Quantum torus symmetries of the CKP and multi-component CKP hierarchies

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

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A class of Noether modules for quantum torus C<sub>-1</sub>

Scientia Sinica Mathematica ◽

10.1360/012012-131 ◽

2013 ◽

Vol 43 (2) ◽

pp. 199-210

Author(s):

ZhiPing LIN ◽

WeiQiang LIN

Keyword(s):

Quantum Torus

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Modules of highest weight type for the algebra of derivations over a rational quantum torus

Journal of Mathematical Physics ◽

10.1063/1.5018344 ◽

2019 ◽

Vol 60 (4) ◽

pp. 041701

Author(s):

Chengkang Xu ◽

Wenqi Luo ◽

Fen Zhang

Keyword(s):

Quantum Torus ◽

Highest Weight

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A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

Canadian Mathematical Bulletin ◽

10.4153/cmb-2002-060-9 ◽

2002 ◽

Vol 45 (4) ◽

pp. 672-685 ◽

Cited By ~ 5

Author(s):

S. Eswara Rao ◽

Punita Batra

Keyword(s):

Lie Algebras ◽

Primitive Root ◽

Irreducible Module ◽

Affine Lie Algebras ◽

Quantum Torus ◽

New Class ◽

Root Of Unity ◽

Quantum Tori ◽

Finite Dimensional

AbstractWe study the representations of extended affine Lie algebras where q is N-th primitive root of unity (ℂq is the quantum torus in two variables). We first prove that ⊕ for a suitable number of copies is a quotient of . Thus any finite dimensional irreducible module for ⊕ lifts to a representation of . Conversely, we prove that any finite dimensional irreducible module for comes from above. We then construct modules for the extended affine Lie algebras which is integrable and has finite dimensional weight spaces.

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Class of representations of skew derivation Lie algebra over quantum torus

Frontiers of Mathematics in China ◽

10.1007/s11464-008-0003-3 ◽

2007 ◽

Vol 3 (1) ◽

pp. 119-131 ◽

Cited By ~ 1

Author(s):

Nina Yu

Keyword(s):

Lie Algebra ◽

Quantum Torus ◽

Skew Derivation

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Verma Modules over Quantum Torus Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-022-1 ◽

2010 ◽

Vol 62 (2) ◽

pp. 382-399 ◽

Cited By ~ 3

Author(s):

Rencai Lü ◽

Kaiming Zhao

Keyword(s):

Lie Algebras ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Linear Functions ◽

Verma Modules ◽

Central Extensions ◽

Quantum Torus ◽

Infinite Dimensional ◽

Quantum Tori ◽

Necessary And Sufficient

AbstractRepresentations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

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