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Noncommutative deformations of type-\(A\) Kleinian singularities. (English) Zbl 0807.16029
Let \(k\) be an algebraically closed field of characteristic zero. The author studies noncommutative \(k\)-algebras generated by the elements \(a,b\) and \(h\) satisfying the relations \(ha - ah = a\), \(hg - bh = -b\), \(ba = v(h)\), \(ab = v(h - 1)\), where \(v(x)\) is a polynomial with coefficients in \(k\). Such an algebra denoted by \(T(v)\) is a Noetherian ring of Krull dimension one. For example, if \(v(x)\) is quadratic then \(T(v)\) is isomorphic to an infinite dimensional primitive factor of the enveloping algebra of the Lie algebra \(sl(2,k)\). The author proves that the associated graded ring of \(T(v)\) is isomorphic to the coordinate ring of type-A Kleinian singularities. In the case of the complex ground field such a singularity may be presented as the quotient of an affine complex plane by a finite cyclic subgroup of \(SL_ 2(\mathbb{C})\). It is proved that \(T(v)\) is an Auslander-Gorenstein ring and its global dimension is infinite if and only if the polynomial \(v\) has repeated roots. In case the roots of \(v\) are distinct there are only the following two possibilities. When any two roots of \(v\) differ by a nonzero integer then the global dimension of \(T(v)\) is equal to 2, otherwise it is equal to 1. Furthermore, in this case it turns out that the Grothendieck group \(K_ 0(T(v))\) is a free abelian group of rank equal to the degree of \(v\). In conclusion the author formulates some open problems related to the subject.

16S80 Deformations of associative rings
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
14H20 Singularities of curves, local rings
16S30 Universal enveloping algebras of Lie algebras
14J70 Hypersurfaces and algebraic geometry
16D90 Module categories in associative algebras
16E20 Grothendieck groups, \(K\)-theory, etc.
16E10 Homological dimension in associative algebras
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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