| 1. |
- Agerholm, Troels, et al.
(author)
-
On selfadjoint functors satisfying polynomial relations
- 2011
-
In: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 330:1, s. 448-467
-
Journal article (peer-reviewed)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.
|
|
| 2. |
- Ahmed, Chwas, et al.
(author)
-
On the number of principal ideals in d-tonal partition monoids
- 2021
-
In: Annals of Combinatorics. - : Springer. - 0218-0006 .- 0219-3094. ; 25:1, s. 79-113
-
Journal article (peer-reviewed)abstract
- For a positive integer d, a non-negative integer n and a non-negative integer h <= n, we study the number C-n((d)) of principal ideals; and the number C-n,h((d)) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.
|
|
| 3. |
- Ahmed, Chwas, et al.
(author)
-
Tonal partition algebras : fundamental and geometrical aspects of representation theory
- 2024
-
In: Communications in Algebra. - : Taylor & Francis. - 0092-7872 .- 1532-4125. ; 52:1, s. 233-271
-
Journal article (peer-reviewed)abstract
- For l, n is an element of N we define tonal partition algebra P-l (n) over Z[delta]. We construct modules {Delta mu} mu for P-l (n) over Z[delta], and hence over any integral domain containing Z[delta] (such as C[delta]), that pass to a complete set of irreducible modules over the field of fractions. We show that P-l (n) is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a "geometrical" index set for the Delta-modules, we give an order with respect to which the decomposition matrix over C (with d. C-x) is upper-unitriangular. We establish several crucial properties of the Delta-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.
|
|
| 4. |
- Andersen, Henning Haahr, et al.
(author)
-
Category O for quantum groups
- 2015
-
In: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 17:2, s. 405-431
-
Journal article (peer-reviewed)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.
|
|
| 5. |
- Batra, Punita, et al.
(author)
-
Blocks and modules for Whittaker pairs
- 2011
-
In: Journal of Pure and Applied Algebra. - 0022-4049 .- 1873-1376. ; 215:7, s. 1552-1568
-
Journal article (peer-reviewed)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.
|
|
| 6. |
- Baur, Karin, et al.
(author)
-
Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras
- 2012
-
In: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 372, s. 85-107
-
Journal article (peer-reviewed)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.
|
|
| 7. |
|
|
| 8. |
|
|
| 9. |
|
|
| 10. |
- Chan, Aaron, et al.
(author)
-
Diagrams and discrete extensions for finitary 2-representations
- 2019
-
In: Mathematical proceedings of the Cambridge Philosophical Society (Print). - : Cambridge University Press. - 0305-0041 .- 1469-8064. ; 166:2, s. 325-352
-
Journal article (peer-reviewed)abstract
- In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.
|
|
| 11. |
- Chen, Chih-Whi, et al.
(author)
-
Simple supermodules over lie superalgebras
- 2021
-
In: Transactions of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9947 .- 1088-6850. ; 374:2, s. 899-921
-
Journal article (peer-reviewed)abstract
- We show that, for many Lie superalgebras admitting a compatible Z-grading, the Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebras of type I, in particular, to the general linear Lie superalgebra gl(m broken vertical bar n,). In the latter case we also show that the rough structure of simple gl(m broken vertical bar n,)-supermodules and also that of Kac supermodules depends only on the annihilator of the gl(m) circle plus gl(n)-input and hence can be computed using the combinatorics of BGG category O.
|
|
| 12. |
- Chen, Chih-Whi, et al.
(author)
-
Some Homological Properties of Category O for Lie Superalgebras
- 2023
-
In: Journal of the Australian Mathematical Society. - : Cambridge University Press. - 1446-7887 .- 1446-8107. ; 114:1, s. 50-77
-
Journal article (peer-reviewed)abstract
- For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule Delta(lambda) to be such that every nonzero homomorphism from another Verma supermodule to Delta(lambda) is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras pe(n) and, furthermore, to reduce the problem of description of Ext(O)(1)(L(mu), Delta(lambda)) for pe(n) to the similar problem for the Lie algebra gl(n). Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category O-p for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra pe(n) and the orthosymplectic Lie superalgebra osp(2 vertical bar 2n).
|
|
| 13. |
- Chen, Chih-Whi, et al.
(author)
-
Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras
- 2021
-
In: Mathematische Zeitschrift. - : Springer Nature. - 0025-5874 .- 1432-1823. ; 297:1-2, s. 255-281
-
Journal article (peer-reviewed)abstract
- We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.
|
|
| 14. |
- Chen, Chih-Whi, et al.
