Graded Commutative Algebras: Examples, Classification, Open Problems

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

The general solution of the multiplicative symmetric functional equation $$f(f(x)y)=f(xf(y))$$, in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres’s solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups).

The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

Radicals in Abelian Groups

Affine semigroups—discrete analogues of convex polyhedral cones—mark the cross-roads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [17] and Kempf, Knudsen, Mumford and Saint-Donat [33] in the early 70s, toric geometry is still a very active area of research. However, the last decade has clearly witnessed the extensive study of affine semigroups from the other two perspectives. No doubt, this is due to the tremendously increased computational power in algebraic geometry, implemented through the theory of Grobner bases, and, of course, to the modern computers. In this article we overview those aspects of this development that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different “sizes” of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects—polytopal semigroups and algebras— provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting (Section 4). Our counterexample to certain covering conjectures (Section 3) was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

Let G be an Abelian group whose p-component G p is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that G p is a direct factor of S(FG) with totally projective complement, provided G p is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if G p is summable. Moreover, if FG≅ FH as F-algebras for some group H and if G p is summable, then H p is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000ab20012004).

An expansion algorithm for constructing axial algebras

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

Biased graphs. VII. Contrabalance and antivoltages