Expand all Collapse all | Results 1 - 25 of 39 |
1. CJM Online first
Outer Partial Actions and Partial Skew Group Rings We extend the classicial notion of an outer action
$\alpha$ of a group $G$ on a unital ring $A$
to the case when $\alpha$ is a partial action
on ideals, all of which have local units.
We show that if $\alpha$ is an outer partial
action of an abelian group $G$,
then its associated partial skew group
ring $A \star_\alpha G$ is simple if and only if
$A$ is $G$-simple.
This result is applied to partial skew group rings associated with two different types of partial dynamical systems.
Keywords:outer action, partial action, minimality, topological dynamics, partial skew group ring, simplicity Categories:16W50, 37B05, 37B99, 54H15, 54H20 |
2. CJM 2014 (vol 67 pp. 28)
Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.
Keywords:derived category, Grothendieck duality, representation of quivers, reflection functor Categories:18E30, 16G20, 18E40, 16D90, 18A40 |
3. CJM 2014 (vol 66 pp. 902)
Corrigendum to Example in "Quantum Drinfeld Hecke Algebras" The last example of the article contains an error which we correct.
We also indicate some indices in Theorem 11.1 that were accidently transposed.
Keywords:quantum/skew polynomial rings, noncommutative Groebner bases Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
4. CJM 2013 (vol 66 pp. 453)
A Remark on BMW algebra, $q$-Schur Algebras and Categorification We prove that the 2-variable BMW algebra
embeds into an algebra constructed from the HOMFLY-PT polynomial.
We also prove that the $\mathfrak{so}_{2N}$-BMW algebra embeds in the $q$-Schur algebra
of type $A$.
We use these results
to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$-BMW algebra.
Keywords:tangle algebras, BMW algebra, HOMFLY-PT Skein algebra, q-Schur algebra, categorification Categories:57M27, 81R50, 17B37, 16W99 |
5. CJM 2013 (vol 66 pp. 874)
Quantum Drinfeld Hecke Algebras We consider finite groups acting on
quantum (or skew) polynomial rings. Deformations of the
semidirect product of the quantum polynomial ring with the acting group
extend symplectic reflection algebras and graded Hecke algebras
to the quantum setting over a field
of arbitrary characteristic.
We give necessary and sufficient conditions for such algebras to satisfy a
PoincarÃ©-Birkhoff-Witt property using the theory of noncommutative
GrÃ¶bner bases.
We include applications to the case of abelian groups
and the case of groups acting on coordinate rings of quantum planes.
In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology
gives an elegant description of the PBW conditions.
Keywords:skew polynomial rings, noncommutative GrÃ¶bner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
6. CJM 2013 (vol 66 pp. 625)
Classifying the Minimal Varieties of Polynomial Growth Let $\mathcal{V}$ be a variety of associative algebras generated by
an algebra with $1$ over a field of characteristic zero. This
paper is devoted to the classification of the varieties
$\mathcal{V}$ which are minimal of polynomial growth (i.e., their
sequence of codimensions growth like $n^k$ but any proper subvariety
grows like $n^t$ with $t\lt k$). These varieties are the building
blocks of general varieties of polynomial growth.
It turns out that for $k\le 4$ there are only a finite number of
varieties of polynomial growth $n^k$, but for each $k \gt 4$, the
number of minimal varieties is at least $|F|$, the cardinality of
the base field and we give a recipe of how to construct them.
Keywords:T-ideal, polynomial identity, codimension, polynomial growth, Categories:16R10, 16P90 |
7. CJM 2013 (vol 66 pp. 481)
On the Hadamard Product of Hopf Monoids Combinatorial structures that compose and decompose give rise to Hopf monoids
in Joyal's category of species. The Hadamard product of two Hopf monoids
is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states
that if one factor is connected and the other is free as a monoid,
their Hadamard product is free (and connected).
The second provides an explicit basis for the Hadamard
product when both factors are free.
The first main result is obtained by showing the existence of a one-parameter deformation
of the comonoid structure and appealing to a rigidity result of Loday and Ronco
that applies when the parameter is set to zero.
