On the variety of Lagrangian subalgebras, II

When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety ${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally isotropic with respect to $K_1 - K_2$, where $K_i$ is the Killing form on the ith factor. We show the irreducible components of ${\mathcal L}$ are smooth, classify them in terms of the generalized Belavin-Drinfeld triples introduced by Schiffmann, and relate them to orbits of the adjoint group $G\times G$. Building on ideas of Yakimov, we give a new proof of Karolinsky's classification of the diagonal $G$-orbits in ${\mathcal L}$. Our proof enables us to compute of the normalizer in ${\frak g}$ of a subalgebra in ${\mathcal L}$ under the diagonal action. As a consequence, we recover the classification of Belavin-Drinfeld triples. By results of math.DG/9909005, ${\mathcal L}$ is a Poisson variety and we determine the rank of the symplectic leaf at each point of ${\mathcal L}$ in terms of combinatorial data and relate the symplectic leaves to intersections of orbits of subgroups of $G\times G$. As a consequence, an intrinsically defined Poisson structure on each conjugacy class on $G$ has an open symplectic leaf and we determine the rank at each point of the conjugacy class.Comment: revised version, 32 pages, some proofs have been made more efficient and some comments were removed. To appear in Ann. EN

Thompson's conjecture for real semi-simple Lie groups

A proof of Thompson's conjecture for real semi-simple Lie groups has been given by Kapovich, Millson, and Leeb. In this note, we give another proof of the conjecture by using a theorem of Alekseev, Meinrenken, and Woodward from symplectic geometry.Comment: 18 pages, LaTe

Poisson geometry of the Grothendieck resolution of a complex semisimple group

We study a Poisson structure $\pi$ on the Grothendieck resolution $X$ of a complex semi-simple group $G$ and prove that the desingularization map $\mu:(X,\pi) \to (G,\pi_0)$ is Poisson, where $\pi_0$ is a Poisson structure such that intersections of conjugacy classes and opposite Bruhat cells $BwB_-$ are Poisson subvarieties. We compute the symplectic leaves of $X$ and show that $(X, \pi)$ resolves singularities of $(G, \pi_0)$.Comment: Final version for publication in MMJ, added reference

The Child Play Behavior and Activity Questionnaire: A Parent-Report Measure of Childhood Gender-Related Behavior in China

Boys and girls establish relatively stable gender stereotyped behavior patterns by middle childhood. Parent-report questionnaires measuring children’s gender-related behavior enable researchers to conduct large-scale screenings of community samples of children. For school-aged children, two parent-report instruments, the Child Game Participation Questionnaire (CGPQ) and the Child Behavior and Attitude Questionnaire (CBAQ), have long been used for measuring children’s sex-dimorphic behaviors in Western societies, but few studies have been conducted using these measures for Chinese populations. The current study aimed to empirically examine and modify the two instruments for their applications to Chinese society. Parents of 486 Chinese boys and 417 Chinese girls (6–12 years old) completed a questionnaire comprising items from the CGPQ and CBAQ, and an additional 14 items specifically related to Chinese gender-specific games. Items revealing gender differences in a Chinese sample were identified and used to construct a Child Play Behavior and Activity Questionnaire (CPBAQ). Four new scales were generated through factor analysis: a Gender Scale, a Girl Typicality Scale, a Boy Typicality Scale, and a Cross-Gender Scale (CGS). These scales had satisfactory internal reliabilities and large effect sizes for gender. The CPBAQ is believed to be a promising instrument for measuring children’s gender-related behavior in China

Safety Voicing: The impact of job insecurity and the differences in severity of safety concerns.

The aim of this current research was two-fold; one aim was to develop a deeper understanding of job insecurity and its association with safety voicing. The perception of job insecurity was specifically examined in relation to job insecurity and its association with safety voice. The second aim was to examine safety voice with possible antecedents based on past research. This involved investigating the proposition that safety voice falls within a safety severity concern continuum. This continuum suggests that the severity of the safety concern may play a role in an employee’s willingness to voice these concerns. A questionnaire was devised to investigate perceptions of 47 employees from a single organisation. This organisation is undergoing a planned future job redundancy process (within a few years as of 2014). Correlational and univariate analyses were used to investigate any associations and differences in means between the different measures. Results from this research found partial support for the hypothesis that perceived organisational support, safety climate, and perceived co-worker support were positively associated with safety voice. The hypothesis that higher job insecurity would be associated with lower safety voice concerns was tested, and was not found not to be statistically significant to support the idea. This dissertation offers a preliminary indication that safety voice varies according to severity, and that job insecurity may affect employees’ likelihood to voice safety concerns. Practical implications and directions for further research are discussed