scholarly journals COMPILATION OF RELATIONS FOR THE ANTISYMMETRIC TENSORS DEFINED BY THE LIE ALGEBRA COCYCLES OF su(n)

2001 ◽  
Vol 16 (08) ◽  
pp. 1377-1405 ◽  
Author(s):  
J. A. DE AZCÁRRAGA ◽  
A. J. MACFARLANE
Keyword(s):  
Lie Algebra ◽  
Essential Role ◽  
Lie Algebra Cohomology ◽  
Powerful Means ◽  
Casimir Operators ◽  
Definition Of ◽  
Comprehensive Compilation

This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of su (n), and that play an essential role in the optimal definition of Racah–Casimir operators of su (n). Since the Omega tensors occur naturally within the algebra of totally antisymmetrized products of λ-matrices of su (n), relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of su (n). Various key derivations are given to illustrate the methods employed.

scholarly journals THE VAN EST SPECTRAL SEQUENCE FOR HOPF ALGEBRAS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 33-48 ◽  
Author(s):  
E. J. BEGGS ◽  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Spectral Sequences ◽  
De Rham Cohomology ◽  
Lie Algebra Cohomology ◽  
De Rham ◽  
Definition Of

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.

scholarly journals The Politics of Peoplehood

2015 ◽  
Vol 45 (4) ◽  
pp. 439-465 ◽  
Author(s):  
Jonathan White ◽  
Lea Ypi
Keyword(s):  
Political Theory ◽  
Essential Role ◽  
Political Legitimacy ◽  
Political Conflict ◽  
The People ◽  
Definition Of

Contemporary political theory has made the question of the “people” a topic of sustained analysis. This article identifies two broad approaches taken—norm-based and contestation-based—and, noting some problems left outstanding, goes on to advance a complementary account centred on partisan practice. It suggests the definition of “the people” is closely bound up in the analysis of political conflict, and that partisans engaged in such conflict play an essential role in constructing and contesting different principled conceptions. The article goes on to show how such an account does not lead to a normatively hollow, purely historical conception of “the people,” but rather highlights the normative importance of practices that, at the minimum, de-naturalise undesirable conceptions of the people and, at their best, give political legitimacy and a representative basis to those one might wish to see prosper.

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.

ʿAdālat al-Ṣaḥāba: The Construction of a Religious Doctrine

Arabica ◽  
2013 ◽  
Vol 60 (3-4) ◽  
pp. 272-305 ◽  
Author(s):  
Amr Osman
Keyword(s):  
Civil Wars ◽  
Essential Role ◽  
Conclusive Evidence ◽  
Presumption Of Innocence ◽  
Textual Evidence ◽  
Religious Doctrine ◽  
Definition Of

Abstract This article investigates the development of ʿadālat al-ṣaḥāba, a central doctrine in Sunnī orthodoxy that stresses the integrity of the Prophet Muḥammad’s Companions. The examination of relevant Sunnī works indicates that the doctrine crystalized in the 5th/11th century, by which time the basic tenets of the doctrine had been developed. These include, among other things, the definition of Companions and their essential role in securing the authenticity of Islam. Furthermore, it was around that time that medieval Sunnī scholars developed an epistemological—rather than a historical or theological—basis for the doctrine. Establishing the integrity of the Companions during the Prophet’s lifetime on the presumption of innocence that is further confirmed by textual evidence, they argued that good Muslims must continue to accept that integrity given the lack of conclusive evidence that they lost it at a later time, particularly when they participated in civil wars. I argue that this epistemological ground was furnished by Murğiʾism, as the examination of some Murğiʾī texts demonstrates.1

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.

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