scholarly journals Generalized negligible morphisms and their tensor ideals

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
Thorsten Heidersdorf ◽  
Hans Wenzl
Keyword(s):  
Maximal Ideal ◽  
Monoidal Category ◽  
Single Element ◽  
Trace Function ◽  
Geometric Description ◽  
Prime Characteristic ◽  
Tilting Modules ◽  
Root Of Unity ◽  
Simply Connected ◽  
Deligne Categories

AbstractWe introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

scholarly journals THE CENTER OF SL2 TILTING MODULES

2021 ◽  
pp. 1-20
Author(s):  
DANIEL TUBBENHAUER ◽  
PAUL WEDRICH
Keyword(s):  
Quantum Group ◽  
Prime Characteristic ◽  
Complex Root ◽  
Tilting Modules ◽  
Root Of Unity

Abstract In this note, we compute the centers of the categories of tilting modules for G = SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective G g T-modules when g = 1, 2.

scholarly journals Chern characters, reduced ranks and -modules on the flag variety

1994 ◽  
Vol 37 (3) ◽  
pp. 477-482 ◽  
Author(s):  
T. J. Hodges ◽  
M. P. Holland
Keyword(s):  
Lie Algebra ◽  
Euler Characteristic ◽  
Maximal Ideal ◽  
Chern Character ◽  
Flag Variety ◽  
Central Character ◽  
Geometric Description ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Chern Characters

Let D be the factor of the enveloping algebra of a semisimple Lie algebra by its minimal primitive ideal with trival central character. We give a geometric description of the Chern character ch: K0(D)→HC0(D) and the state (of the maximal ideal m) s: K0(D)→K0(D/m) = ℤ in terms of the Euler characteristic χ:K0()→ℤ, where is the associated flag variety.

scholarly journals QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

2006 ◽  
Vol 15 (10) ◽  
pp. 1245-1277 ◽  
Author(s):  
STEPHEN F. SAWIN
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Roots Of Unity ◽  
Quantum Field ◽  
Root Of Unity ◽  
Basic Representation ◽  
Topological Quantum ◽  
Quantum Version ◽  
Simply Connected ◽  
Connected Lie Group

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

scholarly journals DIAGRAMMATIC CONSTRUCTION OF REPRESENTATIONS OF SMALL QUANTUM $$ \mathfrak{sl} $$2

2021 ◽  
Author(s):  
C. BLANCHET ◽  
M. DE RENZI ◽  
J. MURAKAMI
Keyword(s):  
Quantum Group ◽  
Monoidal Category ◽  
Fundamental Representation ◽  
Combinatorial Description ◽  
Root Of Unity ◽  
Small Quantum ◽  
Odd Order

AbstractWe provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $$ \mathfrak{sl} $$ sl 2 at a root of unity q of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley–Lieb category specialized at δ = −q − q−1.

scholarly journals Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 295
Author(s):  
Muhammad Anwar
Keyword(s):  
Borel Subgroup ◽  
Three Dimensional ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Simply Connected ◽  
The Given

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when ⟨ λ , α ∨ ⟩ = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or ⟨ λ , α ∨ ⟩ = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

Cohomological annihilators

1982 ◽  
Vol 91 (3) ◽  
pp. 345-350 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Maximal Ideal ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Local Rings ◽  
Intersection Theorem ◽  
Prime Characteristic ◽  
Local Cohomology Modules ◽  
System Of Parameters ◽  
Cohomology Modules ◽  
Unique Maximal Ideal

The local cohomology theory introduced by Grothendieck(1) is a useful tool for attacking problems in commutative algebra and algebraic geometry. Let A denote a local ring with its unique maximal ideal m. For an ideal I ⊂ A and a finitely generated A-module M we consider the local cohomology modules HiI (M), i є ℤ, of M with respect to I, see Grothendieck(1) for the definition. In particular, the vanishing resp. non-vanishing of the local cohomology modules is of a special interest. For more subtle considerations it is necessary to study the cohomological annihilators, i.e. aiI(M): = AnnΔHiI(M), iєℤ. In the case of the maximal ideal I = m these ideals were used by Roberts (6) to prove the ‘New Intersection Theorem’ for local rings of prime characteristic. Furthermore, we used this notion (7) in order to show the amiability of local rings possessing a dualizing complex. Note that the amiability of a system of parameters is the key step for Hochster's construction of big Cohen-Macaulay modules for local rings of prime characteristic, see Hochster(3) and (4).

