scholarly journals Irreducible representations of a simple Lie algebra admitting a one-dimensional weight space

1968 ◽  
Vol 19 (5) ◽  
pp. 1161-1161 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
Weight Space ◽  
One Dimensional ◽  
Simple Lie Algebra

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.

Note on Weight Spaces of Irreducible Linear Representations

1968 ◽  
Vol 11 (3) ◽  
pp. 399-403 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Weight Function ◽  
Wide Class ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
One Dimensional ◽  
Highest Weight ◽  
Linear Representations ◽  
Finite Dimensional

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

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 $${\mathcal{T}}({\mathfrak{g}},e)$$ .

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).

Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras

1971 ◽  
Vol 14 (1) ◽  
pp. 113-115 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
Space Decomposition ◽  
Finite Dimensional

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

H-Finite Irreducible Representations of Simple Lie Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1084-1106 ◽  
Author(s):  
F. Lemire ◽  
M. Pap
Keyword(s):  
Irreducible Representation ◽  
Lie Algebras ◽  
Complex Number ◽  
Cartan Subalgebra ◽  
Irreducible Representations ◽  
Weight Space ◽  
Algebra Homomorphism ◽  
One Dimensional ◽  
Enveloping Algebra ◽  
Complex Number Field

Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.

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