Let g be a complex reductive Lie algebra and U(g) the universal enveloping algebra of g. Associated to a faithful irreducible finite dimensional representation of g, a square matrix F with entries in U(g) naturally arises and if we consider the entries of F are elements in End(M) of a given U(g)-module M, the minimal polynomial of F is defined as the usual one for an associative algebra over the complex field. Suppose M is a generalized Verma module induced from a character of a parabolic subalgebra of g. In this paper a polynomial q(x) with the parameter of the character is constructed, which equals the minimal polynomial for the generic parameter. Then the two-sided ideal of U(g) generated by the entries of q(F) is studied. We give a sufficient condition for the parameter such that the ideal describes the difference of two left ideals related to M and the corresponding Verma module. The result has many applications. For example we can explicitly give a generator system of the annihilator of M for the generic parameter. This paper also deals with many concrete examples.
In [ZH], R.B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras "à la" Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie superalgebra osp(1,2l) is generated by its intersection with the centralizer of the even part of the enveloping algbra.
A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such "singular" Verma modules for osp(1,2l) cases. In this article we give a…
We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.
In this paper we study the scalar generalized Verma module $M$ associated to a character of a parabolic subgroup of $\operatorname{SL}(E)$. Here $E$ is a finite dimensional vector space over an algebraically closed field $K$ of characteristic zero. The Verma module $M$ has a canonical simple quotient $L$ with a canonical filtration $F$. In the case when the quotient $L$ is finite dimensional we use left annihilator ideals in $U(\mathfrak{sl}(E))$ and geometric results on jet bundles to generalize to an algebraically closed field of characteristic zero a classical formula of W. Smoke on the str…
Let G be a semi simple linear algebraic group over a field of characteristic zero and let V be a finite dimensional irreducible G-module with highest weight vector v. Let P in G be the parabolic subgroup fixing v and let g=Lie(G). We get a canonical filtration of V by P-modules U^k(g)v where U^k(g) is the filtration of the universal enveloping algebra U(g). This filtration was in a previous paper studied in the case where P in G=SL(E) is the subgroup fixing an m-dimensional subspace. The aim of this paper is to use higher direct images of G-linearized sheaves, filtrations of generalized Verma …
We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.
In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebra…
We study general properties of quantisations of Slodowy slices and discuss in detail the case of the minimal nilpotent orbit. Associated varieties of related primitive ideals of U(g) are determined, in the general case, and the de Vos-van Driel conjecture on Verma modules for finite W-algebras is proved in the minimal nilpotent case.
Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced $\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 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 ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ 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,…
This is the author's diploma thesis. In the first part of the thesis the algebra structure on the Ext-spaces Ext^k(M(x), M(y)) of Verma modules M(x) and M(y) in the parabolic category O for the case of the parabolic subalgebras gl(n) x gl(m) for n=1 and n=2 is computed and expressed in terms of quivers. For arbitrary n, more general results about Hom-spaces of projective modules are achieved. The second part of the thesis deals with A-infinity structures on the algebras described above. An explicit construction for a minimal model is given and all higher multiplications are determined in the c…
We study the Ext-algebra of the direct sum of all parabolic Verma modules in the principal block of the Bernstein-Gelfand-Gelfand category O for the hermitian symmetric pair $(\mathfrak{gl}_{n+m}, \mathfrak{gl}_{n} \oplus \mathfrak{gl}_m)$ and present the corresponding quiver with relations for the cases n=1, 2. The Kazhdan-Lusztig combinatorics is used to deduce a general vanishing result for the higher multiplications in the A infinity-structure of a minimal model. An explicit example of the higher multiplications with non-vanishing $m_3$ is included.
We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.
In this paper we establish a connection between the cohomology of a modular Lie algebra and its p-envelopes. We also compute the cohomology of Zassenhaus algebras and their minimal p-envelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p>2.
We construct an L^2-model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If $V$ is split and G is not of type A_n, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n,1)_0. A distinguished feature of these representations in all c…
The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal gra…
In this paper we discuss the structure of the tensor product V'_{α,β}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion, and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operator for modules over vertex operator algebra yields reducibility …
In Lie theory the partial sum property (PSP) says that for a root system in any Kac-Moody algebra, every positive root is an ordered sum of simple roots whose partial sums are all roots. In this paper, we present two generalizations: 1) "Parabolic generalization": if $I$ is a subset of simple roots, every root with positive $I$-height is an ordered sum of roots of $I$-height 1, whose partial sums are all roots. In fact we show this on the Lie algebra level, by showing that every root space is spanned by the Lie words formed from root vectors of $I$-height 1. As an application, we provide a "mi…
Let $\mathcal{O}_c$ be the category of finite-length modules for the Virasoro Lie algebra at central charge $c$ whose composition factors are irreducible quotients of reducible Verma modules. For any $c\in\mathbb{C}$, this category admits the vertex algebraic braided tensor category structure of Huang, Lepowsky, and Zhang. Here, we begin the detailed study of $\mathcal{O}_{c_{p,q}}$ where $c_{p,q} = 1-\frac{6(p-q)^2}{pq}$ for relatively prime integers $p, q \geq 2$; in conformal field theory, $\mathcal{O}_{c_{p,q}}$ corresponds to a logarithmic extension of the central charge $c_{p,q}$ Virasor…
