scholarly journals KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 7-26
Author(s):  
KONSTANTIN ARDAKOV ◽  
IAN GROJNOWSKI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Krull Dimension ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Complex Semisimple

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

scholarly journals TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

2009 ◽  
Vol 20 (03) ◽  
pp. 339-368 ◽  
Author(s):  
MINORU ITOH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Central Elements ◽  
Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

Decomposition Varieties in Semisimple Lie Algebras

1998 ◽  
Vol 50 (5) ◽  
pp. 929-971 ◽  
Author(s):  
Abraham Broer
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Orbits ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Rational Singularities ◽  
Common Generalization ◽  
Simultaneous Resolution ◽  
Decomposition Class

AbstractThe notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

Uniqueness theorems for left-symmetric enveloping algebras

1996 ◽  
Vol 120 (2) ◽  
pp. 193-206
Author(s):  
J. R. Bolgar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Uniqueness Theorems ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Uniqueness Results ◽  
Symmetric Algebras ◽  
Algebra Over A Field

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

scholarly journals Separation of Variables for

2005 ◽  
Vol 48 (4) ◽  
pp. 587-600 ◽  
Author(s):  
Samuel A. Lopes
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Analogous Result ◽  
Separation Of Variables ◽  
Positive Part ◽  
Semisimple Lie Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Quantized Enveloping Algebra

AbstractLet be the positive part of the quantized enveloping algebra . Using results of Alev–Dumas and Caldero related to the center of , we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra U(g) of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra Ŭq(g). Of greater importance to its representation theory is the fact that is free over a larger polynomial subalgebra N in n variables. Induction from N to provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

scholarly journals Vanishing resonance and representations of Lie algebras

2015 ◽  
Vol 2015 (706) ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Classical Representation ◽  
Semisimple Lie Algebra ◽  
Representations Of Lie Algebras ◽  
Complex Semisimple ◽  
Resonance Varieties

AbstractWe explore a relationship between the classical representation theory of a complex, semisimple Lie algebra 𝔤 and the resonance varieties

scholarly journals On Automorphisms of Enveloping Algebras

2019 ◽  
Vol 2020 (21) ◽  
pp. 8183-8196 ◽  
Author(s):  
Akaki Tikaradze
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Subgroup ◽  
Derived Category ◽  
Canonical Homomorphism ◽  
Group Of Automorphisms ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Simple Lie Algebra

Abstract Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathcal{L}_{\infty }(\mathfrak{g})$ defined over $\mathbb{C}_{\infty }$, the reduction of $\mathbb{C}$ modulo the infinitely large prime, and show that for a class of Lie algebras, $\mathcal{L}_{\infty }(\mathfrak{g})$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a non-trivial semi-simple automorphism. As a corollary, we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ is isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$

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