Identities of irreducible representations of a three-dimensional simple lie algebra

1983 ◽  
Vol 22 (3) ◽  
pp. 229-250
Author(s):  
I. M. Trishin
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Simple Lie Algebra

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

A New Family of Irreducible Representations of An

1975 ◽  
Vol 18 (4) ◽  
pp. 543-546 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
Weight Space ◽  
Complex Numbers ◽  
One Dimensional ◽  
Simple Lie Algebra ◽  
New Family ◽  
Highest Weight ◽  
The Family

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

MONOMIAL BASES FOR SOME IRREDUCIBLE 𝔤2-MODULES

2010 ◽  
Vol 09 (05) ◽  
pp. 705-715 ◽  
Author(s):  
DONG-IL LEE
Keyword(s):  
Lie Algebra ◽  
Explicit Construction ◽  
Irreducible Representations ◽  
Simple Lie Algebra ◽  
Finite Dimensional

We find relations for finite-dimensional irreducible representations of the simple Lie algebra 𝔤2, resulting in an explicit construction of Gröbner–Shirshov pairs and monomial bases for 𝔤2-modules V(nΛ2).

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

ALGEBRAIC DEFINITION OF TOPOLOGICAL W GRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 5193-5211
Author(s):  
SHINOBU HOSONO
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Simple Lie Algebra ◽  
Topological Gravity ◽  
Definition Of ◽  
Extended Theories

We propose a definition of the topological W gravity using some properties of the principal three-dimensional subalgebra of a simple Lie algebra due to Kostant. In our definition, structures of the two-dimensional topological gravity are naturally embedded in the extended theories. In accordance with the definition, we will present some explicit calculations for the W3 gravity.

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