Noncommutative homology of quantum tori. (English. Russian original) Zbl 0708.19003

Takhtadzhyan, L. A.

Funct. Anal. Appl. 23, No. 2, 147-149 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 75-76 (1989).

Let \(T^ n_ q\) be the algebra of formal Laurent series \(\sum_{k\in {\mathbb{Z}}^ n}a(k) x_ 1^{k_ 1}...x_ n^{k_ n}\), where a: \({\mathbb{Z}}^ n\to {\mathbb{C}}\) is rapidly decreasing and \(x_ i\) are noncommutative variables satisfying \(x_ ix_ j=qx_ jx_ i\) \((1\leq i<j\leq n)\) for a fixed \(q\in {\mathbb{C}}\), \(| q| =1\). It is shown that the noncommutative de Rham cohomology of \(T^ n_ q\) is isomorphic to the exterior algebra of the n-dimensional quantum vector space: it is generated by elements \(\xi_ i\) (1\(\leq i\leq n)\) of degree one satisfying the relations \(\xi_ i^ 2=0\), \(\xi_ i\xi_ j=-q^{-1}\xi_ j\xi_ i\) \((1\leq i<j\leq n)\). Some consequences for the quantum algebra \({\mathbb{C}}^ n_ q={\mathbb{C}}<x_ 1,...,x_ n>/(x_ ix_ j-qx_ jx_ i)_{i\leq j}\) are discussed.

Reviewer: A.Kufner

MSC:

19D55	\(K\)-theory and homology; cyclic homology and cohomology
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Keywords:

algebra of formal Laurent series; noncommutative de Rham cohomology; exterior algebra of the n-dimensional quantum vector space; quantum algebra

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