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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.

##### MSC:
 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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