16,287 research outputs found

### Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(2|n)]

Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so
contains a \textit{universal $R$-matrix} in the tensor product algebra which
satisfies the Yang-Baxter equation. Applying the vector representation $\pi$,
which acts on the vector module $V$, to one side of a universal $R$-matrix
gives a Lax operator. In this paper a Lax operator is constructed for the
$C$-type quantum superalgebras $U_q[osp(2|n)]$. This can in turn be used to
find a solution to the Yang-Baxter equation acting on $V \otimes V \otimes W$
where $W$ is an arbitrary $U_q[osp(2|n)]$ module. The case $W=V$ is included
here as an example.Comment: 15 page

### R-matrices and Tensor Product Graph Method

A systematic method for constructing trigonometric R-matrices corresponding
to the (multiplicity-free) tensor product of any two affinizable
representations of a quantum algebra or superalgebra has been developed by the
Brisbane group and its collaborators. This method has been referred to as the
Tensor Product Graph Method. Here we describe applications of this method to
untwisted and twisted quantum affine superalgebras.Comment: LaTex 7 pages. Contribution to the APCTP-Nankai Joint Symposium on
"Lattice Statistics and Mathematical Physics", 8-10 October 2001, Tianjin,
Chin

### WFIRST Ultra-Precise Astrometry II: Asteroseismology

WFIRST microlensing observations will return high-precision parallaxes,
sigma(pi) < 0.3 microarcsec, for the roughly 1 million stars with H<14 in its
2.8 deg^2 field toward the Galactic bulge. Combined with its 40,000 epochs of
high precision photometry (~0.7 mmag at H_vega=14 and ~0.1 mmag at H=8), this
will yield a wealth of asteroseismic data of giant stars, primarily in the
Galactic bulge but including a substantial fraction of disk stars at all
Galactocentric radii interior to the Sun. For brighter stars, the astrometric
data will yield an external check on the radii derived from the two
asteroseismic parameters, and nu_max, while for the fainter ones, it
will enable a mass measurement from the single measurable asteroseismic
parameter nu_max. Simulations based on Kepler data indicate that WFIRST will be
capable of detecting oscillations in stars from slightly less luminous than the
red clump to the tip of the red giant branch, yielding roughly 1 million
detections.Comment: 13 pages, 6 figures, submitted to JKA

### Casimir invariants and characteristic identities for $gl(\infty )$

A full set of (higher order) Casimir invariants for the Lie algebra
$gl(\infty )$ is constructed and shown to be well defined in the category
$O_{FS}$ generated by the highest weight (unitarizable) irreducible
representations with only a finite number of non-zero weight components.
Moreover the eigenvalues of these Casimir invariants are determined explicitly
in terms of the highest weight. Characteristic identities satisfied by certain
(infinite) matrices with entries from $gl(\infty )$ are also determined and
generalize those previously obtained for $gl(n)$ by Bracken and Green.$^{1,2}$Comment: 10 pages, PlainTe

### Unitarity and Complete Reducibility of Certain Modules over Quantized Affine Lie Algebras

Let $U_q(\hat{\cal G})$ denote the quantized affine Lie algebra and
$U_q({\cal G}^{(1)})$ the quantized {\em nontwisted} affine Lie algebra. Let
${\cal O}_{\rm fin}$ be the category defined in section 3. We show that when
the deformation parameter $q$ is not a root of unit all integrable
representations of $U_q(\hat{\cal G})$ in the category ${\cal O}_{\rm fin}$ are
completely reducible and that every integrable irreducible highest weight
module over $U_q({\cal G}^{(1)})$ corresponding to $q>0$ is equivalent to a
unitary module.Comment: 17 pages (minor errors corrected

### A class of quadratic deformations of Lie superalgebras

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree
2 which are a special class of deformations of Lie superalgebras, which we call
quadratic Lie superalgebras. Starting from the formal definition, we discuss
the generalised Jacobi relations in the context of the Koszul property, and
give a proof of the PBW basis theorem. We give several concrete examples of
quadratic Lie superalgebras for low dimensional cases, and discuss aspects of
their structure constants for the `type I' class. We derive the equivalent of
the Kac module construction for typical and atypical modules, and a related
direct construction of irreducible modules due to Gould. We investigate in
detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie
superalgebra sl(n/1). We formulate the general atypicality conditions at level
1, and present an analysis of zero-and one-step atypical modules for a certain
family of Kac modules.Comment: 26pp, LaTeX. Original title: "Finite dimensional quadratic Lie
superalgebras"; abstract re-worded; text clarified; 3 references added;
rearrangement of minor appendices into text; new subsection 4.

### Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras

The Perk--Schultz model may be expressed in terms of the solution of the
Yang--Baxter equation associated with the fundamental representation of the
untwisted affine extension of the general linear quantum superalgebra
$U_q[sl(m|n)]$, with a multiparametric co-product action as given by
Reshetikhin. Here we present analogous explicit expressions for solutions of
the Yang-Baxter equation associated with the fundamental representations of the
twisted and untwisted affine extensions of the orthosymplectic quantum
superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the
Perk--Schultz model.Comment: 10 pages, 2 figure

### Punctuated Equilibrium in Software Evolution

The approach based on paradigm of self-organized criticality proposed for
experimental investigation and theoretical modelling of software evolution. The
dynamics of modifications studied for three free, open source programs Mozilla,
Free-BSD and Emacs using the data from version control systems. Scaling laws
typical for the self-organization criticality found. The model of software
evolution presenting the natural selection principle is proposed. The results
of numerical and analytical investigation of the model are presented. They are
in a good agreement with the data collected for the real-world software.Comment: 4 pages, LaTeX, 2 Postscript figure

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