Generalized Twisted Quantum Doubles and the McKay Correspondence

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions H = \mb{C}[\bar{G}] \bowtie \mb{C}[G] ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if $G$ is a subgroup of SU_2(\mb{C}) then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of $G$ stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when $\bar{G} = 1$.Comment: 5 figure

Structure of the module of vector-valued modular forms

Let $V$ be a representation of the modular group $\Gamma$ of dimension $p$. We show that the $\mathbb{Z}$-graded space $\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{H}$ considered as a functor from $\Gamma$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincar\'{e} of $\mathcal{H}(V)$ in some cases

Using Whole-Group Metabolic Rate and Behaviour to Assess the Energetics of Courtship in Red-Sided Garter Snakes

Reproductive effort is an important aspect of life history as reproductive success is arguably the most important component of fitness. Males tend to compete for access to females and, in the process, expend their energetic capital on mate searching, maleemale competition and courtship rather than directly on offspring. Red-sided garter snakes, Thamnophis sirtalis parietalis, are an exceptional model for studying energetic costs of courtship and mating as they fast during the spring mating season, which segregates the cost of energy acquisition from the cost of courtship and mating. However, measuring an individual male\u27s metabolic rate during courtship is complicated by the fact that male courtship behaviour in redsided garter snakes is dependent on both the detection of a female sexual attractiveness pheromone and on facilitated courtship (i.e. vigorous courtship is only exhibited in the presence of other males). Thus, traditional techniques of placing a mask over the head of individuals would prevent male courtship behaviour, and single animals placed in a flow-through chamber would not yield ecologically realistic levels of courtship, which are only seen in the context of a mating aggregation in this species. Because of these difficulties, we placed groups of males in a flow-through metabolic chamber together with a single female whose respiratory gases were vented outside the chamber to yield a whole-group metabolic rate during competitive courtship. We also measured the standard metabolic rates (SMR) of the males individually for comparison with active metabolic rates. Conservative estimates of peak group metabolic rates during courtship are 10e20 times higher than resting group metabolic rate, which was 1.88 times higher than SMR. These measurements, coupled with the fact that these males are aphagous during the breeding, indicates that costs of courtship may be high for males and has implications for the male mating tactics in this system

Einstein Supergravity and New Twistor String Theories

A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. The theories include one with the spectrum of N=8 supergravity, another with the spectrum of N=4 supergravity coupled to N=4 super-Yang-Mills, and a family with $N\ge 0$ supersymmetries with the spectra of self-dual supergravity coupled to self-dual super-Yang-Mills. The non-supersymmetric string with N=0 gives self-dual gravity coupled to self-dual Yang-Mills and a scalar. A three-graviton amplitude is calculated for the N=8 and N=4 theories and shown to give a result consistent with the cubic interaction of Einstein supergravity.Comment: LaTeX, 69 pages, no figures; v2: minor corrections made, footnotes and references adde

On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group H by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle w is gauge equivalent to a twisted quantum double of an elementary abelian 2-group if, and only if, w^2 is trivial; second, we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8, and classify the braided tensor equivalence classes of these quasi-triangular quasi-bialgebras. It turns out that there are exactly 20 such equivalence classes.Comment: 27 pages, LateX, a few of typos in v2 correcte

Pulling back the curtain: Modeling the motivational process underlying the contemporary engagement concept

Researchers and practitioners appear to have captured something of importance in the concept of engagement. Engagement surveys have been shown to be effective predictors of job performance and other important organizational outcomes, and yet there is clearly no single unifying description or understanding of the engagement concept in the extant literature or applied domain. This research sought to clearly define, operationalize, and model the construct that best represents what researchers and practitioners today intend when they use the term “engagement.” Specifically, the purpose of this research was to make five unique contributions to the current discussion of the engagement concept: 1) identify the primary sources of confusion surrounding the contemporary engagement concept and summarize current areas of agreement; 2) identify the nomological network of engagement; 3) describe the relationship between engagement, motivation, and performance in order to provide a more complete definition of engagement 4) present a new theoretical model of the engagement process; and 5) empirically test components of the new engagement model. A new comprehensive theoretical model of job engagement was presented that outlined the manner in which the constructs within engagement’s nomological network interact and impact an active cognitive motivational mechanism underlying job engagement. The practical effect of this model was to essentially “pull back the curtain” on the engagement concept to reveal the motivation mechanism underlying the process. Engagement was more precisely defined in this research as job engagement, a state of active motivation to perform in one’s job, characterized by an ongoing willingness to expend effort in the service of sustaining job performance. Hypotheses examining specific aspects of the new job engagement model were analyzed using data from a typical applied engagement survey conducted in March of 2008 at a Fortune 500 company with employees located in the United States and Canada. Results demonstrated that job engagement is influenced by the four broad antecedent factors of incentives, directives, enablers, and feedback. These four factors impact job engagement through four input points into the process by impacting a worker’s commitment to job performance, task goal identification, task efficacy, and assessment of goal achievement respectively. The primary implication of this research for theoretical research is to clarify areas of confusion in prior research and provide a theoretically sound model of job engagement to guide future research. Practitioners can draw upon the simple four factor structure of the antecedents of job engagement to conduct analyses of survey data and design targeted interventions at all levels within an organization