Thin coverings of modules

Billig, Yuly; Lau, Michael

doi:10.1016/j.jalgebra.2007.01.002

Cited by 11 publications

(20 citation statements)

References 12 publications

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“…Here U (L) is considered with the G-grading induced by the quotient map G → G (which is a coarsening of the G-grading), and the existence of the mentioned G-grading on End(V ) can also be seen from the fact that the kernel of ρ is a G-graded ideal. Recalling the description of the graded Brauer group from the previous section, we see that Br(λ) is determined by the "commutation factor" of the operators u χ (χ ∈ K λ ), i.e., the alternating bicharacterβ of K λ given by (2). Proof.…”

Section: 2mentioning

confidence: 90%

Graded modules over classical simple Lie algebras with a grading

Elduque

Kochetov

2015

Isr. J. Math.

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Abstract. Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D 4 , where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.

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Section: 2mentioning

confidence: 90%

Graded modules over classical simple Lie algebras with a grading

Elduque

Kochetov

2015

Isr. J. Math.

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“…When Q is finite, Billig and Lau described quasi-finite Q-graded-simple modules over Q-graded associative (or Lie) algebras in [3]. With respect to this, we will study Q-graded-simple modules over Q-graded Lie algebras without assuming qasifiniteness of modules or finiteness of Q.…”

Section: General Overviewmentioning

confidence: 99%

“…Note that the set {X α : α ∈ Q} is called a Q-covering of V in [3]. Now we can obtain some necessary and sufficient conditions for the Q-graded module M(Q, P, V ) to be Q-graded simple.…”

Section: For Example M(q Q V ) = V ⊗ Kq (With the Obvious Q-gradinmentioning

confidence: 99%

Graded simple Lie algebras and graded simple representations

Mazorchuk

Zhao

2017

manuscripta math.

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Abstract. Let Q be an abelian group and k a field. We prove that any Q-graded simple Lie algebra g over k is isomorphic to a loop algebra in case k has a primitive root of unity of order |Q|, if Q is finite, or k is algebraically closed and dim g < |k| (as cardinals). For Q-graded simple modules over any Q-graded Lie algebra g, we propose a similar construction of all Q-graded simple modules over any Q-graded Lie algebra over k starting from nonextendable gradings of simple g-modules. We prove that any Q-graded simple module over g is isomorphic to a loop module in case k has a primitive root of unity of order |Q| if Q is finite, or k is algebraically closed and dim g < |k| as above. The isomorphism problem for simple graded modules constructed in this way remains open. For finite-dimensional Q-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.

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“…In order to specify the spaces on which g T (σ) acts, we recall the definition of thin covering of a module [7]. Let L = g∈G L g be a Lie algebra graded by a finite abelian group G, and let U be a (not necessarily graded) module for L. A covering of U is a collection of subspaces U g (g ∈ G) satisfying the following axioms…”

Section: Representations Ofmentioning

confidence: 99%

Irreducible modules for extended affine Lie algebras

Billig

Lau

2011

Journal of Algebra

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We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.Extended affine Lie algebras (EALAs) are natural generalizations of the affine Kac-Moody algebras. They come equipped with a non-degenerate symmetric invariant bilinear form, a finite-dimensional Cartan subalgebra, and a discrete root system. Originally introduced in the contexts of singularity theory and mathematical physics, their structure theory has been extensively studied for over 15 years. (See [2,3,20] and the references therein.)Their representations are much less well understood. Early attempts to replicate the highest weight theory of the affine setting were stymied by the lack of a triangular decomposition; later work considered only the untwisted toroidal Lie algebras and a few other isolated examples. As a result of major breakthroughs announced in [3] and [20], it is now clear that, except for (extensions of) matrix algebras over non-cyclotomic quantum tori, every extended affine Lie algebra can be constructed as

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Thin coverings of modules

Cited by 11 publications

References 12 publications

Graded modules over classical simple Lie algebras with a grading

Graded modules over classical simple Lie algebras with a grading

Graded simple Lie algebras and graded simple representations

Irreducible modules for extended affine Lie algebras

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