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1. CJM 2013 (vol 66 pp. 453)

Vaz, Pedro; Wagner, Emmanuel
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

2. CJM 2010 (vol 63 pp. 413)

Konvalinka, Matjaž; Skandera, Mark
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
Categories:15A15, 20C08, 81R50

3. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
Closed and Exact Functions in the Context of Ginzburg--Landau Models
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
Categories:42B05, 81Q50, 42A16

4. CJM 2008 (vol 60 pp. 241)

Alexandrova, Ivana
Semi-Classical Wavefront Set and Fourier Integral Operators
Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions.

Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysis
Categories:35S30, 35A27, 58J40, 81Q20

5. CJM 2007 (vol 59 pp. 897)

Bruneau, Laurent
The Ground State Problem for a Quantum Hamiltonian Model Describing Friction
In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.

Categories:81Q10, 46N50

6. CJM 2001 (vol 53 pp. 756)

Froese, Richard
Correction to: Upper Bounds for the Resonance Counting Function of Schrödinger Operators in Odd Dimensions
The proof of Lemma~3.4 in [F] relies on the incorrect equality $\mu_j (AB) = \mu_j (BA)$ for singular values (for a counterexample, see [S, p.~4]). Thus, Theorem~3.1 as stated has not been proven. However, with minor changes, we can obtain a bound for the counting function in terms of the growth of the Fourier transform of $|V|$.

Categories:47A10, 47A40, 81U05

7. CJM 2000 (vol 52 pp. 503)

Gannon, Terry
The Level 2 and 3 Modular Invariants for the Orthogonal Algebras
The `1-loop partition function' of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\SL_2(\bbZ)$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_r^{(1)}$ and $D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2---the $B_r^{(1)}$, $D_r^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$. The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal E}_6$-exceptional and $\E_8$-exceptional, respectively, in the A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine~$u(1)$.

Keywords:Kac-Moody algebra, conformal field theory, modular invariants
Categories:17B67, 81T40

8. CJM 1999 (vol 51 pp. 816)

Hall, Brian C.
A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type
I consider a two-parameter family $B_{s,t}$ of unitary transforms mapping an $L^{2}$-space over a Lie group of compact type onto a holomorphic $L^{2}$-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s \rightarrow \infty$ and $s \rightarrow t/2$.

Categories:22E30, 81S30, 58G11

9. CJM 1998 (vol 50 pp. 1298)

Milson, Robert
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables
Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the $A_2$ root system.

Categories:35Q40, 53C30, 81R05

10. CJM 1998 (vol 50 pp. 756)

Brydges, D.; Dimock, J.; Hurd, T. R.
Estimates on renormalization group transformations
We consider a specific realization of the renormalization group (RG) transformation acting on functional measures for scalar quantum fields which are expressible as a polymer expansion times an ultra-violet cutoff Gaussian measure. The new and improved definitions and estimates we present are sufficiently general and powerful to allow iteration of the transformation, hence the analysis of complete renormalization group flows, and hence the construction of a variety of scalar quantum field theories.

Categories:81T08, 81T17

11. CJM 1998 (vol 50 pp. 538)

Froese, Richard
Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions
The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schr\"odinger operator in odd dimensions. At the same time we generalize the result to the class of super-exponentially decreasing potentials.

Categories:47A10, 47A40, 81U05

12. CJM 1997 (vol 49 pp. 916)

Brylinski, Ranee
Quantization of the $4$-dimensional nilpotent orbit of $\SL(3,\R)$
We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit $\OR$ of $\SL(3,\R)$. The space of quantization is the space of holomorphic functions on ${\C}^2-\{0\})$ which are square integrable with respect to a signed measure defined by a Meijer $G$-function. We construct the quantization out a non-flat Kaehler structure on ${\C}^2-\{0\})$ (the universal cover of $\OR$) with Kaehler potential $\rho=|z|^4$.

Categories:81S10, 32C17, 22E70

13. CJM 1997 (vol 49 pp. 160)

Rieffel, Marc A.
The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$
We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit.

Categories:46L60, 46l55, 81S30

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