To obtain the second result, we introduce an operation on species that is intertwined
by the free monoid functor with the Hadamard product.
As an application of the first result, we deduce that the Boolean transform
of the dimension sequence of a connected Hopf monoid is nonnegative.
Keywords:species, Hopf monoid, Hadamard product, generating function, Boolean transform Categories:16T30, 18D35, 20B30, 18D10, 20F55 |
8. CJM 2013 (vol 66 pp. 205)
Generalized Frobenius Algebras and Hopf Algebras "Co-Frobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
left-right symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasi-co-Frobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) co-Frobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)co-Frobenius coalgebra $C$. We
give many examples of co-Frobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
q-homological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, co-Frobenius Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10 |
9. CJM 2012 (vol 64 pp. 1222)
Normality of Maximal Orbit Closures for Euclidean Quivers Let $\Delta$ be an Euclidean quiver. We prove that the closures of
the maximal orbits in the varieties of representations of $\Delta$
are normal and Cohen--Macaulay (even complete intersections).
Moreover, we give a generalization of this result for the tame
concealed-canonical algebras.
Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra Categories:16G20, 14L30 |
10. CJM 2012 (vol 65 pp. 241)
Lagrange's Theorem for Hopf Monoids in Species Following Radford's proof of Lagrange's theorem for pointed Hopf algebras,
we prove Lagrange's theorem for Hopf monoids in the category of
connected species.
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence of a Hopf monoid
in the form of certain polynomial inequalities, and of
a set-theoretic Hopf monoid in the form of certain linear inequalities.
The latter express that the binomial transform of the sequence must be nonnegative.
Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarÃ©-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement Categories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35 |
11. CJM 2010 (vol 63 pp. 3)
Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 |
12. CJM 2009 (vol 61 pp. 904)
The Face Semigroup Algebra of a Hyperplane Arrangement This article presents a study of an algebra spanned by the faces of a
hyperplane arrangement. The quiver with relations of the algebra is
computed and the algebra is shown to be a Koszul algebra.
It is shown that the algebra depends only on the intersection lattice of
the hyperplane arrangement. A complete system of primitive orthogonal
idempotents for the algebra is constructed and other algebraic structure
is determined including: a description of the projective indecomposable
modules, the Cartan invariants, projective resolutions of the simple
modules, the Hochschild homology and cohomology, and the Koszul dual
algebra. A new cohomology construction on posets is introduced, and it is
shown that the face semigroup algebra is isomorphic to the cohomology
algebra when this construction is applied to the intersection lattice of
the hyperplane arrangement.
Categories:52C35, 05E25, 16S37 |
13. CJM 2009 (vol 61 pp. 315)
Injective Representations of Infinite Quivers. Applications In this article we study injective representations of infinite
quivers. We classify the indecomposable injective representations of
trees and describe Gorenstein injective and projective
representations of barren trees.
Categories:16G20, 18A40 |
14. CJM 2008 (vol 60 pp. 923)
Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field The Kronecker modules $\mathbb{V}(m,h,\alpha)$, where $m$ is a positive integer, $h$ is
a height function, and $\alpha$ is a $K$-linear functional on the
space $K(X)$ of rational functions in one variable $X$ over an
algebraically closed field $K$, are models for the family of all
torsion-free rank-2 modules that are extensions of finite-dimensional
rank-1 modules. Every such module comes with a regulating polynomial
$f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}(m,h,\alpha)$ is
commutative and non-trivial, the regulator $f$ must be quadratic in
$Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra
is the trivial extension $K\ltimes S$ for some vector space $S$. If
$f$ has distinct roots in $K(X)$, then the endomorphisms form a
structure that we call a bridge. These include the coordinate rings
of some curves. Regardless of the number of roots in the regulator,
those $\End\mathbb{V}(m,h,\alpha)$ that are domains have zero radical. In addition,
each semi-local $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension
$K\ltimes S$ or the product $K\times K$.
