Abstract
AbstractGiven 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).
Funder
Ministero dell’Istruzione, dell’Università e della Ricerca
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference13 articles.
1. Brilli, D.: A bound on the degree of singular vectors for the exceptional Lie superalgebra $$E(5,10)$$. arXiv:2006.16196
2. Cantarini, N., Caselli, F.: Low Degree Morphisms of $$E(5,10)$$-generalized Verma Modules. Alg. Rep. Theory 23, 2131–2165 (2020)
3. Cantarini, N., Caselli, F., Kac, V.G.: Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras. Adv. Math. 378, article 107523 (2021)
4. Cantarini, N., Kac, V.G.: Infinite dimensional primitive linearly compact Lie superalgebras. Adv. Math. 207, 328–419 (2006)
5. Cheng, S.-J., Kac, V.G.: Structure of some Z-graded Lie superalgebras of vector fields. Transf. Groups 4, 219–272 (1999)
Cited by
5 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献