(author)
-
Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras
- 2023
-
In: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 401:1, s. 717-768
-
Journal article (peer-reviewed)abstract
- We study various categories of Whittaker modules over a type I Lie super algebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor gamma(zeta) .We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category O and of certain singular categories of Harish-Chandra (g, g(0)& macr;)-bimodules. We also show that gamma(zeta) is a realization of the Serre quotient functor. We further investigate a q-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor gamma(zeta) and various realizations of Serre quotients and Serre quotient functors categorify this q-symmetrized Fock space and its q-symmetrizer. In this picture, the canonical and dual canonical bases in this q-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.
|
|
| 15. |
- Cheng, Shun-Jen, et al.
(author)
-
Equivalence of Blocks for the General Linear Lie Superalgebra
- 2013
-
In: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 103:12, s. 1313-1327
-
Journal article (peer-reviewed)abstract
- We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category for a general linear Lie superalgebra to an integral block of for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of .
|
|
| 16. |
- Coulembier, Kevin, et al.
(author)
-
Dualities and derived equivalences for category O
- 2017
-
In: Israel Journal of Mathematics. - : HEBREW UNIV MAGNES PRESS. - 0021-2172 .- 1565-8511. ; 219:2, s. 661-706
-
Journal article (peer-reviewed)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.
|
|
| 17. |
- Coulembier, Kevin, et al.
(author)
-
Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules
- 2015
-
In: Symmetry, Integrability and Geometry. - 1815-0659. ; 11
-
Journal article (peer-reviewed)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.
|
|
| 18. |
- Coulembier, Kevin, et al.
(author)
-
Indecomposable manipulations with simple modules in category O
- 2019
-
In: Mathematical Research Letters. - : INT PRESS BOSTON, INC. - 1073-2780 .- 1945-001X. ; 26:2, s. 447-499
-
Journal article (peer-reviewed)abstract
- We study the problem of indecomposability of translations of simple modules in the principal block of BGG category O for sl(n), as conjectured in [KiM1]. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for n <= 6. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results.
|
|
| 19. |
- Coulembier, Kevin, et al.
(author)
-
Primitive ideals, twisting functors and star actions for classical Lie superalgebras
- 2016
-
In: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; 718, s. 207-253
-
Journal article (peer-reviewed)abstract
- We 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 O. The third topic addresses deformed orbits of the Weyl group. These take over the role of the usual Weyl group orbits for Lie algebras, in the study of primitive ideals and twisting functors for Lie superalgebras.
|
|
| 20. |
- Coulembier, Kevin, et al.
(author)
-
Some homological properties of category O. III
- 2015
-
In: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 283, s. 204-231
-
Journal article (peer-reviewed)abstract
- We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish-Chandra bimodules, but disprove it for singular blocks.
|
|
| 21. |
- Coulembier, Kevin, et al.
(author)
-
Some homological properties of category O. IV
- 2017
-
In: Forum mathematicum. - : Walter de Gruyter GmbH. - 0933-7741 .- 1435-5337. ; 29:5, s. 1083-1124
-
Journal article (peer-reviewed)abstract
- We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.
|
|
| 22. |
- Coulembier, Kevin, et al.
(author)
-
The G-Centre and Gradable Derived Equivalences
- 2018
-
In: Journal of the Australian Mathematical Society. - : CAMBRIDGE UNIV PRESS. - 1446-7887 .- 1446-8107. ; 105:3, s. 289-315
-
Journal article (peer-reviewed)abstract
- We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group G. Our generalisation, which we call the G-centre, is designed to control the endomorphism category of the grading shift functors. We show that the G-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory.
|
|
| 23. |
- Drozd, Yuriy, et al.
(author)
-
Koszul duality for extension algebras of standard modules
- 2007
-
In: Journal of Pure and Applied Algebra. - : Elsevier BV. - 0022-4049 .- 1873-1376. ; 211:2, s. 484-496
-
Journal article (peer-reviewed)abstract
- We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category Omicron, and some quasi-hereditary algebras with Cartan decomposition in the sense of Konig.
|
|
| 24. |
- Drozd, Yuriy, et al.
(author)
-
Representation type of H-infinity(lambda)mu(1)
- 2006
-
In: Quarterly Journal of Mathematics. - : Oxford University Press (OUP). - 0033-5606 .- 1464-3847. ; 57:3, s. 319-338
-
Journal article (peer-reviewed)abstract
- For a semi-simple finite-dimensional complex Lie algebra, we classify the representation type of the associative algebras associated with the categories H-infinity(lambda)mu(1) of Harish-Chandra bimodules for g.
|
|
| 25. |
- Dubsky, Brendan Frisk, et al.