Categories:16S50, 15A27 |
15. CJM 2008 (vol 60 pp. 266)
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables We introduce a natural Hopf algebra structure on the space of noncommutative
symmetric functions.
The bases for this algebra are indexed
by set partitions. We show that there exists a natural inclusion of the Hopf
algebra of noncommutative symmetric functions
in this larger space. We also consider this algebra as a subspace of
noncommutative polynomials and use it to
understand the structure of the spaces of harmonics and coinvariants
with respect to this collection of noncommutative polynomials and conclude
two analogues of Chevalley's theorem in the noncommutative setting.
Categories:16W30, 05A18;, 05E10 |
16. CJM 2008 (vol 60 pp. 379)
Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated
singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth.
This paper proves a non-commutative analogue. Namely, if $A$ is a
(non-commutative) graded Artin--Schelter \CM\ algebra which is fully
bounded Noetherian and
has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by
$A$ is smooth.
Keywords:Artin--Schelter Cohen--Macaulay algebra, Artin--Schelter Gorenstein algebra, Auslander's theorem on finite Cohen--Macaulay type, Cohen--Macaulay ring, fully bounded Noetherian algebra, isolated singularity, maximal Cohen--Macaulay module, non-commutative Categories:14A22, 16E65, 16W50 |
17. CJM 2007 (vol 59 pp. 1260)
Generic Extensions and Canonical Bases for Cyclic Quivers We use the monomial basis theory developed by Deng and Du to
present an elementary algebraic construction of the canonical
bases for both the Ringel--Hall algebra of a cyclic quiver and the
positive part $\bU^+$ of the quantum affine $\frak{sl}_n$. This
construction relies on analysis of quiver representations and the
introduction of a new integral PBW-like basis for the Lusztig
$\mathbb Z[v,v^{-1}]$-form of~$\bU^+$.
Categories:17B37, 16G20 |
18. CJM 2007 (vol 59 pp. 880)
Radical Ideals in Valuation Domains An ideal $I$ of a ring $R$ is called a radical ideal if
$I={\mathcalR}(R)$ where ${\mathcal R}$ is a radical in the sense of
Kurosh--Amitsur. The main theorem of this paper asserts that if $R$
is a valuation domain, then a proper ideal $I$ of $R$ is a radical
ideal if and only if $I$ is a distinguished ideal of $R$ (the
latter property means that if $J$ and $K$ are ideals of $R$ such
that $J\subset I\subset K$ then we cannot have $I/J\cong K/I$ as
rings) and that such an ideal is necessarily prime. Examples are
exhibited which show that, unlike prime ideals, distinguished
ideals are not characterizable in terms of a property of the
underlying value group of the valuation domain.
Categories:16N80, 13A18 |
19. CJM 2007 (vol 59 pp. 658)
Division Algebras of Prime Degree and Maximal Galois $p$-Extensions Let $p$ be an odd prime number, and let $F$
be a field of characteristic not $p$ and not containing
the group $\mu_p$ of $p$-th roots of unity.
We consider cyclic $p$-algebras over $F$ by descent from
$L = F(\mu_p)$. We generalize a theorem of Albert by
showing that if $\mu_{p^n} \subseteq L$, then a division
algebra $D$ of degree $p^n$ over $F$ is a cyclic
algebra if and only if there is $d\in D$ with $d^{p^n}\in
F - F^p$. Let $F(p)$ be the maximal $p$-extension
of $F$. We show that $F(p)$ has a noncyclic algebra
of degree $p$ if and only if a certain eigencomponent of the
$p$-torsion of $\Br(F(p)(\mu_p))$ is nontrivial.
To get a better understanding of $F(p)$, we consider
the valuations on $F(p)$ with residue characteristic
not $p$, and determine what residue fields and value
groups can occur. Our results support the conjecture
that the $p$ torsion in $\Br(F(p))$ is always trivial.
Category:16K20 |
20. CJM 2007 (vol 59 pp. 332)
Endomorphism Rings of Finite Global Dimension For a commutative local ring $R$, consider (noncommutative)
$R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$
where $M$ is a reflexive $R$-module with nonzero free direct summand.
