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## An investigation into the effects of introducing algebra using a function-based approach

Sterritt, Nicola
Fonte: University of Limerick Publicador: University of Limerick
Tipo: info:eu-repo/semantics/masterThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations
ENG
Relevância na Pesquisa
45.76%
peer-reviewed; Ireland is currently witnessing a major overhaul of its mathematics syllabus for second level education. This syllabus is known as ‘Project Maths’ and came about as a results of concerns relating to the mathematics performance of students in Ireland in international comparative studies such as the PISA (Program for International Student Assessment) tests (Close & Oldham 2005; Cosgrove, Shiel, Sofroniou, Zastrutzki & Shortt 2005; Perkins, Moran, Cosgrove and Shiel 2010; Oldham 2002, 2006). The author found inspiration for this research when she identified concerns in her own classroom. These concerns were two-fold; firstly the author found that first year students began secondary school with a poor attitude towards mathematics and secondly, the author found that first year students had a lot of difficulty grasping and retaining basic algebraic concepts. The author followed an action research approach to implementing an intervention in her classroom aimed at overcoming these problems. In the first phase of this research, the author carried out a comprehensive review of literature on affect pertaining to mathematics education and on the teaching and learning of algebra. As a result of this review, the author decided to use a function-based approach to teaching algebra as a means of improving students understanding of basic algebra. A collaborative peer learning environment was chosen as the main pedagogical tool for improving attitude towards mathematics. The second phase of this research saw the development and implementation of an intervention in the author’s classroom during which fourth year students tutored first year students. Quantitative and qualitative data was gathered during this phase. The third phase comprised of an analysis of data...

## On the Strong Homotopy Associative Algebra of a Foliation

Vitagliano, Luca
Tipo: Artigo de Revista Científica
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An involutive distribution $C$ on a smooth manifold $M$ is a Lie-algebroid acting on sections of the normal bundle $TM/C$. It is known that the Chevalley-Eilenberg complex associated to this representation of $C$ possesses the structure $\mathbb{X}$ of a strong homotopy Lie-Rinehart algebra. It is natural to interpret $\mathbb{X}$ as the (derived) Lie-Rinehart algebra of vector fields on the space $\mathbb{P}$ of integral manifolds of $C$. In this paper, I show that $\mathbb{X}$ is embedded in a strong homotopy associative algebra $\mathbb{D}$ of (normal) differential operators. It is natural to interpret $\mathbb{D}$ as the (derived) associative algebra of differential operators on $\mathbb{P}$. Finally, I speculate about the interpretation of $\mathbb{D}$ as the universal enveloping strong homotopy algebra of $\mathbb{X}$.; Comment: 28 pages, comments welcome

## Homological Algebra for Superalgebras of Differentiable Functions

Carchedi, David; Roytenberg, Dmitry
Tipo: Artigo de Revista Científica
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45.72%

## Yang-Mills algebra

Connes, Alain; Dubois-Violette, Michel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.72%
Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s=1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin-Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra A and for the dimension in degree n of the graded Lie algebra of which A is the universal enveloping algebra. In the 4-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie-algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.; Comment: 14 pages; appendix added

## An $E_8$-approach to the moonshine vertex operator algebra

Shimakura, Hiroki
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$ associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex operator algebra analogue of "An $E_8$-approach to the Leech lattice and the Conway group" by Lepowsky and Meurman. Moreover, we find new analogies among the moonshine vertex operator algebra, the Leech lattice and the extended binary Golay code.; Comment: 25 pages

## The cohomology ring of the 12-dimensional Fomin-Kirillov algebra

Stefan, Dragos; Vay, Cristian
Tipo: Artigo de Revista Científica
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The $12$-dimensional Fomin-Kirillov algebra $\mathcal{FK}_3$ is defined as the quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations $a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. By a result of A. Milinski and H.-J. Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter-Drinfeld module $V$, over the symmetric group $\mathbb{S}_3$, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring $\operatorname{Ext}_{\mathcal{FK}_3}^*(\Bbbk, \Bbbk)$, showing that it is a polynomial ring $S[X]$ with coefficients in the symmetric braided algebra of $V$. As an application we also compute the cohomology rings of the bosonization $\mathcal{FK}_3\#\Bbbk\mathbb{S}_3$ and of its dual, which are $72$-dimensional ordinary Hopf algebras.; Comment: v2: minor typos are corrected and new results are added

## Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

Morita, Shigeyuki; Sakasai, Takuya; Suzuki, Masaaki
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.73%
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.; Comment: 30 pages, 18 figures. Title modified, final version, to appear in Duke Math. J

## A Hopf algebra associated to a Lie pair

Chen, Zhuo; Stiénon, Mathieu; Xu, Ping
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.72%
The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $\alpha_{L/A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the Lie algebra object $L/A[-1]$ and we prove that it is a Hopf algebra object in $D^b(\mathscr{A})$.; Comment: 6 pages

## Classification of simple weight modules over the 1-spatial ageing algebra

Lu, Rencai; Mazorchuk, Volodymyr; Zhao, Kaiming
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
In this paper we use Block's classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra age(1). Most of these modules have infinite dimensional weight spaces and so far the algebra age(1) is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrodinger algebra S that have a simple age(1)-submodule thus constructing many new simple weight S-modules.; Comment: 18 pages

