Introducing Crystalline Graded Algebras

Nauwelaerts, Erna; Oystaeyen, Freddy Van

doi:10.1007/s10468-007-9078-x

Cited by 22 publications

(33 citation statements)

References 14 publications

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“…In [8] it was shown that for pre-crystalline graded rings, the elements α(s, t) are normalizing elements of A 0 , i.e. A 0 α(s, t) = α(s, t) A 0 for each s, t ∈ G. For a pre-crystalline graded ring A 0 ♦ α σ G, we let S(G) denote the multiplicative set in A 0 generated by {α(g, g −1 ) | g ∈ G} and let S(G × G) denote the multiplicative set generated by {α(g, h) | g, h ∈ G}.…”

Section: 3mentioning

confidence: 99%

“…Lemma 2.5 (see [8]). If A = A 0 ♦ α σ G is a pre-crystalline graded ring, then the following assertions are equivalent:…”

Section: 3mentioning

confidence: 99%

“…The first Weyl algebra is an example of a crystalline graded ring, with G = (Z, +) and A e = A 0 = C[xy] (see e.g. [8] for details). By Remark 5.3 the ring A 0 is not simple.…”

Section: Where R E Is Commutative If R Is a Simple Ring Then R E Ismentioning

confidence: 99%

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Simple group graded rings and maximal commutativity

Öinert

2009

Operator Structures and Dynamical Systems

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Abstract. In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring R = L g∈G Rg the grading group G acts, in a natural way, as automorphisms of the commutant of the neutral component subring Re in R and of the center of Re. We show that if R is a strongly G-graded ring where Re is maximal commutative in R, then R is a simple ring if and only if Re is G-simple (i.e. there are no nontrivial G-invariant ideals). We also show that if Re is commutative (not necessarily maximal commutative) and the commutant of Re is G-simple, then R is a simple ring. These results apply to G-crossed products in particular. A skew group ring Re ⋊σ G, where Re is commutative, is shown to be a simple ring if and only if Re is G-simple and maximal commutative in Re ⋊σ G. As an interesting example we consider the skew group algebra C(X) ⋊h Z associated to a topological dynamical system (X, h). We obtain necessary and sufficient conditions for simplicity of C(X) ⋊h Z with respect to the dynamics of the dynamical system (X, h), but also with respect to algebraic properties of C(X) ⋊h Z. Furthermore, we show that for any strongly G-graded ring R each nonzero ideal of R has a nonzero intersection with the commutant of the center of the neutral component. IntroductionThe aim of this paper is to highlight the important role that maximal commutativity of the neutral component subring plays in a strongly group graded ring when investigating simplicity of the ring itself. The motivation comes from the theory of C * -crossed product algebras associated to topological dynamical systems. To each topological dynamical system, (X, h), consisting of a compact Hausdorff space X and a homeomorphism h : X → X, 2000 Mathematics Subject Classification. 13A02, 16S35.

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Section: 3mentioning

confidence: 99%

“…Lemma 2.5 (see [8]). If A = A 0 ♦ α σ G is a pre-crystalline graded ring, then the following assertions are equivalent:…”

Section: 3mentioning

confidence: 99%

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Simple group graded rings and maximal commutativity

Öinert

2009

Operator Structures and Dynamical Systems

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“…We shall begin by recalling some basic definitions and properties following [20]. For a thorough exposition of the theory of graded rings we refer to [1,19].…”

Section: Definitions and Backgroundmentioning

confidence: 99%

“…Under some conditions on the crossed products the two statements are in fact equivalent, but not in general. In the recent paper [20], by E. Nauwelaerts and F. Van Oystaeyen, so called crystalline graded rings, which generalize algebraic crossed products, were defined. In the paper [21], by T. Neijens, F. Van Oystaeyen and W.W. Yu, the structure of the center of special classes of crystalline graded rings and generalized Clifford algebras was studied.…”

Section: Introductionmentioning

confidence: 99%