Calculation of Fuchsian groups associated to billiards in a rational triangle

1998 ◽  
Vol 18 (4) ◽  
pp. 1019-1042 ◽  
Author(s):  
CLAYTON C. WARD
Keyword(s):  
Algebraic Equation ◽  
Fuchsian Group ◽  
Riemann Surfaces ◽  
Regular Polygon ◽  
Fuchsian Groups ◽  
Geometric Description ◽  
Fermat Curves ◽  
Simply Connected

We define, following Veech, the Fuchsian group $\Gamma(P)$ of a rational polygon $P$. If $P$ is simply-connected, then ‘rational’ is equivalent to the condition that all interior angles of $P$ be rational multiples of $\pi$. Should it happen that $\Gamma(P)$ has finite covolume in $\mathop{\rm PSL}\nolimits (2, {\Bbb R})$ (and is thus a {\it lattice}), then a theorem of Veech states that every billiard path in $P$ is either finite or uniformly distributed in $P$.We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {\it not\/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation $x^n + y^n = 1$.

On hearing the shape of a drum: further results

1971 ◽  
Vol 69 (2) ◽  
pp. 353-363 ◽  
Author(s):  
K. Stewartson ◽  
R. T. Waechter
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Finite Number ◽  
Convex Curve ◽  
Trace Function ◽  
Smooth Convex ◽  
Dirichlet Conditions ◽  
Image Position ◽  
Simply Connected ◽  
Uniform Membrane

1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some progress is possible on establishing the leading terms of the asymptotic expansion of the trace function for small positive t. In particular, for a simply connected membrane Ω bounded by a smooth convex curve Γ for which the displacement satisfies the wave equation ∇2ø = ∂2ø/∂t2 and Dirichlet conditions on Γwhere |Ω| = area of Ω, L = length of Γ, and the constant ⅙ is determined (apart from a factor) on integrating the curvature of the boundary; moreover, if Ω is permitted to have a finite number of smooth convex holes then the constant becomes ⅙(1–r), where r = number of holes.

scholarly journals Generators for a maximally differential ideal in positive characteristic

1993 ◽  
Vol 132 ◽  
pp. 37-41 ◽  
Author(s):  
Alok Kumar Maloo
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Positive Characteristic ◽  
Proper Ideal ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Minimal Set ◽  
Noetherian Local Ring ◽  
Differential Ideal

In this note we give the structure of maximally differential ideals in a Noetherian local ring of prime characteristic p > 0, in terms of their generators. More precisely, we prove the following result:THEOREM 4. Let A be a Noetherian local ring of prime characteristic p > 0 with maximal ideal m. Let I be a proper ideal of A. Suppose n= emdim(A) and r = emdim(A/l). If I is maximally differential under a set of derivations of A then there exists a minimal set xl,…,xn of generators of m such that I = (xρl, …,xρr, xr+1,…xn).

A standard for transmission electron spectroscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 530-531 ◽  
Author(s):  
Dennis Maher ◽  
David Joy ◽  
Peggy Mochel
Keyword(s):  
Thin Films ◽  
Electron Spectroscopy ◽  
Host Lattice ◽  
Single Element ◽  
Multicomponent Systems ◽  
Transmission Electron ◽  
Major Interest ◽  
Standard Specimens ◽  
Forms Of Carbon ◽  
Electron Energy Loss

A variety of standard specimens is needed in order to systematically investigate the instrumentation, specimen, data reduction and quantitation variables in electron energy-loss spectroscopy (EELS). Pure single element specimens (e.g. various forms of carbon) have received considerable attention to date but certain elements of interest cannot be prepared directly as thin films. Since studies of the first and second row elements in two- or multicomponent systems will be of considerable importance in microanalysis using EELS, there is a need for convenient standards containing these species. For many investigations a standard should contain the desired element, or elements, homogeneously dispersed through a suitable matrix and at an accurately known concentration. These conditions may be met by the technique of implantation.Silicon was chosen as the host lattice since its principal ionization energies, EL23 = 98 eV and Ek = 1843 eV, are well removed from the K-edges of most elements of major interest such as boron (Ek = 188 eV), carbon (Ek = 283 eV), nitrogen (Ek = 400 eV) and oxygen (Ek = 532 eV).

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