Expand all Collapse all | Results 1 - 13 of 13 |
1. CJM Online first
Corrigendum to "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 |
2. 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 |
3. CJM Online first
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 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |
9. 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 |
10. CJM 2002 (vol 54 pp. 897)
The Valuative Theory of Foliations This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
Categories:12J20, 13F30, 16W60, 37F75, 34M25 |
11. CJM 2002 (vol 54 pp. 595)
Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups
to the case of pro-affine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the pro-discrete
topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine
algebraic group $G$ over $K$, which is discrete in the
finite-dimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 |
12. CJM 1999 (vol 51 pp. 881)
The Representation Ring and the Centre of a Hopf Algebra When $H$ is a finite dimensional, semisimple, almost cocommutative
Hopf algebra, we examine a table of characters which extends the
notion of the character table for a finite group. We obtain a
formula for the structure constants of the representation ring in
terms of values in the character table, and give the example of the
quantum double of a finite group. We give a basis of the centre of
$H$ which generalizes the conjugacy class sums of a finite group,
and express the class equation of $H$ in terms of this basis. We
show that the representation ring and the centre of $H$ are dual
character algebras (or signed hypergroups).
Categories:16W30, 20N20 |
13. CJM 1999 (vol 51 pp. 488)
Homological Aspects of Semigroup Gradings on Rings and Algebras This article studies algebras $R$ over a simple artinian ring $A$,
presented by a quiver and relations and graded by a semigroup $\Sigma$.
Suitable semigroups often arise from a presentation of $R$.
Throughout, the algebras need not be finite dimensional. The graded
$K_0$, along with the $\Sigma$-graded Cartan endomorphisms and Cartan
matrices, is examined. It is used to study homological properties.
A test is found for finiteness of the global dimension of a
monomial algebra in terms of the invertibility of the Hilbert
$\Sigma$-series in the associated path incidence ring.
The rationality of the $\Sigma$-Euler characteristic, the Hilbert
$\Sigma$-series and the Poincar\'e-Betti $\Sigma$-series is studied
when $\Sigma$ is torsion-free commutative and $A$ is a division ring.
These results are then applied to the classical series. Finally, we
find new finite dimensional algebras for which the strong no loops
conjecture holds.
Categories:16W50, 16E20, 16G20 |