Artin Schelter Regular Algebras Research Articles

Global Dimension 4 Extensions of Artin–Schelter Regular Algebras

This paper classifies central and normal extensions from global dimension 3 Artin–Schelter regular algebras to global dimension 4 Artin–Schelter regular algebras. Let A be an AS regular algebra of global dimension 3, and let D be an extension of A by a normal graded element z, i.e., D/〈z〉=A. The algebra A falls under a classification due to Artin, Schelter, Tate, and Van den Bergh [Artin and Schelter, Adv. Math.66 (1987), pp. 171–216; Artin et al., in “The Grothendieck Festschrift,” Vol. 1, pp. 33–85, Birkhäuser, Basel, 1990; Artin et al., Invent Math.106 (1991), pp. 335–388] and is either quadratic or cubic. The quadratic algebras A are Koszul, and this fact was used by Le Bruyn, Smith, and Van den Bergh [Le Bruyn et al., Math. Z.222 (1996), 171–212] to classify the four-dimensional AS regular algebras D when A is quadratic and deg(z)=1. Alternative methods are needed when A is cubic or deg(z)>1. We prove in all such cases that the regularity of D and z is equivalent to the regularity of z in low degree (e.g., 2 or 3) and this is equivalent to easily verifiable matrix conditions on the relations for D.

The homogenization of the three dimensional skew polynomial algebras of type I

This paper studies a special case of graded central extensions of three dimen-sional Artin-Schelter regular algebras, see [9, §3]. The algebras are homoge-nizations of two classes of three dimensional skew polynomial algebras. We refer to these algebras as Type I and Type II algebras. We describe the non-commutative projective geometry and compute the finite dimensional simple modules for the homogenization of Type I algebras in the case that α is not a primitive root of unity. In this case, all finite dimensional simple modules are quotients of line modules that are homogenizations of Verma modules.

Embedding a quantum rank three quadric in a quantum P3

We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P3. 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.

Normalizing Extensions of the Two-Veronese of a Three Dimensional Artin–Schelter Regular Algebra on Two Generators

We consider certain regular algebras of global dimension four that map surjectively onto the two-Veronese of a regular algebra of global dimension three on two generators. We also study the point modules.

The Center of Some Quantum Projective Planes

The homogeneous coordinate ring of a quantum projective plane is a 3-dimensional Artin–Schelter regular algebra with the same Hilbert series as the polynomial ring in three variables; such an algebraAis a graded noncommutative analogue of the polynomial ring in three variables. WhenAis a finite module over its centerZ(A), we define the schemeS=Proj(Z(A)) and the sheaf A of OS-algebras by A(S(f))=A[f−1]0. The center Z of A is defined by Z(S(f))=Z(A[f−1]0) and following Grothendieck we may define the scheme Spec(Z). The algebrasAfall into several families, and for many of these it has been shown that Spec(Z)≅P2whenAis finite over its center. This paper shows that Spec(Z)≅P2for two more families.

Graded Modules of Gelfand–Kirillov Dimension One over Three-Dimensional Artin–Schelter Regular Algebras

LetAbe a three dimensional Artin–Schelter regular algebra. We give a description of the category of finitely generatedA-modules of Gelfand–Kirillov dimension one (modulo those of finite dimension over the ground field). The proof is based upon a result by Gabriel which says that locally finite categories can be described by module categories over topological rings.

Algebras associated to elliptic curves

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

Artin–Schelter Regular Algebras of Global Dimension Three

Artin–Schelter Regular Algebras of Global Dimension Three

Central extensions of three dimensional Artin–Schelter regular algebras

Central extensions of three dimensional Artin–Schelter regular algebras

Central extensions of three dimensional Artin-Schelter regular algebras

Central extensions of three dimensional Artin-Schelter regular algebras

Segre Product of Artin–Schelter Regular Algebras of Dimension 2 and Embeddings in Quantum [formula omitted]3's

The Segre embedding of P1×P1as a smooth quadricQin P3corresponds to the surjection of the four-dimensional polynomial ring onto the Segre productSof two copies of the homogeneous coordinate ring of P1. We study Segre products of noncommutative algebras. If in particularAandBare two copies of a quantum P1thenS=⊕i(Ai⊗kBi) is a twisted homogeneous coordinate ring of the quadricQ. The main result of this paper is the classification of all embeddings of the Segre product of two quantum planes into so-called quantum P3's. These are (the Proj of) Artin–Schelter regular algebrasRof global dimension four with the Hilbert series of a commutative polynomial ring and which map ontoS. IfRis not a twist of a polynomial ring, then the point scheme ofReither is the union of the quadricQwith a line or is only the quadricQ. In the first case,Ris a central extension of a three-dimensional Artin–Schelter regular algebra and a twist of an algebra mapping onto the (commutative) homogeneous coordinate ring ofQ; in the second case, such an algebraRis the first known example of a four-dimensional Artin–Schelter regular algebra which is not determined by its point scheme.

On Gelfand-Kirillov Transcendence Degree

We study some basic properties of the Gelfand-Kirillov transcendence degree and compute the transcendence degree of various infinite-dimensional division algebras including quotient division algebras of quantized algebras related to quantum groups, 3-dimensional Artin-Schelter regular algebras and the 4-dimensional Sklyanin algebra.

On Quantum Spaces of Lie Algebras

The homogenization H(g) of the enveloping algebra of a finite dimensional Lie algebra g is an Artin-Schelter regular algebra. We characterize d-dimensional linear subspaces in the corresponding quantum space P q (g) as homogenizations of induced representations from codimension d Lie subalgebras. Furthermore we prove that the point variety has an embedded component iff there is a line, not contained in this point variety

On quantum spaces of Lie algebras

The homogenization H ( g ) H(\mathfrak {g}) of the enveloping algebra of a finite dimensional Lie algebra g \mathfrak {g} is an Artin-Schelter regular algebra. We characterize d d -dimensional linear subspaces in the corresponding quantum space P q ( g ) {\mathbb {P}_q}(\mathfrak {g}) as homogenizations of induced representations from codimension d d Lie subalgebras. Furthermore we prove that the point variety has an embedded component iff there is a line, not contained in this point variety.

Koszul duality and the Hochschild cohomology of Artin–Schelter regular algebras

We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.