On Functionals and outside Corners of Monomial Ideals a Lifted Grr Obner Basis for the Enveloping Algebra

Tatiana Gateva-Ivanova, American University in BulgariaNoncommutative Grobner Bases in Skew-polynomial RingsWe study a class of graded standard nitely presented quadratic algebrasA over a xed eld K, called binomial skew polynomial rings. Followinga classical tradition (and recent trend), we take a combinatorial approachto study the algebraic properties of A. Our purpose is to exhibit a classof non-commutative algebras with good algebraic properties among whichNoetherianness, being a domain, regularity, which can be read o a presen-tation A = K hXi =(F ), where X = fx1; :::; xng is a set of indeterminatesof degree 1, K hXi is the unitary free associative algebra generated by Xand (F ) is the two-sided ideal, generated by a nite set of homogeneouspolynomials.Following [2], a binomial skew polynomial ring is a graded algebraA = K hXi =(F ), which has precisely n(n 1)=2 de ning relations F =fxjxi cijuij j 1 i < j ng; such that (a) cij are non-zero coe cients; (b)uij =xi0xj0; for 1 i < j n, with i0 < j, i0 < j 0; (c) the set F is a reducedGrobner basis (w.r.t. the deg-lex ordering on the free semigroup hXi) of theideal (F ).We give necessary and su cient condition for Noetherianness of A in casethat the set of relations F is square-free. We show that the notion of Grobnerbasis can be extended for modules over strictly ordered algebras, so that ananalogue of Bergman's Diamond Lemma holds (cf. [2], 2.2) and apply thisfor Noetherian skew polynomial rings with binomial relations. The followingresult can be extracted from [2] and [3].Theorem Let A = K hXi =(F ), be a binomial skew-polynomial ring. Sup-pose the set of relations F is square-free. Then the following conditions areequivalent:(1) each xixj; with 1 i < j n, appears in the right hand side of somerelation in F ;(2) A is left Noetherian;(3) A is right Noetherian;(4) Any left ideal of A has a nite Grobner basis (with respect to somegrading of A;)(5) Any right ideal of A has a nite Grobner basis (with respect to somegrading of A; Furthermore, any of the conditions (1) (5) implies that(a) the membership problem for nitely generated one-sided ideals in A isdecidable;2 (b) A is a domain;(c) A is regular in the sense of Artin-Schelter (cf.[2]).(d) the operator on X2 naturally de ned by the quadratic relations F sat-is es the setheoretic Yang Baxter equations.References[1] M. Artin, W. Schelter, Graded algebras of global dimension 3, Advancesof Math. 66 (1987), 171-216.[2] T. Gateva-Ivanova, Skew polynomial rings, J. Algebra, 185 (1996), 710-753.[3] T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I-type Preprint.************************************************************Susan Hermiller, New Mexico State UniversityRewriting systems and 3-manifoldsThe fundamental groups of most (conjecturally, all) closed 3-manifolds withuniform geometries have nite complete rewriting systems. The fundamentalgroups of a large class of amalgams of circle bundles also have nite completerewriting systems. The general case remains open. This is joint work withMike Shapiro.************************************************************Anne Heyworth, University of WalesComputerised left Knuth-Bendix procedure for enumerating cosetsUsing the presentation hX,Ri of a group G, and the generators of a subgroupH in G, and a monomial ordering on F(X) one can do a left Knuth-Bendixcompletion. This gives a \left-complete" system that can be applied to thewords in the free group F(X) to enumerate the right cosets of H in G. Ihave computerised the process (using Axiom). It is a valid alternative to theTodd-Coxeter process.************************************************************3 Xenia Kramer, New Mexico State UniversityThe Noetherian Property in Some Quadratic AlgebrasWe introduce a new class of noncommutative rings called pseudopolyno-mial rings and give su cient conditions for such a ring to be Noetherian.Pseudopolynomial rings are standard nitely presented algebras over a eldwith some additional restrictions on their de ning relations|namely thatthe polynomials in a Grobner basis for the ideal of relations must be ho-mogeneous of degree 2|and on the Ufnarovskii graph (A). The class ofpseudopolynomial rings properly includes the generalized skew polynomialrings introduced by M. Artin and W. Schelter. We use the graph (A) tode ne a weaker notion of almost commutative, which we call almost com-mutative on cycles. We show as our main result that a pseudopolynomialring which is almost commutative on cycles is Noetherian. A counterex-ample