbete var att undersöka vilka svårigheter elever som går första året på gymnasieskolans yrkesprogram har när de arbetar med att formulera algebraiska uttryck och formler samt vad svårigheterna beror på. För att kunna besvara mina frågeställningar har jag intervjuat sex elever i grupp om två och två medan de löste några utvalda uppgifter. Resultatet visade att elever som ingick i studien hade svårt med att uppfatta generella uttryck vilket kan bero på bl.a. att eleverna inte har utvecklat tillräckligt hög abstraktionsnivån för att kunna hantera den symboliska algebran. För att hjälpa eleverna till att bättre förstå meningen med detta ställde jag mig en fråga om man genom att betona uppgifternas underliggande aritmetiska strukturer kan få en bättre ingång till algebran. Undersökningen som jag gjorde tydde på att denna åtgärd hjälpte eleverna att få en naturlig övergång till att använda bokstäver i generella uttryck samt till en bättre förståelse för vad algebra är bra för.

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Andres, Wolf Daniel [Verfasser]. "Noncommutative computer algebra with applications in algebraic analysis / Wolf Daniel Andres." Aachen : Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2014. http://d-nb.info/1049821475/34.

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Harker, Hayden. "Cohomology of a sub-Hopf algebra of the Steenrod alebra /." view abstract or download file of text, 2005. http://wwwlib.umi.com/cr/uoregon/fullcit?p3181104.

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Thesis (Ph. D.)--University of Oregon, 2005.
Typescript. Includes vita and abstract. Includes bibliographical references (leaf 61). Also available for download via the World Wide Web; free to University of Oregon users.

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EI-Chaar, Caroline. "The Onsager Algebra." Thesis, University of Ottawa (Canada), 2010. http://hdl.handle.net/10393/28699.

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In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.

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Mason, Ralph T. "Learning algebra personally." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/nq21596.pdf.

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Farlow, Kasie Geralyn. "Max-Plus Algebra." Thesis, Virginia Tech, 2009. http://hdl.handle.net/10919/32191.

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In max-plus algebra we work with the max-plus semi-ring which is the set ℝ_max=[-∞)∪ℝ together with operations 𝑎⊕𝑏 = max(𝑎,𝑏) and 𝑎⊗𝑏= 𝑎+𝑏. The additive and multiplicative identities are taken to be ε=-∞ and ε=0 respectively. Max-plus algebra is one of many idempotent semi-rings which have been considered in various fields of mathematics. Max-plus algebra is becoming more popular not only because its operations are associative, commutative and distributive as in conventional algebra but because it takes systems that are non-linear in conventional algebra and makes them linear. Max-plus algebra also arises as the algebra of asymptotic growth rates of functions in conventional algebra which will play a significant role in several aspects of this thesis. This thesis is a survey of max-plus algebra that will concentrate on max-plus linear algebra results. We will then consider from a max-plus perspective several results by Wentzell and Freidlin for finite state Markov chains with an asymptotic dependence.
Master of Science

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De, Carufel Jean-Lou. "Demonic Kleene Algebra." Doctoral thesis, Québec : Université Laval, 2009. http://www.theses.ulaval.ca/2009/25981/25981.pdf.

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Esslamzadeh, Gholam Hossein. "Banach algebra structure and amenability of a class of matrix algebras with applications." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0002/NQ29033.pdf.

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Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1182/1/PhdThesisFaccinPaolo.pdf.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

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Russell, Ewan. "Prime ideals in quantum algebras." Thesis, University of Edinburgh, 2009. http://hdl.handle.net/1842/3450.

