S. P. Smith

Central extensions of three dimensional Artin-Schelter regular algebras

For a 3-dimensional Artin-Schelter-regular algebra A with Hilbert series (1 − t)−3 we study central extensions; that is, graded algebras D with a regular central element z in degree 1, such that D/(z) = A. We classify such D and we also classify certain D-modules (point modules and line modules) which proved to be important in the study of 3-dimensional Artin-Schelter-regular algebras.

Auslander-gorenstein rings

We study basic properties of Auslander-Gorenstein rings related to duality, localization and purity of minimal injective resolutions.

Subspaces of non-commutative spaces

This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space.

Homogenized sl(2)

This note studies a special case of Artins projective geometry (Geometry of quantum planes, MIT, preprint, 1990) for noncommutative graded algebras. It is shown that (most of) the line modules over the homogenization of the enveloping algebra U(sl(2, C)) are in bijection with the lines lying on the quadrics that are the (closures of the) conjugacy classes in sl(2, C). Furthermore, these line modules are the homogenization of the Verma modules for sl((2, C)

The minimal nilpotent orbit, the Joseph ideal, and differential operators

Abstract Fix a simple complex Lie algebra g , not of type G 2 , F 4 , or E 8 . Let Ō min denote the Zariski closure of the minimal non-zero nilpotent orbit in g , and let g = n + ⊕ h ⊕ n − be a triangular decomposition. We prove THEOREM. (1) If g is not of type A n then there exists an irreducible component X of Ō min ∩ n + such that U(g)/J o = D(X), where J o is the Joseph ideal and D(X) denotes the ring of differential operators on X. (2) If g is of type A n then for n − 2 of the n irreducible components X i of Ō min ∩ n + there exist (distinct) maximal ideals J i of U(g) such that U(g)/J i = D(X i ) .

Krull dimension of the enveloping algebra of sl(2, C

Let U denote the enveloping algebra of the simple Lie algebra ~42, C). In this paper it is shown that the Krull dimension of U (denoted ] U]) is two. If U(g) is the enveloping algebra of a finite-dimensional solvable Lie algebra g then it is straightforward to show that ) U(g)] = dim g [5, 3.8.111. The problem as to the Krull dimension of U was first mentioned by Gabriel and Nouazt [9] they show that CJ has a chain of prime ideals of length two, and none of length greater than two. From this they conclude that the Krull dimension of U is two, although the correct conclusion is only that ] Uj > 2. Subsequent to [9], both Arnal and Pinczon [l] and Roos [lo] established that if R were a non-artinian simple primitive factor ring of U then ] R I= 1. More recently the author [ 111 proved that if R were a nonartinian primitive factor ring of U which was not simple then again IR ) = 1. The result in the present paper implies those in [ 1, 10, 111. The fundamental tool in the proof that (Ul = 2 is Gelfand-Kirillov dimension (GK-dimension). The proof is in two parts. In Section 2 a number of preliminary results (already known) concerning GK-dimension are recalled. In particular, Lemma 2.3 provides the basic connection between GK-dimension and Krull dimension. The more detailed analysis of U is carried out in Section 3. The crucial result is that any finitely generated U module of Krull dimension 1 has GK-dimension 2 the result then quickly follows from Lemma 2.3. The author would like to thank J. C. McConnell both for bringing this problem to his attention, and for many helpful conversations.

Injective resolutions of some regular rings

Abstract Let A be a noetherian Auslander regular ring and δ the canonical dimension function on A -modules, which is defined as δ(M)=d−j(M) where d is the global dimension of A and j(M) is the grade of M. An A -module is s - pure if δ(N)=s for all its non-zero noetherian submodules N , and is essentially s - pure if it contains an essential submodule which is s -pure. Consider a minimal injective resolution of A as an A -module 0→A→I 0 →I 1 →⋯→I d →0. We say A has a pure (resp. essentially pure ) injective resolution if I i is (d−i) -pure (resp. essentially (d−i) -pure). We show that several classes of Auslander regular rings with global dimension at most 4 have pure or essentially pure injective resolutions.

The Four-Dimensional Sklyanin Algebras

The four-dimensional Sklyanin algebras are certain noncommutative graded algebras having the same Hilbert series as the polynomial ring on four indeterminates. Their structure and representation theory is intimately connected with the geometry of an elliptic curve (and a fixed translation) embedded in p3. This is an account of the work done on these algebras over the past four years.

A remark on Gelfand-Kirillov dimension

Let A be a finitely generated non-PI Ore domain and Q the quotient division algebra of A. If C is the center of Q, then GKdimC ≤ GKdimA− 2. Throughout k is a commutative field and dimk is the dimension of a k-vector space. Let A be a k-algebra and M a right A-module. The Gelfand-Kirillov dimension of M is GKdimM = sup V,M0 lim n→∞ logn dimkM0V n where the supremum is taken over all finite dimensional subspaces V ⊂ A and M0 ⊂ M . If F ⊃ k is another central subfield of A, we may also consider the Gelfand-Kirillov dimension of M over F which will be denoted by GKdimF to indicate the change of the field. We refer to [BK], [GK] and [KL] for more details. Let Z be a central subdomain of A. Then A is localizable over Z and the localization is denoted by AZ . For any right A-module M , M ⊗ AZ is denoted by MZ . Let F be the quotient field of Z. The first author [Sm, 2.7] proved the following theorem: Let A be an almost commutative algebra and Z a central subdomain. Suppose M is a right A-module such that MZ 6= 0. Then GKdimM ≥ GKdimF MZ + GKdimZ. As a consequence of this, if A is almost commutative but non-PI and Z is a central subalgebra such that every nonzero element in Z is regular in A, then GKdimZ ≤ GKdimA− 2. It is natural to ask if the above theorem (and hence the consequence) is true for all algebras. In this paper we will precisely prove this. Theorem 1. Let A be an algebra and Z a central subdomain. Suppose M is a right A-module such that MZ 6= 0. Then GKdimM ≥ GKdimF MZ + GKdimZ. An algebra is called locally PI if every finitely generated subalgebra is PI. As a consequence of Theorem 1, we have Received by the editors July 12, 1996 and, in revised form, August 20, 1996. 1991 Mathematics Subject Classification. Primary 16P90.

Krull dimension of factor rings of the enveloping algebra of a semi-simple Lie algebra

Let g be a semi-simple complex Lie algebra with enveloping algebra E/(g). It is shown that the Krull dimension of t/(g) is bounded above by dimg — r, where r is half the minimal dimension of a non-trivial G orbit in g* (G is the adjoint group of g).