## Frobenius character formula and spin generic degrees for Hecke-Clifford algebra

Wan, Jinkui; Wang, Weiqiang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
The spin analogues of several classical concepts and results for Hecke algebras are established. A Frobenius type formula is obtained for irreducible characters of the Hecke-Clifford algebra. A precise characterization of the trace functions allows us to define the character table for the algebra. The algebra is endowed with a canonical symmetrizing trace form, with respect to which the spin generic degrees are formulated and shown to coincide with the spin fake degrees. We further provide a characterization of the trace functions and the symmetrizing trace form on the spin Hecke algebra which is Morita super-equivalent to the Hecke-Clifford algebra.; Comment: 34 pages, v2, updated references, mild corrections of typos, to appear in Proc. London Math. Soc

## Analog of Lie Algebra and Lie Group for Quantum Non-Hamiltonian Systems

Tarasov, Vasily E.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup (loop). The author derives that analog of Lie algebra for quantum non-Hamiltonian systems is commutant Lie algebra and analog of Lie group for these systems is analytic commutant associative loop (Valya loop). A commutant Lie algebra is an algebra such that commutant (a subspace which is generated by all commutators) is a Lie subalgebra. Valya loop is a non-associative loop such that the commutant of this loop is associative subloop (group). We prove that a tangent algebra of Valya loop is a commutant Lie algebra. It is shown that generalized Heisenberg-Weyl algebra, suggested by the author to describe quantum non-Hamiltonian (dissipative) systems, is a commutant Lie algebra. As the other example of commutant Lie algebra, it is considered a generalized Poisson algebra for differential 1-forms. Note that non-Hamiltonian (dissipative) quantum theory has a broad range of application for non-critical strings in "coupling constant" phase space and bosonic string in non-Riemannian (for example, affine-metric) curved space which are non-Hamiltonian (dissipative) systems.; Comment: 7 pages...

## Lifting homotopy T-algebra maps to strict maps

Johnson, Niles; Noel, Justin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends to a monad on the homotopy category and defines a category of homotopy $T$-algebras. In this setting there is a forgetful functor from the homotopy category of $T$-algebras to the category of homotopy $T$-algebras. Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy $T$-algebra map to a strict map of $T$-algebras. Once we have a map of $T$-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of $T$-algebra maps and the edge homomorphism on $\pi_0$ is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the $E_2$-term to be calculable in terms of Quillen cohomology groups. We provide worked examples in $G$-spaces, $G$-spectra...

## Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

Gorbounov, V.; Rimanyi, R.; Tarasov, V.; Varchenko, A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag variety F_\lambda parametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i, as the Yangian Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) as the algebra of functions on fibers of a discrete Wronski map. In particular this gives generators and relations of that algebra. This identification also gives us hypergeometric solutions of the associated quantum differential equation. That fact manifests the Landau-Ginzburg mirror symmetry for the cotangent bundle of the flag variety.; Comment: Latex, 45 pages, references added, Conjecture 7.10 is now Theorem 7.10, Theorem 7.13 added

## On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths

Bulacu, Daniel; Torrecillas, Blas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobenius algebra and the restriction at $R$ of its Nakayama automorphism is equal to the Nakayama automorphism of $R$. As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable.; Comment: 42 pages, many figures

## The multiple zeta value algebra and the stable derivation algebra

Furusho, Hidekazu
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.74%
The MZV algebra is the graded algebra over ${\bold Q}$ generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck-Teichm\"{u}ller group. We shall show that there is a canonical surjective $\bold Q$-linear map from the graded dual vector space of the stable derivation algebra over $\bold Q$ to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier's talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the new-zeta space with the $l$-adic Galois image Lie algebra which is associated with so the Galois representation on the pro-$l$ fundamental group of $\bold P ^1_ {\bar{\bold Q}}-{0,1,\infty}$ .; Comment: 20 pages, to be appeared in Publ. Res. Inst. Math. Sci

## On the algebra of cornered Floer homology

Douglas, Christopher L.; Manolescu, Ciprian
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.73%
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is satisfied. Moreover, with a view toward the structure of a potential Floer homology theory of 3-manifolds with codimension-two corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.; Comment: a few minor revisions

## On the cohomology of the Weyl algebra, the quantum plane, and the q-Weyl algebra

Gerstenhaber, Murray; Giaquinto, Anthony
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler-Poincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.; Comment: 16 pages

## The pop-switch planar algebra and the Jones-Wenzl idempotents

Grano, Ellie; Bigelow, Stephen
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.71%
The Jones-Wenzl idempotents are elements of the Temperley-Lieb planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra, which contains the Temperley-Lieb planar algebra. It is motivated by Jones' idea of the graph planar algebra of type $A_n$. In the tensor category of idempotents of the pop-switch planar algebra, the $n$th Jones-Wenzl idempotent is isomorphic to a direct sum of $n+1$ diagrams consisting of only vertical strands.

## Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

Aguiar, Marcelo; Sottile, Frank
We define and study the category $Coh_n(\Pone)$ of normal coherent sheaves on the monoid scheme $\Pone$ (equivalently, the $\mathfrak{M}_0$-scheme $\Pone / \fun$ in the sense of Connes-Consani-Marcolli \cite{CCM}). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of $Coh_n(\Pone)$. We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra $L {\mathfrak{gl}}_2$ and an abelian Lie algebra on infinitely many generators. This should be viewed as a $(q=1)$ version of Kapranov's result relating (a certain subalgebra of) the Ringel-Hall algebra of $\mathbb{P}^1$ over $\mathbb{F}_q$ to a non-standard quantum Borel inside the quantum loop algebra $\mathbb{U}_{\nu} (\slthat)$, where $\nu^2=q$.; Comment: Several corrections. Calculation of K_0 corrected