shows that a Noetherian pseudopolynomial ring need not be almostcommutative on cycles.Xenia Kramer, New Mexico State UniversityCombinatorial Homotopy of Simplicial ComplexesCombinatorial homotopy is an analog of the usual homotopy theory of alge-braic topology which is intended to re ect combinatorial features of a sim-plicial complex. The combinatorial version of a path is a nite sequence ofsimplices 1; 2; : : : ; n in a complex where for 1 i < n, the dimensionof i \ i+1 is at least q, for some previously set value q. Two paths arehomotopic in this combinatorial sense if one can be deformed to the otherwhile keeping the dimension of the intersection of adjacent simplices at leastq.The motivation for the study of combinatorial homotopy comes from theanalysis of complex information systems. The ow of tra c in such a systemis constrained by the underlying structure of the system; that is, the dynamicbehavior of a system is in uenced by its static structure. Combinatorialhomotopy was introduced to describe features of that static structure.In this talk, we will focus on the mathematics of combinatorial homotopy.After de ning it, we give an algorithm for computing it which uses Grobnerbases in a free associate algebra. This work was done in collaboration withReinhard Laubenbacher.************************************************************4 Peter Malcolmson, Wayne State UniversityGrobner-Shirshov bases for quantum enveloping algebrasWe give a method for nding Grobner-Shirshov bases for the quantum en-veloping algebras of Drinfel'd and Jimbo, show how the methods can beapplied to Kac-Moody algebras, and explicitly nd the bases for quantumenveloping algebras of type A (for the quantizing coe cient not an eighthroot of unity).************************************************************Eduardo do Nascimento Marcos, IME-USP (Dept. de Matematica)BrazilGraded Rings of Local Finite Representation TypeGraded Artin algebras whose category of graded modules is locally of niterepresentation type are introduced. The representation theory of such al-gebras is studied. In the hereditary case and in the stably equivalent tohereditary case, such algebras are classi ed.************************************************************Mike Newman, Australian National UniversitySome use of Knuth-Bendix in the study of groupsI will describe the use of Knuth-Bendix and other software for proving thatcertain Engel groups are nilpotent.************************************************************Gretchen Ostheimer, Tufts UniversityFinding Matrix Representations for Polycyclic GroupsIt has been known since the 1960's that a solvable group G can be embed-ded in GL (n;Z) for some n if and only if G is polycyclic. In 1990, Segaldescribed an algorithm for constructing such an embedding when the poly-cyclic group is given by a nite presentation; however, that algorithm is notsuitable for computer implementation. We describe an alternative algorithmfor this problem. Our primary tool is a program developed by Lo in 1996 inwhich the Grobner basis method from commutative ring theory is extendedto the group ring of a polycyclic group given by a nite presentation. Pre-liminary experiments indicate that our algorithm will be e cient enough toinvestigate some interesting examples. Further experimentation is needed to5 determine the range of input for which the algorithm is practical with currenttechnology.This work was undertaken as part of the author's Ph.D. thesis under thedirection of Charles Sims and was continued collaboratively with Eddie Lo.************************************************************Deepak Parashar, The Robert Gordon UniversityQuantum Groups and Quantum SpacesDuring the recent past there has been considerable success in the devel-opment of the theory of Quantum Groups, an exciting new generalizationof ordinary Lie Groups. These mathematical structures have a wide vari-ety of applications, primarily in non-commutative geometry and models intwo-dimensional physics. The purpose of this talk is to give a systematicmathematical introduction to the subject of quantum groups and quantumspaces using GLq(2) as our main example.