On graded characterizations of finite dimensionality for algebraic algebras
Permanent link to this recordhttp://hdl.handle.net/20.500.12613/5675
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Abstract© 2017, Springer International Publishing AG. We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring gr R is right noetherian, if and only if gr R has right Krull dimension, if and only if gr R satisfies a polynomial identity.
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Has partArchiv der Mathematik
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EXTENDING ACTIONS OF HOPF ALGEBRAS TO ACTIONS OF THE DRINFEL'D DOUBLELorenz, Martin, 1951-; Walton, Chelsea; Letzter, E. S. (Edward S.), 1958-; Futer, David; Riseborough, Peter (Temple University. Libraries, 2019)Mathematicians have long thought of symmetry in terms of actions of groups, but group actions have proven too restrictive in some cases to give an interesting picture of the symmetry of some mathematical objects, e.g. some noncommutative algebras. It is generally agreed that the right generalizations of group actions to solve this problem are actions of Hopf algebras, the study of which has exploded in the years since the publication of Sweedler's Hopf algebras in 1969. Different varieties of Hopf algebras have been useful in many fields of mathematics. For instance, in his "Quantum Groups" paper, Vladimir Drinfel'd introduced quasitriangular Hopf algebras, a class of Hopf algebras whose modules each provide a solution to the quantum Yang-Baxter equation. Solutions of this equation are a source of knot and link invariants and in physics, determine if a one-dimensional quantum system is integrable. Drinfel'd also introduced the Drinfel'd double construction, which produces for each finite-dimensional Hopf algebra a quasitriangular one in which the original embeds. This thesis is motivated by work of Susan Montgomery and Hans-Jürgen Schneider on actions of the Taft (Hopf) algebras T_n(q) and extending such actions to the Drinfel'd double D(T_n(q)). In 2001, Montgomery and Schneider classified all non-trivial actions of T_n(q) on an n-dimensional associative algebra A. It turns out that A must be isomorphic to the group algebra of grouplike elements kG(T_n(q)). They further determined that each such action extends uniquely to an action of the Drinfel'd double D(T_n(q)) on A, effectively showing that each action has a unique compatible coaction. We generalize Montgomery and Schneider's results to Hopf algebras related to the Taft algebras: the Sweedler (Hopf) algebra, bosonizations of 1-dimensional quantum linear spaces, generalized Taft algebras, and the Frobenius-Lusztig kernel u_q(sl_2). For each Hopf algebra H, we determine 1. whether there are non-trivial actions of H on A, 2. the possible H-actions on A, and 3. the possible D(H)-actions on A extending an H-action and how many there are.
A direct computation of the cohomology of the braces operadDolgushev, V; Willwacher, T (2017-03-01)© 2017 by De Gruyter. We give a self-contained and purely combinatorial proof of the well-known fact that the cohomology of the braces operad is the operad Ger governing Gerstenhaber algebras.
The Problem with Word Problems: An Exploratory Study of Factors Related to Word Problem SuccessByrnes, James P.; Newton, Kristie Jones, 1973-; Ding, Meixia; Booth, Julie L. (Temple University. Libraries, 2016)College Algebra is a gatekeeper course that serves as an obstacle for many students pursuing STEM careers. Lack of success in this course could be a key reason why the United States lags behind other industrialized countries in the number of students graduating with STEM majors and joining the STEM workforce. Of the many topics presented in College Algebra that pose problems, students often have particular difficulty with the application of systems of equations in the form of word problems. The present study aims to identify the factors associated with success and failure on systems of equations word problems. The goal was to identify the factors that remained significant predictors of success in order to build a theory to explain why some students are successful and other have difficulty. Using the Opportunity-Propensity Model of Byrnes and colleagues as the theoretical guide (e.g., Byrnes & Miller-Cotto, 2016), the following questions set the groundwork for the current study: (1) To what extent do antecedent (gender, race/ethnicity, socioeconomic status, and university) and propensity factors (mathematical calculation ability, mathematics anxiety, self-regulation, motivation, and ESL) individually and collectively predict success with systems of equations word problems in College Algebra students, and (2) How do these factors relate to each other? Bivariate correlations and hierarchical multiple regression were used to examine the relationships between the factors and word problem set-up as well as correct completion of the word problems presented. Results indicated after all variables were entered, calculation ability, self-regulation as determined by homework score, and anxiety were the only factors to remain significant predictors of student performance on both levels. All other factors either failed to enter as significant predictors or dropped out when the complete set had been entered. Reasons for this pattern of results are discussed, as are suggestions for future research to confirm and extend these findings.