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| 2. |
- Mazorchuk, Volodymyr, et al.
(författare)
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Radical *-doubles of finite-dimensional algebras
- 2004
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795. ; 390, s. 293-309
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Tidskriftsartikel (refereegranskat)abstract
- The classification of the representation types for radical doubles of finite-dimensional algebras over the field of complex numbers was discussed. For the usal doubling, independent adjoints were added to all elements of original algebra where as for the radical doubling independent adjoints were added to the elements from the jacobsan radical of the algebra. The main advantange of the new construction is the representation type of the radical doubles happens to be a Mortia invariant of the original algebra. The result shows a tame-wild dichotomy for the given problem.
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| 3. |
- Nilsson, Jonathan, 1986-
(författare)
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Simple Modules over Lie Algebras
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Simple modules are the elemental components in representation theory for Lie algebras, and numerous mathematicians have worked on their construction and classification over the last century. This thesis consists of an introduction together with four research articles on the subject of simple Lie algebra modules. In the introduction we give a light treatment of the basic structure theory for simple finite dimensional complex Lie algebras and their representations. In particular we give a brief overview of the most well-known classes of Lie algebra modules: highest weight modules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolically induced modules.The four papers contribute to the subject by construction and classification of new classes of Lie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. modules which are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra. In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and we determine which of these modules are simple. For sl2 we also obtain some additional results on tensor product decomposition. Paper II uses the theory of coherent families to obtain a similar classification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give a proof that U(h)-free modules do not exist for any other simple finite-dimensional algebras which completes the classification. In Paper III we construct a new large family of simple generalized Whittaker modules over the general linear Lie algebra gl2n(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)-modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.
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| 4. |
- Umutabazi, Vincent, 1982-
(författare)
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Smooth Schubert varieties and boolean complexes of involutions
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is composed of two papers both in algebraic combinatorics and Coxeter groups.In Paper I, we concentrate on smoothness of Schubert varieties indexed by involutions from finite simply laced types. We show that if a Schubert variety indexed by an involution of a finite and simply laced Coxeter group is smooth, then that involution must be the longest element of a parabolic subgroup.Given a Coxeter system (W, S), we introduce in Paper II the boolean complex of involutions of W as an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By using discrete Morse Theory, we compute the homotopy type for a large class of W, including all finite Coxeter groups. In all cases, the homotopy type is that of a wedge of spheres of dimension |S| − 1. In addition, we provide a recurrence formula for the number of spheres in the wedge.
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| 5. |
- Agerholm, Troels, et al.
(författare)
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On selfadjoint functors satisfying polynomial relations
- 2011
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 330:1, s. 448-467
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Tidskriftsartikel (refereegranskat)abstract
- We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors. are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products. Serre subcategories, quotients and centralizer subalgebras.
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| 6. |
- Andersen, Henning Haahr, et al.
(författare)
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Category O for quantum groups
- 2015
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 17:2, s. 405-431
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Tidskriftsartikel (refereegranskat)abstract
- 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 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 to determine all irreducible characters as well as the characters of all indecomposable tilting modules in O-q. As a consequence, we also recover the known result that the generic quantum case behaves like the classical category O.
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| 7. |
- Batra, Punita, et al.
(författare)
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Blocks and modules for Whittaker pairs
- 2011
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Ingår i: Journal of Pure and Applied Algebra. - 0022-4049 .- 1873-1376. ; 215:7, s. 1552-1568
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Tidskriftsartikel (refereegranskat)abstract
- Inspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.
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| 8. |
- Baur, Karin, et al.
(författare)
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Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras
- 2012
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 372, s. 85-107
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Tidskriftsartikel (refereegranskat)abstract
- We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type A. The formula involves various aspects of combinatorics of Dyck paths and leads to a new interesting integral sequence.
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| 9. |
- Coulembier, Kevin, et al.
(författare)
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Dualities and derived equivalences for category O
- 2017
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Ingår i: Israel Journal of Mathematics. - : HEBREW UNIV MAGNES PRESS. - 0021-2172 .- 1565-8511. ; 219:2, s. 661-706
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Tidskriftsartikel (refereegranskat)abstract
- We determine the Ringel duals for all blocks in the parabolic versions of the BGG category associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category . We subsequently classify all derived equivalence classes of blocks of category in type A which preserve the Koszul grading.
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| 10. |
- Coulembier, Kevin, et al.
(författare)
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Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules
- 2015
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Ingår i: SIGMA. Symmetry, Integrability and Geometry. - 1815-0659 .- 1815-0659. ; 11
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the categories of Gelfand-Zeitlin modules of g = gl(n), and Whittaker modules associated with a semi-simple complex finite-dimensional algebra g are extension full in the category of all g-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.
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