Whittaker categories and strongly typical Whittaker modules for Lie superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules

Cohomology of quantum groups: An analog of Kostant's Theorem

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.Comment: 12 page

Design advanced algebra courses with significant learning experiences: Using SageMath to enhance Project-based learning activities

In advanced algebra classes, the transition from problem-solving to proof-based writing creates challenges for students. The students are not motivated to excel in these courses because they do not see how the materials can be applied to their real life or other classes. In this proposed project, we aim to create project-based learning activities that provide significant learning experiences for students in these high level classes. In particular, we design mini-projects, including applications of materials from the current classes to topics from some other classes or real life experience. These mini-projects require both the understanding of abstract concepts and the ability to conduct multiple computational steps. To help students complete the mini-projects effectively, we teach students to use SageMath, a free open-source mathematics software system with capabilities of abstract and symbolic computation, as a powerful calculator to complete their mini-projects. This new form of learning also serves as a preparatory step for students in their Senior projects. A pilot study is to implement this project in the Linear Algebra and Discrete Mathematics courses at UNG

Whittaker Categories and Whittaker Modules for Lie Superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, the construction of Whittaker modules from Whittaker modules for the even part. © 2014 Copyright Taylor & Francis Group, LLC

The Oil Yield, Mineral Content, and Fatty Acid Compositions of Some Rye (Secale cereale) Grains

The oil contents of rye grain seeds ranged from 0.70% (type 25) to 3.92 % (type 1). Palmitic acid contents of oils changed between 10.82% (type 26) and 22.43% (type 10). In addition, while oleic acid contents of oil samples vary between 20.61% (type 17) and 37.86% (type 3), linoleic acid contents of oils changed between 18.91% (type 3) to 54.0% (type 13). Also, linolenic acid contents of oil samples were found between 2.43% (type 1) and 8.34% (type 26). Total saturated fatty acid contents of oil samples changed between 15.57% (typ 26) and 34.38% (type 1). K contents of rye grains were found between 3810.31 mg/kg (type 2) and 6148.28 mg/kg (type 17). While P contents of grains vary between 1806.43 mg/kg (5) and 3710.25 mg/kg (18), Mg contents of samples ranged from 962.87 mg/kg (type 5) to 1602.33 mg/kg (type 12). The highest Ca content was determined in 14 samples (1447.96 mg/kg). The crude protein contents of rye grains ranged from 10.08% (type 1) to 15.25% (type 3). As a result, rye grain is rich in minerals and essential fatty acids for human health.Selcuk University Scientific Research Project (S.U.-BAP, Konya-TURKEY)Selcuk UniversityThis work was supported by Selcuk University Scientific Research Project (S.U.-BAP, Konya-TURKEY)