************************************************************F. Leon Pritchard, Rutgers University-NewarkThe Ideal Membership Problem in Non-Commutative PolynomialRingsLet X be a non-commutative monoid with term order; let R be a commu-tative, unital ring; let I be an ideal in the non-commutative polynomialring RhXi; and let f 2 RhXi. In this setting the problem of determiningwhether f 2 I is studied. In a manner analogous to the commutative case,weak Grobner bases are de ned and their basic properties are studied. Wewill see that in the non-commutative setting, when the coe cient ring is nota eld, and when we enlarge the polynomial ring by adding more variables,weak Grobner bases may exhibit unpleasant behavior that has no analog inthe commutative case. Quite in general for f 2 RhXi, it is undecidablewhether f 2 I. This follows from the fact that the word problem for freesemigroups is undecidable. If I is generated by a recursively enumerable set,then we give a semi-decision procedure that halts if and only if f 2 I. Finallywe examine a class of nicely behaved ideals for which weak Grobner basescan be easly computed.************************************************************6 Mark Stankus, California Polytechnic State University Of San Luis ObispoApplying Noncommutative Grobner Basis to Problems in Analysisand Control TheoryThere are a number of theorems in analysis and control theory which have amajor algebraic component. This algebraic component consists of nding asystem of matrix equations which has some desirable property (e.g., amenableto numerical solution) and which is equivalent to a given system of matrixequations.Our experience has shown that transforming the given system of matrixequations into one which has a desirable property cannot be done by usinga noncommutative Grobner Basis algorithm alone. One common concern isthat a noncommutative Grobner Basis can be in nite and so the noncom-mutative Grobner Basis algorithm will never terminate. When one stops thealgorithm, a Grobner Basis is not obtained. Whether or not the set of re-lations obtained is a Grobner Basis, many of the relations obtained may beconceptually redundant from the perspective of the application being con-sidered. For this reason, we consider the following additional algorithm:(1) Find a minimal generating set for an ideal (so that one can removeconceptually redundant information generated by the Grobner Basis algo-rithm).Since allowing for changes of variables can be very useful, we also considerthe following additional algorithm which facilitates the introduction of newvariables:(2) Compute decompositions of a matrix expression into a compositionof matrix expressions which respect certain qualitative information.Our methodology for deriving the algebraic component of a theorem usesthe noncommutative Grobner Basis algorithm and the two algorithms men-tioned above. The methodology consists of running these three algorithmson a computer for a nite amount of time, displaying the relations obtained,having a person view the relations obtained, making any desired conclusionsbased on the non-algebraic components of the theorem, making any desiredchanges of variables and repeating the process as necessary.************************************************************Victor Ufnarovski, Lund UniversityBERGMAN and ANICK programs for non-commutative algebrasThe last versions of two Computer Algebra programs are discussed. One ofthem (author J.Backelin, Stokholm University) is a rather e cient programfor calculating Grobner bases for graded algebras. Another one (authors -7 A.Podoplelov, V.Ufnarovski, Institute of Mathematics, Moldova; LTH, Lund,Sweden) calculates Anick's resolution.************************************************************John J. Wavrik, University of California-San DiegoGrobner Bases, Rewrite Rules, and Matrix ExpressionsThis talk concerns the automated simpli cation of expressions which involvenon-commuting variables. The technology has been applied to the simpli -cation of matrix and operator theory expressions which arise in engineeringapplications. The non-commutative variant of the Grobner Basis Algorithmis used to generate rewrite rules. We will also look at the phenomenon ofin nite bases and implications for automated theorem proving.************************************************************Sergey Yuzvinsky, University of OregonKoszul algebras related to hyperplane arrangements and graphsIn the talk, we discuss the Orlik-Solomon algebras. Such an algebra A is acombinatorial interpretation of the cohomology algebra of the complementof a complex hyperplane arrangement. A has a standard, so called \brokencircuit" basis. For the well-studied class of supersolvable arrangements thisbasis has a very nice property that allows us to deform A to a monomialalgebra A0 de ned by a graph. Studying a basis of A0 we recover that it isKoszul (a theorem by Froberg) and obtain an elegant formula for its Hilbertseries. Then Drinfeld's deformation theory implies that A is Koszul.************************************************************8