Structure of Modules Induced from Simple Modules with Minimal Annihilator

2004 ◽  
Vol 56 (2) ◽  
pp. 293-309 ◽  
Author(s):  
Oleksandr Khomenko ◽  
Volodymyr Mazorchuk
Keyword(s):  
Simple Module ◽  
Verma Modules ◽  
Simple Complex ◽  
Parabolic Subalgebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Levi Factor ◽  
Generalized Verma Modules ◽  
Primitive Ideals ◽  
Irreducibility Criterion

AbstractWe study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Deficient homomorphisms between generalized Verma modules

2019 ◽  
Vol 30 (11) ◽  
pp. 1950056
Author(s):  
Anthony C. Kable
Keyword(s):  
Necessary Condition ◽  
Verma Modules ◽  
Parabolic Subalgebra ◽  
Generalized Verma Modules

A class of homomorphisms between generalized Verma modules that have an unusual degeneracy is identified. Homomorphisms in this class are called deficient homomorphisms. A family of maximally deficient homomorphisms is constructed. A necessary condition on a parabolic subalgebra is identified for the associated category of generalized Verma modules to admit deficient homomorphisms.

scholarly journals Existence and dimensionality of simple weight modules for quantum enveloping algebras

1993 ◽  
Vol 48 (1) ◽  
pp. 35-40
Author(s):  
Zhiyong Shi
Keyword(s):  
Necessary Conditions ◽  
Simple Module ◽  
Weight Space ◽  
Semisimple Lie Algebra ◽  
Simple Modules ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Sufficient And Necessary Conditions

We give sufficient and necessary conditions for simple modules of the quantum group or the quantum enveloping algebra Uq(g) to have weight space decompositions, where g is a semisimple Lie algebra and q is a nonzero complex number. We show that(i) if q is a root of unity, any simple module of Uq(g) is finite dimensional, and hence is a weight module;(ii) if q is generic, that is, not a root of unity, then there are simple modules of Uq(g) which do not have weight space decompositions.Also the group of units of Uq(g) is found.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

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).

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

scholarly journals Primitive ideals, twisting functors and star actions for classical Lie superalgebras

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Kevin Coulembier ◽  
Volodymyr Mazorchuk
Keyword(s):  
Representation Theory ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Primitive Ideals

AbstractWe study three related topics in representation theory of classical Lie superalgebras. The first one is classification of primitive ideals, i.e. annihilator ideals of simple modules, and inclusions between them. The second topic concerns Arkhipov’s twisting functors on the BGG category

scholarly journals Complexity for modules over the classical Lie superalgebra

2012 ◽  
Vol 148 (5) ◽  
pp. 1561-1592 ◽  
Author(s):  
Brian D. Boe ◽  
Jonathan R. Kujawa ◽  
Daniel K. Nakano
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Projective Resolution ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Minimal Projective Resolution ◽  
Finite Dimensional ◽  
Reductive Lie Algebra ◽  
Completely Reducible ◽  
Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

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