English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

If $\mathfrak{g}$ is a contragredient Lie superalgebra and $\gamma$ is a rootof $\mathfrak{g},$ we prove the existence and uniqueness of \v{S}apovalovelements for $\gamma$ and give upper bounds on the degrees of theircoefficients. Then we use \v{S}apovalov elements to define some new highestweight modules. If $X$ is a set of orthogonal isotropic roots and $\lambda \in\mathfrak{h}^*$ is such that $\lambda +\rho$ is orthogonal to all roots in $X$,we construct a highest weight module $M^X(\lambda)$ with character$\epsilon^\lambda {p}_X$. Here $p_X$ is a function that counts partitions notinvolving roots in $X$. Examples of such modules can be constructed viaparabolic induction provided $X$ is contained in the set of simple roots ofsome Borel subalgebra. However our construction works without this conditionand provides a highest weight module for the distinguished Borel subalgebra.The main results are analogs of the \v{S}apovalov determinant and the Jantzensum formula for $M^X(\lambda)$ when $\mathfrak{g}$ has type A.For the proof itis enough to study the behavior for certain "relatively general" highestweights. Using an equivalence of categories due to Cheng, Mazorchuk and Wang,the information we require is deduced from the behavior of the modules$M^X(\lambda)$ when $\mathfrak{g}=\mathfrak{gl}(2,1)$ or $\mathfrak{gl}(2,2)$.These low dimensional cases are studied in detail in an appendix.

\vS{a}povalov elements and the Jantzen sum formula for contragredient Lie superalgebras