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  • 1.
    Darpö, Erik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Classification of pairs of rotations in finite-dimensional Euclidean space2009Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, nr 2-5, s. 333-342Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A rotation in a Euclidean space V is an orthogonal map δ∈ ∈O(V) which acts locally as a plane rotation with some fixed angle a(δ)∈ ∈[0,π]. We give a classification of all finite-dimensional representations of the real algebra ℝ (X,Y) that are given by rotations, up to orthogonal isomorphism.

  • 2.
    Darpö, Erik
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Real commutative division algebras2007Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 10, nr 2, s. 179-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane R->0 x R. The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in R-2 which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with R they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497 - 529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625 - 635, 1983, who achieved partial results on the classification of the real commutative division algebras.

  • 3.
    Greenstein, Jacob
    et al.
    Univ Calif Riverside, Dept Math, Riverside, CA 92521 USA..
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Koszul Duality for Semidirect Products and Generalized Takiff Algebras2017Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 3, s. 675-694Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.

  • 4.
    Herschend, Martin
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the representation ring of a quiver2009Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, nr 6, s. 513-541Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Clebsch-Gordan problem for quiver representations is the problem of   decomposing the tensor product of any two representations of a quiver,   where the tensor product is defined point-wise and arrow-wise. We   introduce the so-called characteristic representations and decompose   their tensor product. This is then applied to solve the Clebsch-Gordan   problem for quivers of type A(n) and D-n. In both cases we also provide   an explicit description of the representation ring.

  • 5.
    Kildetoft, Tobias
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Decomposition of Tensor Products Involving a Steinberg Module2017Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 4, s. 951-975Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G (r) or a finite Chevalley group . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G (r) or . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.

  • 6.
    Kåhrström, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
    Tensoring with infinite-dimensional modules in O_02010Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 13, nr 5, s. 561-587Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the principal block O-0 of the BGG category O for a semi-simple Lie algebra g acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category O. We study such functors, describe their adjoints and show that they give rise to a natural (co) monad structure on O-0. Furthermore, all this generalises to parabolic subcategories of O-0. As an example, we present some explicit computations for the algebra sl(3).

  • 7.
    Külshammer, Julian
    Univ Stuttgart, Inst Algebra & Number Theory, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.
    Pro-species of algebras I: Basic properties2017Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 5, s. 1215-1238Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.

  • 8. Lu, Rencai
    et al.
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Zhao, Kaiming
    Classification of Simple Weight Modules Over the 1-Spatial Ageing Algebra2015Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 18, nr 2, s. 381-395Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we use Block's classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra . Most of these modules have infinite dimensional weight spaces and so far the algebra is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrodinger algebra that have a simple -submodule thus constructing many new simple weight S-modules.

  • 9.
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Applications of the Category of Linear Complexes of Tilting Modules Associated with the Category O2009Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, nr 6, s. 489-512Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category O(p, Lambda).

  • 10.
    Pasquali, Andrea
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Self-injective algebras from Postnikov diagramsIngår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079Artikel i tidskrift (Övrigt vetenskapligt)
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