Quantum Planes of Weight (1, 1,n)

Stephenson, Darin R.

doi:10.1006/jabr.2000.8626

Cited by 10 publications

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“…This work was stimulated by Darin Stephenson's paper [11]. I would like to thank him for explaining his results, and also suggesting that Proposition 4.10 should be true.…”

Section: Acknowledgementsmentioning

confidence: 92%

Maps between non-commutative spaces

Smith

2003

Trans. Amer. Math. Soc.

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Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A 0 ⊕ A 1 ⊕• • • in which dim k A i < ∞ for all i. We show that the map A → A/J induces a closed immersion i : Proj nc A/J → Proj nc A between the non-commutative projective spaces with homogeneous coordinate rings A and A/J. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism φ : A → B between not necessarily commutative N-graded rings induces an affine map Proj nc B ⊃ U → Proj nc A from a non-empty open subspace U ⊂ Proj nc B. Second, if A is a right noetherian connected graded algebra (not necessarily generated in degree one), and A (n) is a Veronese subalgebra of A, there is a map Proj nc A → Proj nc A (n) ; we identify open subspaces on which this map is an isomorphism. Applying these general results when A is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.

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“…This work was stimulated by Darin Stephenson's paper [11]. I would like to thank him for explaining his results, and also suggesting that Proposition 4.10 should be true.…”

Section: Acknowledgementsmentioning

confidence: 92%

Maps between non-commutative spaces

Smith

2003

Trans. Amer. Math. Soc.

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“…If deg(f ) = 1, L is of type S 1 as classified in [1], and if deg(f ) = 2, L is of type S 1 . When deg(f ) = n + 1 > 2, L is a quantum planes of type (1, 1, n) as studied by Stephenson in [19] and [20].…”

Section: Introductionmentioning

confidence: 97%

Basic properties of generalized down–up algebras

Cassidy

Shelton

2004

Journal of Algebra

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We introduce a large class of infinite dimensional associative algebras which generalize down-up algebras. Let K be a field and fix f ∈ K[x] and r, s, γ ∈ K. Define L = L(f, r, s, γ ) to be the algebra generated by d, u and h with defining relations:Included in this family are Smith's class of algebras similar to U(sl 2 ), Le Bruyn's conformal sl 2 enveloping algebras and the algebras studied by Rueda. The algebras L have Gelfand-Kirillov dimension 3 and are Noetherian domains if and only if rs = 0. We calculate the global dimension of L and, for rs = 0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules.  2004 Elsevier Inc. All rights reserved.

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“…Acknowledgements. This work was stimulated by Darin Stephenson's paper [11]. I would like to thank him for explaining his results, and also suggesting that Proposition 4.10 should be true.…”

Section: Introductionmentioning

confidence: 92%

Corrigendum to “Maps between non-commutative spaces”

Smith

2016

Trans. Amer. Math. Soc.

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Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A 0 ⊕ A 1 ⊕ · · · in which dim k A i < ∞ for all i. We show that the map A → A/J induces a closed immersion i : Proj nc A/J → Proj nc A between the non-commutative projective spaces with homogeneous coordinate rings A and A/J. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism φ : A → B between not necessarily commutative N-graded rings, induces an affine map Proj nc B ⊃ U → Proj nc A from a non-empty open subspace U ⊂ Proj nc B. Second, if A is a right noetherian connected graded algebra (not necessarily generated in degree one), and A (n) is a Veronese subalgebra of A, there is a map Proj nc A → Proj nc A (n) ; we identify open subspaces on which this map is an isomorphism. Applying these general results when A is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.

show abstract

Quantum Planes of Weight (1, 1,n)

Cited by 10 publications

References 0 publications

Maps between non-commutative spaces

Maps between non-commutative spaces

Basic properties of generalized down–up algebras

Corrigendum to “Maps between non-commutative spaces”

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