Classification of Degenerate Verma Modules for E(5, 10)
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220_2021_Article_4031.pdf
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Author(s)
Cantarini, Nicoletta
Caselli, Fabrizio
Kac, Victor
Date Issued
March 13, 2021
Publisher
Springer Berlin Heidelberg
Version
Final published version
Abstract
Abstract
Given a Lie superalgebra
$${\mathfrak {g}}$$
g
with a subalgebra
$${\mathfrak {g}}_{\ge 0}$$
g
≥
0
, and a finite-dimensional irreducible
$${\mathfrak {g}}_{\ge 0}$$
g
≥
0
-module F, the induced
$${\mathfrak {g}}$$
g
-module
$$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$
M
(
F
)
=
U
(
g
)
⊗
U
(
g
≥
0
)
F
is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra
$${\mathfrak {g}}=E(5,10)$$
g
=
E
(
5
,
10
)
with the subalgebra
$${\mathfrak {g}}_{\ge 0}$$
g
≥
0
of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00220-021-04031-z
