The Global Dimensions of Ore Extensions and Weyl Algebras

We consider associative algebras filtered by the additive monoid ℕp. We prove that, under quite general conditions, the study of Gelfand-Kirillov dimension of modules over a multi-filtered algebra R can be reduced to the associated ℕp-graded algebra G(R). As a consequence, we show the exactness of the Gelfand-Kirillov dimension when the multi-filtration is finite-dimensional and G(R) is a finitely generated noetherian algebra. Our methods apply to examples like iterated Ore extensions with arbitrary derivations and “homothetic” automorphisms (e.g. quantum matrices, quantum Weyl algebras) and the quantum enveloping algebra of sl(v + 1)

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Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

Acta Applicandae Mathematicae ◽

10.1007/s10440-008-9371-7 ◽

2008 ◽

Vol 108 (1) ◽

pp. 73-81 ◽

Cited By ~ 1

Author(s):

Candis Holtz ◽

Kenneth Price

Keyword(s):

Weyl Algebras ◽

Ore Extensions

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Primitive Ore extensions

Glasgow Mathematical Journal ◽

10.1017/s0017089500003086 ◽

1977 ◽

Vol 18 (1) ◽

pp. 93-97 ◽

Cited By ~ 15

Author(s):

D. A. Jordan

Keyword(s):

Polynomial Ring ◽

Polynomial Algebra ◽

Laurent Polynomial ◽

Ore Extension ◽

Enveloping Algebra ◽

Weyl Algebras ◽

Solvable Lie Algebra ◽

Ore Extensions ◽

Laurent Polynomial Ring ◽

Necessary And Sufficient

Apart from simple Ore extensions such as the Weyl algebras, the best known example of a primitive Ore extension is the universal enveloping algebra U(g) of the 2-dimensional solvable Lie algebra g over a field k of characteristic zero, see [4, p. 22]. U(g) is a polynomial algebra over k in two indeterminates x and y with multiplication subject to the relation xy – yx = y, and may be regarded either as an Ore extension of k [x] by the k-automorphism which maps x to x – 1 or as an Ore extension of k[y] by the derivation yd/dy. The argument suggested in [4, p. 22] to prove the primitivity of U(g) can easily be generalised [6] to show that, if α is an automorphism of the ring R then the following conditions are sufficient for R[x, α] to be primitive: (i) no power αs, s ≧ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are necessary and sufficient for the simplicity of the skew Laurent polynomial ring R[x, x–1, α] but are not necessary for the primitivity of R[x, α] (the ordinary polynomial ring D[x] over a division ring D not algebraic over its centre is easily seen to be primitive).

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Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

Algebra Colloquium ◽

10.1142/s1005386716000596 ◽

2016 ◽

Vol 23 (04) ◽

pp. 701-720 ◽

Cited By ~ 2

Author(s):

Xiangui Zhao ◽

Yang Zhang

Keyword(s):

Computational Method ◽

Finitely Generated ◽

Finitely Generated Module ◽

Polynomial Algebras ◽

Ore Extensions ◽

Basis Method ◽

Finitely Generated Modules ◽

Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

Algebras and Representation Theory ◽

10.1023/b:alge.0000026785.03997.60 ◽

2004 ◽

Vol 7 (2) ◽

pp. 159-172 ◽

Cited By ~ 1

Author(s):

Adriana Nenciu

Keyword(s):

Hopf Algebras ◽

Pointed Hopf Algebras ◽

Ore Extensions

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Ore extensions of quasitriangular Hopf group coalgebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500169 ◽

2014 ◽

Vol 13 (06) ◽

pp. 1450016 ◽

Cited By ~ 1

Author(s):

Daowei Lu ◽

Dingguo Wang

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Baxter Equation ◽

Ore Extension ◽

Ore Extensions ◽

Necessary And Sufficient

In this paper, we mainly consider some special Ore extension of quasitriangular Hopf group coalgebra, and give the necessary and sufficient conditions when the Ore extension of quasitriangular Hopf group coalgebras will preserve the same quasitriangular structure. Furthermore, in the two examples given at the end, we construct new solutions of Yang–Baxter equation of Hopf group coalgebras version.

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ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

Journal of Algebra and Its Applications ◽

10.1142/s021949881250079x ◽

2012 ◽

Vol 11 (04) ◽

pp. 1250079 ◽

Cited By ~ 7

Author(s):

A. ALHEVAZ ◽

A. MOUSSAVI

Keyword(s):

Polynomial Rings ◽

Trivial Extension ◽

Skew Polynomial Ring ◽

Matrix Rings ◽

Skew Polynomial Rings ◽

Armendariz Rings ◽

Ore Extensions ◽

Necessary And Sufficient ◽

Skew Polynomial ◽

Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

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