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Register a new userTranslated simple modules for Lie algebras and simple supermodules for Lie superalgebras
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We prove that the tensor product of a simple and a finite dimensional $mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $mathfrak{q}(n)$-supermodules to that of simple $mathfrak{sl}_n$-modules. Rough structure of simple $mathfrak{q}(n)$-supermodules, considered as $mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $mathcal{O}$.
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We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $mathfrak g$.
In this paper, a family of non-weight modules over Lie superalgebras $S(q)$ of Block type are studied. Free $U(eta)$-modules of rank $1$ over Ramond-Block algebras and free $U(mathfrak{h})$-modules of rank $2$ over Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the sufficient and necessary conditions for such modules to be simple are presented, and their isomorphism classes are also determined. The results cover some existing results.
In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.
Let $min N$, $P(t)in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $mathcal{R}_m(P)=Der(R_m(P))$ and $mathcal{S}_m(P)=Der(S_m(P))$ are called the $m$-th superelliptic Lie algebras associated to $P(t)$. In this paper we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
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