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The central objects of study in this thesis are quantized coordinate algebras. These algebras originated in the 1980s in the work of Drinfeld and Jumbo and are noncommutative analogues of coordinate rings of algebraic varieties. The organic nature by which these algebras arose is of great interest to algebraists. In particular, investigating ring theoretic properties of these noncommutative algebras in comparison to the properties already known about their classical (commutative) counterparts proves to be a fruitful process. The prime spectrum of an algebra has always been seen as an important key to understanding its fundamental structure. The search for prime spectra is a central focus of this thesis. Our focus is mainly on Quantum Grassmannian subalgebras of quantized coordinate rings of Matrices of size m x n (denoted Oq(Mm;n)). Quantum Grassmannians of size m x n are denoted Gq(m; n) and are the subalgebras generated by the maximal quantum minors of Oq(Mm;n). In Chapter 2 we look at the simplest interesting case, namely the 2 x 4 Quantum Grassmannian (Gq(2; 4)), and we identify the H-primes and automorphism group of this algebra. Chapter 3 begins with a very important result concerning the dehomogenisation isomorphism linking Gq(m; n) and Oq(Mm;n¡m). This result is applied to help to identify H-prime spectra of Quantum Grassmannians. Chapter 4 focuses on identifying the number of H-prime ideals in the 2xn Quan- tum Grassmannian. We show the link between Cauchon fillings of subpartitions and H-prime ideals. In Chapter 5, we look at methods of ordering the generating elements of Quantum Grassmannians and prove the result that Quantum Grassmannians are Quantum Graded Algebras with a Straightening Law is maintained on using one of these alternative orderings. Chapter 6 looks at the Poisson structure on the commutative coordinate ring, G(2; 4) encoded by the noncommutative quantized algebra Gq(2; 4). We describe the symplectic ideals of G(2; 4) based on this structure. Finally in Chapter 7, we present an analysis of the 2 x 2 Reflection Equation Algebra and its primes. This algebra is obtained from the quantized coordinate ring of 2 x 2 matrices, Oq(M2;2).

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Elce, Kimberly R. "The Dieudonne ring for ordinary homology /." view abstract or download file of text, 2002. http://wwwlib.umi.com/cr/uoregon/fullcit?p3061942.

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Thesis (Ph. D.)--University of Oregon, 2002.
Typescript. Includes vita and abstract. Includes bibliographical references (leaf 77). Also available for download via the World Wide Web; free to University of Oregon users.

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43

Quinlan, Isis Angelina Marie. "Miki Images of Quantum Toroidal Algebra Generators in the Shuffle Algebra." Thesis, Virginia Tech, 2020. http://hdl.handle.net/10919/98753.

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Through composition of isomorphisms from results by Miki and Negut, this paper seeks to simplify calculations of the images of generators of the quantum toroidal algebra. We will be working in the small shuffle algebra, which is isomorphic to the positive part of the quantum toroidal algebra. There, we will be computing commutators, which are equal to images under the Miki automorphism, though are much simpler to compute.
Master of Science
Computing is hard, even for computers. The fewer computations we have to do, the more time we can save to do more math. This paper accomplishes just that. By looking at the quantum toroidal algebra through an automorphism followed by an isomorphism to a small shuffle algebra, we find a way to compute images of generators under the automorphism relatively easily.

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Stedall, Jacqueline Anne. "A large discourse concerning algebra : John Wallis's 1685 "Treatise of algebra."." Thesis, [n.p.], 2000. http://ethos.bl.uk/.

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Ahlgren, Joyce Christine. "Ideals, varieties, and Groebner bases." CSUSB ScholarWorks, 2003. https://scholarworks.lib.csusb.edu/etd-project/2282.

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The topics explored in this project present and interesting picture of close connections between algebra and geometry. Given a specific system of polynomial equations we show how to construct a Groebner basis using Buchbergers Algorithm. Gröbner bases have very nice properties, e.g. they do give a unique remainder in the division algorithm. We use these bases to solve systems of polynomial quations in several variables and to determine whether a function lies in the ideal.

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Townsend, Brian E. "Examining secondary students algebraic reasoning flexibility and strategy use /." Diss., Columbia, Mo. : University of Missouri-Columbia, 2005. http://hdl.handle.net/10355/4131.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2005.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (November 14, 2006) Vita. Includes bibliographical references.

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Nickerson, Susan Denise. "Supporting students' understanding of algebra : symbolizing in a technology-enhanced classroom /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC IP addresses, 2001. http://wwwlib.umi.com/cr/ucsd/fullcit?p3022703.

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at, michor@esi ac. "The Generalized Cayley Map from an Algebraic Group to its Lie Algebra." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1066.ps.

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Stutsman, Michelle Diane. "Semisimplicity for Hopf algebras." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1247.

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Lundkvist, Christian. "Moduli spaces of zero-dimensional geometric objects." Doctoral thesis, Stockholm : Matematik, Kungliga Tekniska högskolan, 2009. http://www.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:223079.

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