(author)
-
Category O for the Schrodinger algebra
- 2014
-
In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 460, s. 17-50
-
Journal article (peer-reviewed)abstract
- We study category O for the (centrally extended) Schrodinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and relations for the finite dimensional part of O. We use this to determine the center of the universal enveloping algebra and annihilators of Verma modules. Finally, we classify primitive ideals of the universal enveloping algebra which intersect the center of the centrally extended Schrodinger algebra trivially.
|
|
| 26. |
- Early, Nick, et al.
(author)
-
Canonical Gelfand-Zeitlin Modules over Orthogonal Gelfand-Zeitlin Algebras
- 2020
-
In: International mathematics research notices. - : OXFORD UNIV PRESS. - 1073-7928 .- 1687-0247. ; 2020:20, s. 6947-6966
-
Journal article (peer-reviewed)abstract
- We prove that every orthogonal Gelfand-Zeitlin algebra U acts (faithfully) on its Gelfand-Zeitlin subalgebra Gamma. Considering the dual module, we show that every Gelfand-Zeitlin character of Gamma is realizable in a U-module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on Gamma gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules, which realize the given Gelfand-Zeitlin characters.
|
|
| 27. |
|
|
| 28. |
- Forsberg, Love
(author)
-
Semigroups, multisemigroups and representations
- 2017
-
Doctoral thesis (other academic/artistic)abstract
- This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n.A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative.A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup.Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver.Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero.Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities).Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.
|
|
| 29. |
- Frisk, Anders, 1977-
(author)
-
On Stratified Algebras and Lie Superalgebras
- 2007
-
Doctoral thesis (other academic/artistic)abstract
- This thesis, consisting of three papers and a summary, studies properties of stratified algebras and representations of Lie superalgebras.In Paper I we give a characterization when the Ringel dual of an SSS-algebra is properly stratified.We show that for an SSS-algebra, whose Ringel dual is properly stratified, there is a (generalized) tilting module which allows one to compute the finitistic dimension of the SSS-algebra, and moreover, it gives rise to a new covariant Ringel-type duality.In Paper II we give a characterization of standardly stratified algebras in terms of certain filtrations of (left or right) projective modules, generalizing the corresponding theorem of V. Dlab. We extend the notion of Ringel duality to standardly stratified algebras and estimate their finitistic dimension in terms of endomorphism algebras of standard modules.Paper III deals with the queer Lie superalgebra and the corresponding BGG-category O. We show that the typical blocks correspond to standardly stratified algebras, and we generalize Kostant's Theorem to the queer Lie superalgebra.
|
|
| 30. |
|
|
| 31. |
- Frisk, Anders, et al.
(author)
-
Properly stratified algebras and tilting
- 2006
-
In: Proc. London Math. Soc.. - 0024-6115. ; 92:1, s. 29-
-
Journal article (peer-reviewed)abstract
- We study the properties of tilting modules in the context of properlystratified algebras. In particular, we answer the question when theRingel dual of a properly stratified algebra is properly stratifieditself, and show that the class of properly stratified algebras forwhich the characteristic tilting and cotilting modules coincide isclosed under taking the Ringel dual. Studying stratified algebras,whose Ringel dual is properly stratified, we discover a new Ringel-typeduality for such algebras, which we call the two-step duality. Thisduality arises from the existence of a new (generalized) tilting modulefor stratified algebras with properly stratified Ringel dual. Weshow that this new tilting module has a lot of interesting properties,for instance, its projective dimension equals the projectively definedfinitistic dimension of the original algebra, it guarantees that thecategory of modules of finite projective dimension is contravariantlyfinite, and, finally, it allows one to compute the finitistic dimension ofthe original algebra in terms of the projective dimension of thecharacteristic tilting module.
