On the Coxeter transformation of a wild algebra

Martha Takane

doi:10.1007/bf01189885

On the Coxeter transformation of a wild algebra

Archiv der Mathematik ◽

10.1007/bf01189885 ◽

1994 ◽

Vol 63 (2) ◽

pp. 128-135 ◽

Cited By ~ 4

Author(s):

Martha Takane

Keyword(s):

Coxeter Transformation ◽

Wild Algebra

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Salem numbers defined by Coxeter transformation

Linear Algebra and its Applications ◽

10.1016/j.laa.2009.07.037 ◽

2010 ◽

Vol 432 (1) ◽

pp. 144-154 ◽

Cited By ~ 3

Author(s):

Piroska Lakatos

Keyword(s):

Coxeter Transformation ◽

Salem Numbers

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THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sℓ(2,ℂ)

Glasgow Mathematical Journal ◽

10.1017/s0017089514000032 ◽

2014 ◽

Vol 56 (3) ◽

pp. 551-568 ◽

Cited By ~ 1

Author(s):

ROBERTO MARTINEZ-VILLA

Keyword(s):

Lie Algebra ◽

Projective Dimension ◽

Global Dimension ◽

Verma Modules ◽

Regular Component ◽

Koszul Duality ◽

Koszul Algebra ◽

Yoneda Algebra ◽

Enveloping Algebra ◽

Wild Algebra

AbstractIn this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

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