Atomistic subsemirings of the lattice of subspaces of an algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.Comment: 14 page

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group with Lie algebra g. We extend scalars to the power series field in one variable C((π)), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of g ⊗ C((π)), i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig\u27s topological construction of Springer\u27s representations to the affine context

An Explicit basis of lowering Operators for Irreducible Representations of Unitary Groups

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the formalism of lowering operators. Here, one identifies a basis for the algebra of all lowering operators and, for each irreducible representation, gives a prescription for choosing a subcollection of lowering operators that yields a basis upon application to the highest weight vector. Bases obtained through lowering operators are particularly convenient for computing matrix coefficients of observables as the calculations reduce to the commutation relations for the standard matrix units. The best known examples of this approach are the extremal projector construction of the Gelfand-Zetlin basis and the crystal (or canonical) bases of Kashiwara and Lusztig. In this paper, we describe another simple method of obtaining bases for the irreducible representations via lowering operators. These bases do not have the algebraic canonicity of the Gelfand-Zetlin and crystal bases, but the combinatorics involved are much more straightforward, making the bases particularly suited for physical applications. © Lithuanian Physical Society, 2011

Twisted exponents and twisted Frobenius–Schur indicators for Hopf algebras

Classically, the exponent of a group is the least common multiple of the orders of its elements. This notion was generalized by Etingof and Gelaki to Hopf algebras. Kashina, Sommerhäuser, and Zhu later observed that there is a strong connection between exponents and Frobenius–Schur indicators. In this article, we introduce the notion of twisted exponents and show there is a similar relationship between the twisted exponent and the twisted Frobenius–Schur indicators defined in previous work of the authors. In particular, we exhibit a new formula for the twisted indicators and use it to prove periodicity and rationality statements

Inelastic collisions of ultra-cold heteronuclear molecules in an optical trap

Ultra-cold RbCs molecules in high-lying vibrational levels of the a$^3\Sigma^+$ ground electronic state are confined in an optical trap. Inelastic collision rates of these molecules with both Rb and Cs atoms are determined for individual vibrational levels, across an order of magnitude of binding energies. A simple model for the collision process is shown to accurately reproduce the observed scattering rates

The Nature of Nearby Counterparts to Intermediate Redshift Luminous Compact Blue Galaxies II. CO Observations

We present the results of a single-dish beam-matched survey of the three lowest rotational transitions of CO in a sample of 20 local (D < 70 Mpc) Luminous Compact Blue Galaxies (LCBGs). These ~L*, blue, high surface brightness, starbursting galaxies were selected with the same criteria used to define LCBGs at higher redshifts. Our detection rate was 70%, with those galaxies having Lblue<7e9 Lsun no detected. We find the H2 masses of local LCBGs range from 6.6e6 to 2.7e9 Msun, assuming a Galactic CO-to-H2 conversion factor. Combining these results with our earlier HI survey of the same sample, we find that the ratio of molecular to atomic gas mass is low, typically 5-10%. Using a Large Velocity Gradient model, we find that the average gas conditions of the entire ISM in local LCBGs are similar to those found in the centers of star forming regions in our Galaxy, and nuclear regions of other galaxies. Star formation rates, determined from IRAS fluxes, are a few solar masses per year, much higher per unit dynamical mass than normal spirals. If this rate remains constant, the molecular hydrogen depletion time scales are short, 10-200 Myr.Comment: accepted for publication in the ApJ (vol 625

The Archean crust in the Wawa-Chapleau-Timmins region. A field guidebook prepared for the 1983 Archean Geochemistry-Early Crustal Genesis Field Conference

This guidebook describes the characteristics and interrelationships of Archean greenstone-granite and high-grade gneiss terrains of the Superior Province. A 300-km long west to east transect between Wawa and Timmins, Ontario will be used to illustrate regional-scale relationships. The major geological features of the Superior Province are described