Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 141, 32 pages      arXiv:2008.10649      https://doi.org/10.3842/SIGMA.2020.141
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

Extension Quiver for Lie Superalgebra $\mathfrak{q}(3)$

Nikolay Grantcharov a and Vera Serganova b
a) Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
b) Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA

Received August 31, 2020, in final form December 10, 2020; Published online December 21, 2020

Abstract
We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.

Key words: Lie superalgebra; extension quiver; cohomology; flag supermanifold.

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