scholarly journals Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

2018 ◽  
Vol 252 (1203) ◽  
pp. 0-0 ◽  
Author(s):  
Xiao Xiong ◽  
Quanhua Xu ◽  
Zhi Yin
Keyword(s):  
Quantum Tori

scholarly journals Dynamical systems on quantum tori Lie algebras

1993 ◽  
Vol 155 (3) ◽  
pp. 429-448 ◽  
Author(s):  
Jens Hoppe ◽  
Michail Olshanetsky ◽  
Stefan Theisen
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Quantum Tori

HOPF–CYCLIC HOMOLOGY AND RELATIVE CYCLIC HOMOLOGY OF HOPF–GALOIS EXTENSIONS

2006 ◽  
Vol 93 (1) ◽  
pp. 138-174 ◽  
Author(s):  
P. JARA ◽  
D. ŞTEFAN
Keyword(s):  
Cyclic Homology ◽  
Direct Application ◽  
Group Algebras ◽  
Algebra Structure ◽  
Drinfeld Module ◽  
Galois Extensions ◽  
Graded Algebras ◽  
Enveloping Algebra ◽  
Quantum Tori ◽  
Symmetric Algebras

Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module.

scholarly journals Quantum Tori and the Structure of Elliptic Quasi-simple Lie Algebras

1996 ◽  
Vol 135 (2) ◽  
pp. 339-389 ◽  
Author(s):  
Stephen Berman ◽  
Yun Gao ◽  
Yaroslav S. Krylyuk
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Quantum Tori

scholarly journals B(0, N)-graded Lie superalgebras coordinatized by quantum tori

2006 ◽  
Vol 49 (11) ◽  
pp. 1740-1752 ◽  
Author(s):  
Hongjia Chen ◽  
Yun Gao ◽  
Shikui Shang
Keyword(s):  
Lie Superalgebras ◽  
Quantum Tori

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

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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