The tetrahedron algebra and its finite-dimensional irreducible modules

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

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Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091510001604 ◽

2012 ◽

Vol 55 (3) ◽

pp. 697-709 ◽

Cited By ~ 4

Author(s):

Xiangqian Guo ◽

Rencai Lu ◽

Kaiming Zhao

Keyword(s):

Direct Summand ◽

Virasoro Algebra ◽

Additive Subgroup ◽

One Dimensional ◽

Finite Dimensional ◽

Irreducible Modules ◽

Intermediate Series ◽

Complex Number Field ◽

Weight Modules

AbstractLet G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

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Theq-Tetrahedron Algebra and Its Finite Dimensional Irreducible Modules

Communications in Algebra ◽

10.1080/00927870701509180 ◽

2007 ◽

Vol 35 (11) ◽

pp. 3415-3439 ◽

Cited By ~ 36

Author(s):

Tatsuro Ito ◽

Paul Terwilliger

Keyword(s):

Finite Dimensional ◽

Irreducible Modules

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Cohomology of Lie Conformal Algebra Vir⋉Cur g

Algebra Colloquium ◽

10.1142/s1005386721000390 ◽

2021 ◽

Vol 28 (03) ◽

pp. 507-520

Author(s):

Maosen Xu ◽

Yan Tan ◽

Zhixiang Wu

Keyword(s):

Lie Algebra ◽

Conformal Algebra ◽

Cohomology Groups ◽

Simple Lie Algebra ◽

Finite Dimensional ◽

Irreducible Modules

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

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Irreducible Modules for Polycyclic Group Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-071-1 ◽

1981 ◽

Vol 33 (4) ◽

pp. 901-914 ◽

Cited By ~ 2

Author(s):

I. M. Musson

Keyword(s):

Nilpotent Group ◽

Irreducible Module ◽

Polycyclic Group ◽

Group Algebras ◽

Finitely Generated ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Irreducible Modules

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.We introduce some terminology.

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Cyclotomic Nazarov-Wenzl Algebras

Nagoya Mathematical Journal ◽

10.1017/s0027763000026842 ◽

2006 ◽

Vol 182 ◽

pp. 47-134 ◽

Cited By ~ 43

Author(s):

Susumu Ariki ◽

Andrew Mathas ◽

Hebing Rui

Keyword(s):

Irreducible Representations ◽

Arbitrary Field ◽

Dimensional Algebra ◽

Simple Modules ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Irreducible Modules ◽

Brauer Algebras

AbstractNazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank rn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

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