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52
Bialgebra actions, twists, and universal defmroation formulas
 hepth/9411140
"... We introduce the concept of a twisting element based on a bialgebra and show how it can be used to twist a large class of algebras, coalgebras and certain subcategories of their respective module and comodule categories. We prove that this subcategory of modules over the original algebra is equivale ..."
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Cited by 58 (4 self)
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We introduce the concept of a twisting element based on a bialgebra and show how it can be used to twist a large class of algebras, coalgebras and certain subcategories of their respective module and comodule categories. We prove that this subcategory of modules over the original algebra is equivalent to the corresponding category of modules over the twisted algebra. The relation between twisting elements and universal deformation formulas is also given, along with new formulas which are associated to enveloping algebras of nonabelian Lie algebras. The purpose of this paper is to introduce a general notion of “twisting ” algebraic structures based on actions of a bialgebra B and to relate these twists to deformation theory. We show that elements of B ⊗ B can be used to twist the multiplication of any Bmodule algebra or Bmodule coalgebra. Moreover, we focus on a certain subcategory of left Amodules
Generalizing the Notion of Koszul Algebra
"... Abstract. We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and O ..."
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Cited by 28 (3 self)
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Abstract. We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some wellknown examples of algebras that fall into this class. 1.
Regular algebras of dimension 4 and their A∞Extalgebras
 Duke Math. J
"... ABSTRACT. We construct four families of ArtinSchelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of ArtinSchelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noethe ..."
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Cited by 22 (10 self)
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ABSTRACT. We construct four families of ArtinSchelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of ArtinSchelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and CohenMacaulay. One of the main tools is Keller’s highermultiplication theorem on A∞Extalgebras.
Prime spectra of quantized coordinate rings
, 1999
"... This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. ..."
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Cited by 20 (6 self)
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This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The first part of the paper is expository. We survey a number of classes of quantized coordinate rings, as well as some related algebras that share common properties, and we record some of the basic properties known to occur for many of these algebras, culminating in stratifications of the prime spectra by the actions of tori of automorphisms. As our main interest is in the generic case, we assume various parameters are not roots of unity whenever convenient. In the second part of the paper, which is based on [20], we offer some support for the conjecture above, in the form of an axiomatic basis for the observed stratifications and their properties. At present, the existence of a suitable supply of normal elements is taken as one of the axioms; the search for better axioms that yield such normal elements is left as an open problem. I. Quantized Coordinate Rings and Related Algebras This part of the paper is an expository account of the prime ideal structure of algebras on the “quantized coordinate ring ” side of the theory of quantum groups – quantizations of the coordinate rings of affine spaces, matrices, semisimple groups, symplectic and Euclidean spaces, as well as a few related algebras – quantized enveloping algebras of Borel and nilpotent subalgebras of semisimple Lie algebras, and quantized Weyl algebras. These algebras occur widely throughout the quantum groups literature, and different papers
Embedding A Quantum Rank Three Quadric In A Quantum P³
 Comm. Alg
, 1997
"... . We continue the classification, begun in [11], [14] and [12], of quadratic ArtinSchelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurface in P 3 . In this paper, we consider those cases where the quadric has rank 3. We also giv ..."
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Cited by 14 (7 self)
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. We continue the classification, begun in [11], [14] and [12], of quadratic ArtinSchelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurface in P 3 . In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism. Introduction In [2], a notion of regularity, here called ArtinSchelter regularity, was introduced for noncommutative graded algebras. A classification and a general analysis of ArtinSchelter regular algebras of global dimension 3 were carried out in [2], [3] and [4]. A classification theorem for ArtinSchelter regular algebras of global dimension 4 has proved to be much less tractable and is still a long way off. In [11], [12] and [14], the ArtinSchelter regular algebras of global dimension four which map onto a twisted homogeneous coordinate ring of a n...
COHERENT ALGEBRAS AND NONCOMMUTATIVE PROJECTIVE LINES
, 2007
"... Abstract. A wellknown conjecture says that every onerelator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative a ..."
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Cited by 13 (1 self)
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Abstract. A wellknown conjecture says that every onerelator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P1 as a noncommutative scheme based on the coherent noncommutative spectrum
Galgebras, twistings, and equivalences of graded categories
, 2008
"... Given Zgraded rings A and B, we ask when the graded module categories grA and grB are equivalent. Using Zalgebras, we relate the Moritatype results of ÁhnMárki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Zgraded rings, then: (1) A ..."
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Cited by 12 (2 self)
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Given Zgraded rings A and B, we ask when the graded module categories grA and grB are equivalent. Using Zalgebras, we relate the Moritatype results of ÁhnMárki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Zgraded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Zalgebras A = L i,j∈Z Aj−i and B = L i,j∈Z Bj−i are isomorphic. (2) If A and B are connected graded with A1̸ = 0, then grA≃grB if and only if A and B are isomorphic. This simplifies and extends Zhang’s results.
Double Extension Regular Algebras of Type (14641)
 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WASHINGTON
, 2007
"... We construct several families of ArtinSchelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new and are n ..."
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Cited by 10 (4 self)
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We construct several families of ArtinSchelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an ArtinSchelter regular algebra of global dimension three.
The Ground Beneath Her Feet
"... October 13, 2013... inside of a book is there whether you read it or not. Even if ..."
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Cited by 9 (0 self)
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October 13, 2013... inside of a book is there whether you read it or not. Even if
Examples of generic noncommutative surfaces
, 2002
"... We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are noetherian but not strongly noetherian, answering a question of A ..."
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Cited by 9 (2 self)
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We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are noetherian but not strongly noetherian, answering a question of Artin, Small, and Zhang. In addition, these examples are maximal orders and satisfy the χ1 condition but not χi for i ≥ 2, answering a questions of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that