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

Fonte: University of Limerick
Publicador: University of Limerick

Tipo: info:eu-repo/semantics/masterThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations

ENG

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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...

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## On the Strong Homotopy Associative Algebra of a Foliation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Differential Geometry#Mathematical Physics#Mathematics - K-Theory and Homology#Mathematics - Quantum Algebra#53C12, 16S30, 16S32

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

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## Homological Algebra for Superalgebras of Differentiable Functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2012

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#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - Quantum Algebra#16E45

This is the second in a series of papers laying the foundations for a
differential graded approach to derived differential geometry (and other
geometries in characteristic zero). In this paper, we extend the classical
notion of a dg-algebra to define, in particular, the notion of a differential
graded algebra in the world of C-infinity rings. The opposite of the category
of differential graded C-infinity algebras contains the category of
differential graded manifolds as a full subcategory. More generally, this
notion of differential graded algebra makes sense for algebras over any (super)
Fermat theory, and hence one also arrives at the definition of a differential
graded algebra appropriate for the study of derived real and complex analytic
manifolds and other variants. We go on to show that, for any super Fermat
theory S which admits integration, a concept we define and show is satisfied by
all important examples, the category of differential graded S-algebras supports
a Quillen model structure naturally extending the classical one on differential
graded algebras, both in the bounded and unbounded case (as well as
differential algebras with no grading). Finally, we show that, under the same
assumptions, any of these categories of differential graded S-algebras have a
simplicial enrichment...

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## Yang-Mills algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Quantum Algebra#High Energy Physics - Theory#Mathematical Physics#Mathematics - K-Theory and Homology

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/09/2010

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Topology#Mathematics - Algebraic Geometry#Mathematics - Geometric Topology#Mathematics - Quantum Algebra#17B56, 32G15, 55R40, 17B65, 20J06

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

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## A Hopf algebra associated to a Lie pair

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/09/2014

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#Mathematics - Algebraic Geometry#Mathematical Physics#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - Quantum Algebra

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Representation Theory#Mathematical Physics#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras#17B10, 17B30, 17B80

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/1996

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#High Energy Physics - Theory#Mathematics - Functional Analysis#Mathematics - Quantum Algebra#Quantum Physics

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...

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## Lifting homotopy T-algebra maps to strict maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Topology#Mathematics - Geometric Topology#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras#55P99, 55S35, 55T05, 18G55, 13D03, 18C15, 18C10, 55P43, 55P62, 55Q50

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/03/2013

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#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras#Mathematics - Representation Theory#6W30, 16T05, 18D05, 18D10, 16S34

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## On the algebra of cornered Floer homology

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Geometric Topology#Mathematics - Algebraic Topology#Mathematics - Quantum Algebra#Mathematics - Symplectic Geometry#57R56 (Primary) 57R58 (Secondary)

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/08/2012

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#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras#Mathematics - Representation Theory#16E40

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/01/2015

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Combinatorics#Mathematics - Quantum Algebra#Mathematics - Rings and Algebras#05E05, 06A11, 16W30

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of
permutations in detail. We give explicit formulas for its antipode, prove that
it is a cofree coalgebra, determine its primitive elements and its coradical
filtration, and show that it decomposes as a crossed product over the Hopf
algebra of quasi-symmetric functions. In addition, we describe the structure
constants of the multiplication as a certain number of facets of the
permutahedron. As a consequence we obtain a new interpretation of the product
of monomial quasi-symmetric functions in terms of the facial structure of the
cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed
by permutations. Our results are obtained from a combinatorial description of
the Hopf algebraic structure with respect to a new basis for this algebra,
related to the original one via M\"obius inversion on the weak order on the
symmetric groups. This is in analogy with the relationship between the monomial
and fundamental bases of the algebra of quasi-symmetric functions. Our results
reveal a close relationship between the structure of the Malvenuto-Reutenauer
Hopf algebra and the weak order on the symmetric groups.; Comment: 40 pages, 6 .eps figures. Full version of math.CO/0203101. Error in
statement of Lemma 2.17 in published version corrected

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## On the Hall algebra of coherent sheaves on P^1 over F_1

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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