scholarly journals Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

2018 ◽  
Vol 17 (12) ◽  
pp. 1850237
Author(s):  
Seth Shelley-Abrahamson ◽  
Alec Sun
Keyword(s):  
Irreducible Representations ◽  
Infinite Dimensional ◽  
Cherednik Algebras ◽  
Type Formula ◽  
Type D ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra ◽  
Wall Crossing ◽  
Rational Cherednik Algebras

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].

scholarly journals ON THE SMOOTHNESS OF CENTRES OF RATIONAL CHEREDNIK ALGEBRAS IN POSITIVE CHARACTERISTIC

2013 ◽  
Vol 55 (A) ◽  
pp. 27-54 ◽  
Author(s):  
GWYN BELLAMY ◽  
MAURIZIO MARTINO
Keyword(s):  
Deformation Parameter ◽  
Positive Characteristic ◽  
Reflection Group ◽  
Block Decomposition ◽  
Reflection Groups ◽  
Cherednik Algebras ◽  
Cherednik Algebra ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra ◽  
Rational Cherednik Algebras

AbstractIn this paper we study rational Cherednik algebras att= 1 in positive characteristic. We study a finite-dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite seriesG(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

Jacobi–Trudi type formula for a class of irreducible representations of 𝔤𝔩(m|n)

2020 ◽  
pp. 2250015
Author(s):  
Nguyên Luong Thái Bình
Keyword(s):  
Irreducible Representations ◽  
Fundamental Representation ◽  
Symmetric Powers ◽  
Type Formula ◽  
Finite Dimensional

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra [Formula: see text] in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

Classification of irreducible representations of Lie algebra of vector fields on a torus

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Yuly Billig ◽  
Vyacheslav Futorny
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Irreducible Representations ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Standing Problem

AbstractWe solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on

scholarly journals ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

2008 ◽  
Vol 78 (2) ◽  
pp. 261-284 ◽  
Author(s):  
XIN TANG ◽  
YUNGE XU
Keyword(s):  
Quantum Groups ◽  
Irreducible Representations ◽  
Whittaker Model ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
The Relationship ◽  
Natural Families

AbstractWe construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

Standard Representations of Simple Lie Algebras

1968 ◽  
Vol 20 ◽  
pp. 344-361 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Partial Solution ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Dominant Weight ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Algebraic Representations

Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.

scholarly journals MANY FINITE-DIMENSIONAL LIFTING BUNDLE GERBES ARE TORSION

2021 ◽  
pp. 1-16
Author(s):  
DAVID MICHAEL ROBERTS
Keyword(s):  
Analogous Result ◽  
Connected Components ◽  
Fibre Bundles ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Crossed Modules ◽  
Higher Gauge Theory ◽  
Bundle Gerbes ◽  
Simply Connected

Abstract Many bundle gerbes are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson [‘A note on bundle gerbes and infinite-dimensionality’, J. Aust. Math. Soc.90(1) (2011), 81–92] proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$ -class. I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles and finite-dimensional twists of topological K-theory.

scholarly journals Tilting modules over tame hereditary algebras

2012 ◽  
Vol 2013 (682) ◽  
pp. 1-48
Author(s):  
Lidia Angeleri Hügel ◽  
Javier Sánchez
Keyword(s):  
Complete Classification ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Tame Hereditary Algebra

Abstract. We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form where is a union of tubes, and denotes the universal localization of R at in the sense of Schofield and Crawley-Boevey. Here is a direct sum of the Prüfer modules corresponding to the tubes in . Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Prüfer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.

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