Such algebras $\Lambda$ of finite global dimension can be viewed as
potential substitutes for, or analogues of, a resolution of
singularities of $\operatorname{Spec} R$. For example, Van den Bergh
has shown that a three-dimensional Gorenstein normal
$\mathbb{C}$-algebra with isolated terminal singularities has a
crepant resolution of singularities if and only if it has such an
algebra $\Lambda$ with finite global dimension and which is maximal
Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of
singularities''). We produce algebras
$\Lambda=\operatorname{End}_R(M)$ having finite global dimension in
two contexts: when $R$ is a reduced one-dimensional complete local
ring, or when $R$ is a Cohen--Macaulay local ring of finite
Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then
the construction gives a noncommutative crepant resolution of
singularities in the sense of Van den Bergh.
Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules Categories:16G50, 16G60, 16E99 |
21. CJM 2007 (vol 59 pp. 186)
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$,
arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer,
$h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and
$\alpha$ is a $K$-linear functional on the space $\krx$ of rational
functions in one variable $X$. Every pair $(h,\alpha)$ comes with a
polynomial $f$ in $K(X)[Y]$ called the regulator. When the module
${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or
quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and
only if $f$ is an irreducible quadratic. Then the $K$-algebra
$\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For
some height functions $h$ of infinite support $I$, the search for a
functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes
down to having functions $\eta\colon I\ar K$ such that no planar curve
intersects the graph of $\eta$ on a cofinite subset. If $K$ has
characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a
purely-simple Kronecker module ${\mathcal M}$ having non-trivial
endomorphisms, then $h$ attains the value $\infty$ at least once on
$\ktil$ and $h$ is finite-valued at least twice on
$\ktil$. Conversely all these $h$ form part of such triplets. The
proof of this result hinges on the fact that a rational function $r$
is a perfect square in $\krx$ if and only if $r$ is a perfect square
in the completions of $\krx$ with respect to all of its valuations.
Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra Categories:16S50, 15A27 |
22. CJM 2006 (vol 58 pp. 180)
Infinite Dimensional Representations of Canonical Algebras The
aim of this paper is to extend the structure theory for infinitely
generated modules over tame hereditary algebras to the more
general case of modules over concealed canonical algebras. Using
tilting, we may assume that we deal with canonical algebras. The
investigation is centered around the generic and the Pr\"{u}fer
modules, and how other modules are determined by these
modules.
Categories:16D70, 16D90, 16G20, 16G60, 16G70 |
23. CJM 2004 (vol 56 pp. 871)
Lie Elements and Knuth Relations A coplactic class in the symmetric group $\Sym_n$ consists of all
permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may
be described in terms of local relations introduced by Knuth. Any
Lie element in the group algebra of $\Sym_n$ which is constant on
coplactic classes is already constant on descent classes. As a
consequence, the intersection of the Lie convolution algebra
introduced by Patras and Reutenauer and the coplactic algebra
introduced by Poirier and Reutenauer is the direct sum of all
Solomon descent algebras.
Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution product Categories:17B01, 05E10, 20C30, 16W30 |
24. CJM 2003 (vol 55 pp. 766)
Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory |
Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory We develop an explicit skein-theoretical algorithm to compute the
Alexander polynomial of a 3-manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3-manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$-representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$-equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$-action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
Frohman--Nicas theory from the Hard--Lefschetz decomposition of the
$U(1)$-moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
Seiberg--Witten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
Reshetikhin--Turaev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum-$\mathfrak{sl}_2$ is the product of the Reshetikhin--Turaev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 |
25. CJM 2003 (vol 55 pp. 42)
$*$-Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$ Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the
transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let
$\mathfrak{U}$ be a proper subvariety of the variety of algebras with
involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define
two sequences of algebras with involution $\mathcal{R}_p$,
$\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that
$T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$
are $*$-asymptotically equivalent for suitable $p,q$.
Keywords:algebras with involution, asymptotic equivalence Categories:16R10, 16W10, 16R50 |