scholarly journals Colored partitions and a Generalization of the Braid Arrangement

10.37236/1289 ◽  
1996 ◽  
Vol 4 (1) ◽  
Author(s):  
Volkmar Welker
Keyword(s):  
Lie Algebra ◽  
Combinatorial Analysis ◽  
Topological Properties ◽  
Generic Fiber ◽  
Arrangement Of Hyperplanes ◽  
Intersection Lattice ◽  
Colored Partitions ◽  
Rational Cohomology ◽  
Definition Of ◽  
Algebra Homology

We study the topology and combinatorics of an arrangement of hyperplanes in ${\bf C}^n$ that generalizes the classical braid arrangement. The arrangement plays in important role in the work of Schechtman and Varchenko on Lie algebra homology, where it appears in a generic fiber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the definition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of $S_n$ on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the first $s$ integers.

scholarly journals Feynman diagrams and rooted maps

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 149-167 ◽  
Author(s):  
Andrea Prunotto ◽  
Wanda Maria Alberico ◽  
Piotr Czerski
Keyword(s):  
Single Particle ◽  
Feynman Diagram ◽  
Feynman Diagrams ◽  
Topological Properties ◽  
Many Body ◽  
Perturbative Order ◽  
Original Definition ◽  
Many Body Systems ◽  
Definition Of ◽  
Particle Propagator

Abstract The rooted maps theory, a branch of the theory of homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of a Feynman diagram, which totally differs from the usual one, is given.

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

Harish-Chandra Modules over ℤ

2019 ◽  
pp. 126-167
Author(s):  
Günter Harder
Keyword(s):  
Fixed Points ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Reductive Group ◽  
Group Scheme ◽  
Finitely Generated ◽  
Cartan Involution ◽  
Split Maximal Torus ◽  
Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

scholarly journals Cartan Subalgebras of Jordan Algebras

1966 ◽  
Vol 27 (2) ◽  
pp. 591-609 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Cartan Subalgebra ◽  
Jordan Algebras ◽  
Classical Notion ◽  
Cartan Subalgebras ◽  
Definition Of

In this paper we shall give a definition of an analogue for Jordan algebras of the classical notion of a Cartan subalgebra of a Lie algebra. This is based on a notion of associator nilpotency of a Jordan algebra. A Jordan algebra is called associator nilpotent if there exists a positive (odd) integer M such that every associator of order M formed of elements of is 0 (§2).

scholarly journals Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

2016 ◽  
Vol 283 (3-4) ◽  
pp. 979-992 ◽  
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch ◽  
Bernhard Krötz ◽  
Gang Liu
Keyword(s):  
Lie Algebra ◽  
Algebra Homology

scholarly journals Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

2016 ◽  
Vol 283 (3-4) ◽  
pp. 993-994
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch ◽  
Bernhard Krötz ◽  
Gang Liu
Keyword(s):  
Lie Algebra ◽  
Algebra Homology

On unrolled Hopf algebras

2018 ◽  
Vol 27 (10) ◽  
pp. 1850053
Author(s):  
Nicolás Andruskiewitsch ◽  
Christoph Schweigert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Nichols Algebra ◽  
Small Quantum ◽  
Definition Of ◽  
Special Case

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra [Formula: see text] of a Yetter–Drinfeld module [Formula: see text] on which a Lie algebra [Formula: see text] acts by biderivations. As a special case, we find unrolled versions of the small quantum group.

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