|
|
| 32. |
|
|
| 33. |
|
|
| 34. |
- Frisk Dubsky, Brendan
(author)
-
Structure and representations of certain classes of infinite-dimensional algebras
- 2018
-
Doctoral thesis (other academic/artistic)abstract
- We study several infinite-dimensional algebras and their representation theory. In Paper I, we study the category O for the (centrally extended) Schrödinger Lie algebra, which is an analogue of the classical BGG category O. We decompose the category into a direct sum of "blocks", and describe Gabriel quivers of these blocks. For the case of non-zero central charge, we in addition find the relations of these quivers. Also for the finite-dimensional part of O do we find the Gabriel quiver with relations. These results are then used to determine the center of the universal enveloping algebra, the annihilators of Verma modules, and primitive ideals of the universal enveloping algebra which intersect the center of the Schrödinger algebra trivially. In Paper II, we construct a family of path categories which may be viewed as locally quadratic dual to preprojective algebras. We prove that these path categories are Koszul. This is done by constructing resolutions of simple modules, that are projective and linear up to arbitrary position. This is done by using the mapping cone to piece together certain short exact sequences which are chosen so as to fall into three managable families. In Paper III, we consider the category of injections between finite sets, and also the path category of the Young lattice subject to the relations that two boxes added to the same column in a Young diagram yields zero. We construct a new and direct proof of the Morita equivalence of the linearizations of these categories. We also construct linear resolutions of simple modules of the latter category, and show that it is quadratic dual to its opposite. In Paper IV, we define a family of algebras using the induction and restriction functors on modules over the dihedral groups. For a wide subfamily, we decompose the algebras into indecomposable subalgebras, find a basis and relations for each algebra, as well as explicitly describe each center.
|
|
| 35. |
|
|
| 36. |
|
|
| 37. |
- Futorny, Vyacheslav, et al.
(author)
-
Weight Modules over Infinite Dimensional Weyl Algebras
- 2014
-
In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 142:9, s. 3049-3057
-
Journal article (peer-reviewed)abstract
- We classify simple weight modules over infinite dimensional Weyl algebras and realize them using the action on certain localizations of the polynomial ring. We describe indecomposable projective and injective weight modules and deduce from this a description of blocks of the category of weight modules by quivers and relations. As a corollary we establish Koszulity for all blocks.
|
|
| 38. |
|
|
| 39. |
|
|
| 40. |
|
|
| 41. |
|
|
| 42. |
- Ganyushkin, Olexandr, et al.
(author)
-
On the irreducible representations of a finite semigroup
- 2009
-
In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 137:11, s. 3585-3592
-
Journal article (peer-reviewed)abstract
- Work of Clifford, Munn and Ponizovskii parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskii result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.
|
|
| 43. |
|
|
| 44. |
|
|
| 45. |
|
|
| 46. |
|
|
| 47. |
- Greenstein, Jacob, et al.
(author)
-
Koszul Duality for Semidirect Products and Generalized Takiff Algebras
- 2017
-
In: Algebras and Representation Theory. - : Springer Science and Business Media LLC. - 1386-923X .- 1572-9079. ; 20:3, s. 675-694
-
Journal article (peer-reviewed)abstract
- 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.
|
|
| 48. |
- Grensing, Anna-Louise, et al.
(author)
-
Categorification of the Catalan monoid
- 2014
-
In: Semigroup Forum. - : Springer Science and Business Media LLC. - 0037-1912 .- 1432-2137. ; 89:1, s. 155-168
-
Journal article (peer-reviewed)abstract
- We construct a finitary additive 2-category whose Grothendieck ring is isomorphic to the semigroup algebra of the monoid of order-decreasing and order-preserving transformations of a finite chain.
|
|
| 49. |
- Grensing, Anna-Louise, et al.
(author)
-
Finitary 2-categories associated with dual projection functors
- 2017
-
In: Communications in Contemporary Mathematics. - 0219-1997. ; 19:3
-
Journal article (peer-reviewed)abstract
- We study finitary 2-categories associated to dual projection functors for finite-dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A), we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.
|
|
| 50. |
- Grøsfjeld, Tobias, 1991-
(author)
-
The Art of Bad Art : Diagrammatics in Mathematical Physics
- 2024
-
Doctoral thesis (other academic/artistic)abstract
- The purpose of a proof is to reduce the complexity of a statement until it becomes a sequence of trivialities. To this end, the choice of notation, diagrams and overall paradigm can aid in conveying large amounts of information in a simple manner. This compilation thesis focuses on the choice of visual tools to convey algebraic results in the context of mathematical physics, using a categorical paradigm with various topological semantics. The topics range from covering known results in knot theory, abstract diagram categories and low-dimensional topological quantum field theory, to novel results such as the topological rack exclusion principle, tetrahedral symmetry of framed associators and new diagrammatics for graded-monoidal categories based on the Kleisli presentation.We demonstrate how these diagrammatic methods can be used to simplify algebraic proofs and communicate across disciplines.
|
|
