Cubist algebras

Chuang, Joseph; Turner, Will

doi:10.1016/j.aim.2007.06.017

Cited by 14 publications

(10 citation statements)

References 22 publications

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“…Finite truncations V T of Cubist algebras from [CT,Section 6] are balanced. Indeed, V T is standard Koszul by [CT,Proposition 46], and that the Ringel dual of V T is positively graded with respect to the induced grading follows from [CT,Corollary 71]. So, the fact that V T is balanced follows from Remark 9.…”

Section: Examplesmentioning

confidence: 99%

Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras

Mazorchuk

2009

manuscripta math.

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We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules. 1 2. Graded quasi-hereditary algebras By N we denote the set of all positive integers. By a module we always mean a graded left module, and by grading we always mean

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Section: Examplesmentioning

confidence: 99%

Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras

Mazorchuk

2009

manuscripta math.

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“…Originally, we defined the deformations D Q (n, r) in order to compare the algebras D Q (n, r) with the Cubist algebras, another family of algebras which are also related to blocks of symmetric groups [7]. However, it appears these deformations may play a more fundamental role: they provide interesting O-forms for the algebras kD Q (n, r).…”

Section: Dreams and Reflectionsmentioning

confidence: 98%

Tilting equivalences: From hereditary algebras to symmetric groups

Turner

2008

Journal of Algebra

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We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups. MotifLet R be a commutative ring. Let Q be a quiver, whose underlying graph is a tree. We reveal derived equivalences of increasing sophistication, between:I. The path algebra RQ, and its Koszul dual. II. The trivial extension algebra of RQ, and the trivial extension algebra of its Koszul dual. III. The Schur algebra of RQ, and the Schur algebra of its Koszul dual. IV. A double of the Schur algebra of RQ, and a double of the Schur algebra of its Koszul dual.Let Q be a Dynkin quiver, of type A. We lift the derived equivalences of IV to equivalences between: V. A deformation of the double of the Schur algebra of RQ, and a deformation of the double of the Schur algebra of its Koszul dual. VI. A quotient of the deformation of the double of the Schur algebra of RQ, and a quotient of the deformation of the double of the Schur algebra of its Koszul dual.

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“…In this section we construct an example using cubist techniques [4,6], generalising previously described methods ( [14], Section 4).…”

Section: A Cubist Examplementioning

confidence: 99%

Subsets of Scales in Compositions Constructed by Similarity

Turner¹

2021

IJTAM

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We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.

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Cubist algebras

Cited by 14 publications

References 22 publications

Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras

Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras

Tilting equivalences: From hereditary algebras to symmetric groups

Subsets of Scales in Compositions Constructed by Similarity

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