Left-symmetric Superalgebras on Special Linear Lie Superalgebras

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras ${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on ${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit left-symmetric superalgebras for $m \geq 3$, and (iii) ${\mathfrak{sl}}(m+1|m)$ admits a left-symmetric superalgebra for every $m \geq 1$. To prove these results we combine existing results on the existence and classification of left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed analysis of small representations of the Lie superalgebras ${\mathfrak{sl}}(m|1)$. We also conjecture that ${\mathfrak{sl}}(m|n)$ admits left-symmetric superalgebras if and only if $m = n+1$.Comment: 24 pages, 3 table

Parity duality of super rr-matrices via O\mathcal O-operators and pre-Lie superalgebras

This paper studies super $r$-matrices and operator forms of the super classical Yang-Baxter equation. First by a unified treatment, the classical correspondence between $r$-matrices and $\mathcal{O}$-operators is generalized to a correspondence between homogeneous super $r$-matrices and homogeneous $\mathcal{O}$-operators. Next, by a parity reverse of Lie superalgebra representations, a duality is established between the even and the odd $\mathcal{O}$-operators, giving rise to a parity duality among the induced super $r$-matrices. Thus any homogeneous \OO-operator or any homogeneous super $r$-matrix with certain supersymmetry produces a parity pair of super $r$-matrices, and generates an infinite tree hierarchy of homogeneous super $r$-matrices. Finally, a pre-Lie superalgebra naturally defines a parity pair of $\mathcal{O}$-operators, and thus a parity pair of super $r$-matrices.Comment: 26 pages; to appear in Math Research Letter

The value of genotype-specific reference for transcriptome analyses in barley

It is increasingly apparent that although different genotypes within a species share “core” genes, they also contain variable numbers of “specific” genes and different structures of “core” genes that are only present in a subset of individuals. Using a common reference genome may thus lead to a loss of genotype-specific information in the assembled Reference Transcript Dataset (RTD) and the generation of erroneous, incomplete or misleading transcriptomics analysis results. In this study, we assembled genotype-specific RTD (sRTD) and common reference–based RTD (cRTD) from RNA-seq data of cultivated Barke and Morex barley, respectively. Our quantitative evaluation showed that the sRTD has a significantly higher diversity of transcripts and alternative splicing events, whereas the cRTD missed 40% of transcripts present in the sRTD and it only has ∼70% accurate transcript assemblies. We found that the sRTD is more accurate for transcript quantification as well as differential expression analysis. However, gene-level quantification is less affected, which may be a reasonable compromise when a high-quality genotype-specific reference is not available