1,253 research outputs found
G-algebras, twistings, and equivalences of graded categories
Given Z-graded rings A and B, we study when the categories gr-A and gr-B are
equivalent. We relate the Morita-type results of Ahn-Marki and del Rio to the
twisting systems introduced by Zhang. Using Z-algebras, we obtain a simple
proof of Zhang's main result. This makes the definition of a Zhang twist
extremely natural and extends Zhang's results.Comment: 13 pages; typos corrected and revised slightly; to appear in Algebras
and Representation Theor
Classifying birationally commutative projective surfaces
Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3
over an uncountable algebraically closed field. Suppose that the graded
quotient ring of R is a skew-Laurent ring over a field; we say that R is a
birationally commutative projective surface. We classify birationally
commutative projective surfaces and show that they fall into four families,
parameterized by geometric data. This generalizes work of Rogalski and Stafford
on birationally commutative projective surfaces generated in degree 1; our
proof techniques are quite different.Comment: 60 pages; Proceedings of the LMS, 201
Geometric idealizers
Let X be a projective variety, an automorphism of X, L a
-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the
twisted homogeneous coordinate ring , let I be the right
ideal of sections vanishing at Z. We study the subring R = k + I of B. Under
mild conditions on Z and , R is the idealizer of I in B: the maximal
subring of B in which I is a two-sided ideal.
We give geometric conditions on Z and that determine the algebraic
properties of R, and show that if Z and are sufficiently general, in a
sense we make precise, then R is left and right noetherian, has finite left and
right cohomological dimension, is strongly right noetherian but not strongly
left noetherian, and satisfies right (where d = \codim Z) but fails
left . We also give an example of a right noetherian ring with infinite
right cohomological dimension, partially answering a question of Stafford and
Van den Bergh. This generalizes results of Rogalski in the case that Z is a
point in .Comment: 43 pages; comments welcom
The Poisson spectrum of the symmetric algebra of the Virasoro algebra
Let be the Witt algebra of algebraic
vector fields on and let be the Virasoro algebra, the
unique nontrivial central extension of . In this paper, we study the Poisson
ideal structure of the symmetric algebras of and , as well as several
related Lie algebras. We classify prime Poisson ideals and Poisson primitive
ideals of and . In particular, we show that the only functions
in which vanish on a nontrivial Poisson ideal (that is, the only maximal
ideals of with a nontrivial Poisson core) are given by linear
combinations of derivatives at a finite set of points; we call such functions
local. Given a local function , we construct the associated
Poisson primitive ideal through computing the algebraic symplectic leaf of
, which gives a notion of coadjoint orbit in our setting.
As an application, we prove a structure theorem for subalgebras of of
finite codimension and show in particular that any such subalgebra of
contains the central element , substantially generalising a result of Ondrus
and Wiesner on subalgebras of codimension 1. As a consequence, we deduce that
is Poisson simple if and only if .Comment: 51 pages; comments welcome. v2: 52 pages; paper rearranged slightly;
classification of maximal Poisson ideals of added; statements of
some results correcte
A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras
We consider when the symmetric algebra of an infinite-dimensional Lie
algebra, equipped with the natural Poisson bracket, satisfies the ascending
chain condition (ACC) on Poisson ideals. We define a combinatorial condition on
a graded Lie algebra which we call Dicksonian because it is related to
Dickson's lemma on finite subsets of . Our main result is:
Theorem. If is a Dicksonian graded Lie algebra over a field of
characteristic zero, then the symmetric algebra satisfies the
ACC on radical Poisson ideals.
As an application, we establish this ACC for the symmetric algebra of any
graded simple Lie algebra of polynomial growth, and for the symmetric algebra
of the Virasoro algebra. We also derive some consequences connected to the
Poisson primitive spectrum of finitely Poisson-generated algebras.Comment: 29 pages; comments welcome; v2 minor changes to introduction,
submitte
Geometric algebras on projective surfaces
Let X be a projective surface, let \sigma be an automorphism of X, and let L
be a \sigma-ample invertible sheaf on X. We study the properties of a family of
subrings, parameterized by geometric data, of the twisted homogeneous
coordinate ring B(X, L, \sigma). In particular, we find necessary and
sufficient conditions for these subrings to be noetherian. We also study their
homological properties, their associated noncommutative projective schemes, and
when they are maximal orders. In the process, we produce new examples of
maximal orders; these are graded and have the property that no Veronese subring
is generated in degree 1.
Our results are used in a companion paper to give defining data for a large
class of noncommutative projective surfaces.Comment: 39 pages; v2 results largely unchanged, but notation describing
algebras revised significantly. As a result details of many proofs have
changed, and statements of some results. To appear in Journal of Algebr
Generalised Witt algebras and idealizers
Let be an algebraically closed field of characteristic zero, and let
be an additive subgroup of . Results of Kaplansky-Santharoubane
and Su classify intermediate series representations of the generalised Witt
algebra in terms of three families, one parameterised by and two by . In this note, we use the first family to
construct a homomorphism from the enveloping algebra to a
skew extension of . We show that the image of is contained
in a (double) idealizer subring of this skew extension and that the
representation theory of idealizers explains the three families. We further
show that the image of under is not left or right
noetherian, giving a new proof that is not noetherian.
We construct as an application of a general technique to create ring
homomorphisms from shift-invariant families of modules. Let be an arbitrary
group and let be a -graded ring. A graded -module is an
intermediate series module if is one-dimensional for all . Given
a shift-invariant family of intermediate series -modules parametrised by a
scheme , we construct a homomorphism from to a skew-extension of
. The kernel of consists of those elements which annihilate
all modules in .Comment: 9 pages; to appear in J. Algebr
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