On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Patrick Polo

Displaying similar documents to “On the $K$ -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case”

On the composition structure of the twisted Verma modules for $𝔰𝔩 (3, ℂ)$

Libor Křižka, Petr Somberg (2015)

Archivum Mathematicum

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We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $𝔰𝔩 (3, ℂ)$ , including the explicit structure of singular vectors for both $𝔰𝔩 (3, ℂ)$ and one of its Lie subalgebras $𝔰𝔩 (2, ℂ)$ , and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$ -modules on the Schubert cells in the full flag manifold for $SL (3, ℂ)$ .

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors...

Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

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Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_{d e g} N$ such that $^{t} M /^{t + 1} M ≃^{t} N /^{t + 1} N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $d i m_{k} H o m (M, M) = d i m_{k} H o m (P, M)$ if and only if M has no proper degeneration $M <_{d e g} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from...

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

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An $𝒮$ -closed submodule of a module $M$ is a submodule $N$ for which $M / N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$ -closed submodule $N$ of $M$ is a direct summand of $M$ . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$ -modules are projective if and only if all right $R$ -modules are GCS-modules.

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Otmar Venjakob (2002)

Journal of the European Mathematical Society

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This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $Λ$ of a $p$ -adic analytic group $G$ . For $G$ without any $p$ -torsion element we prove that $Λ$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $Λ$ -module. This is classical when $G = ℤ_{p}^{k}$ for some integer $k \geq 1$ , but was previously unknown in the non-commutative case. Then the category...

Loewy coincident algebra and $Q F$ -3 associated graded algebra

Hiroyuki Tachikawa (2009)

Czechoslovak Mathematical Journal

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We prove that an associated graded algebra $R_{G}$ of a finite dimensional algebra $R$ is $Q F$ (= selfinjective) if and only if $R$ is $Q F$ and Loewy coincident. Here $R$ is said to be Loewy coincident if, for every primitive idempotent $e$ , the upper Loewy series and the lower Loewy series of $R e$ and $e R$ coincide. $Q F$ -3 algebras are an important generalization of $Q F$ algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra $R$ , the associated graded...

Separable functors for the category of Doi Hom-Hopf modules

Shuangjian Guo, Xiaohui Zhang (2016)

Colloquium Mathematicae

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Let $̃ (_{k}) {(H)}_{A}^{C}$ be the category of Doi Hom-Hopf modules, $̃ {(_{k})}_{A}$ be the category of A-Hom-modules, and F be the forgetful functor from $̃ (_{k}) {(H)}_{A}^{C}$ to $̃ {(_{k})}_{A}$ . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor, Andrzej Mróz (2007)

Colloquium Mathematicae

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Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{X})}_{X \in} \in ℕ^{}$ such that $M ≅ ⨁_{X \in} X^{m_{X}}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type $̃_{p, q}$ . In the second case detailed algorithms are given (Algorithms 4.5...

Branching problems and $𝔰𝔩 (2, ℂ)$ -actions

Pavle Pandžić, Petr Somberg (2015)

Archivum Mathematicum

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We study certain $𝔰𝔩 (2, ℂ)$ -actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(𝔤, 𝔭)$ , $(𝔤^{'}, 𝔭^{'})$ of Lie algebras and their parabolic subalgebras.

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

Johannes Huebschmann (1998)

Annales de l'institut Fourier

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For any Lie-Rinehart algebra $(A, L)$ , B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$ -algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -module structures on $A$ ; likewise, generators for the Gerstenhaber algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -connections on $A$ . When $L$ is projective as an $A$ -module, given a B-V algebra structure $\partial$ on $Λ_{A} L$ , the homology of the B-V algebra $(Λ_{A} L, \partial)$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A, L)$ -module structure determined...

Structure of central torsion Iwasawa modules

Susan Howson (2002)

Bulletin de la Société Mathématique de France

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We describe an approach to determining, up to pseudoisomorphism, the structure of a central-torsion module over the Iwasawa algebra of a pro- $p$ , $p$ -adic, Lie group containing no element of order $p$ . The techniques employed follow classical methods used in the commutative case, but using Ore’s method of localisation. We then consider the properties of certain invariants which may prove useful in determining the structure of a module. Finally, we describe the case of pro- $p$ subgroups of ${GL}_{2} (ℤ_{p})$ ...

Lifting $D$ -modules from positive to zero characteristic

João Pedro P. dos Santos (2011)

Bulletin de la Société Mathématique de France

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We study liftings or deformations of $D$ -modules ( $D$ is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic $D$ -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given $D$ -module in positive characteristic. At the end we compare...

On twisted group algebras of OTP representation type

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G = G_{p} \times B$ is a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $S^{λ} G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $S^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $S^{λ} G_{p}$ -module V and an irreducible...

Displaying similar documents to “On the K -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case”

Displaying similar documents to “On the $K$ -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case”