Algebraic Conjecture Research Articles

The Elliott conjecture for Villadsen algebras of the first type

We study the class of simple C ∗ -algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C ∗ -algebraic ( Z -stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z -stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C ∗ -algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C ∗ -algebras.

The Steinberg variety and representations of reductive groups

We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck. The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne–Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.

James' conjecture for Hecke algebras of exceptional type, I

In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' conjecture for Iwahori–Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are: • the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham–Lehrer, and • the explicit determination of W-graphs for the irreducible (generic) representations of Hecke algebras of type E 7 and E 8 by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's MeatAxe and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.

A cancellation conjecture for free associative algebras

Ve develop a new method to deal with the Cancellation Conjecture of Zariski in different environments. We prove the conjecture for free associative algebras of rank two. We also produce a new proof of the conjecture for polynomial algebras of rank two over fields of zero characteristic.

An approach to the finitistic dimension conjecture

Let R be a finite dimensional k-algebra over an algebraically closed field k and mod R be the category of all finitely generated left R-modules. For a given full subcategory X of mod R, we denote by pfd X the projective finitistic dimension of X . That is, pfd X : = sup { pd X : X ∈ X and pd X < ∞ } . It was conjectured by H. Bass in the 60's that the projective finitistic dimension pfd ( R ) : = pfd ( mod R ) has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201–204] a function Ψ : mod R → N , which turned out to be useful to prove that pfd ( R ) is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of mod R instead of a class of algebras. That is, we suggest to take the class of categories F ( θ ) , of θ-filtered R-modules, for all stratifying systems ( θ , ⩽ ) in mod R. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension.

Gerstenhaber structure and Deligne’s conjecture for Loday algebras

A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure–Smith to prove an analogue of Deligne’s conjecture for Loday algebras.

Strong no-loop conjecture for algebras with two simples and radical cube zero

Let ${\mit\Lambda}$ be an artinian ring and let ${\mathfrak r}$ denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when ${\mit\Lambda}$ is graded by its radical, with at most two simple modules

Is tame open?

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.

Unambiguous comparison of the states of multiple quantum systems

We consider N quantum systems initially prepared in pure states and address the problem of unambiguously comparing them. One may ask whether or not all $N$ systems are in the same state. Alternatively, one may ask whether or not the states of all N systems are different. We investigate the possibility of unambiguously obtaining this kind of information. It is found that some unambiguous comparison tasks are possible only when certain linear independence conditions are satisfied. We also obtain measurement strategies for certain comparison tasks which are optimal under a broad range of circumstances, in particular when the states are completely unknown. Such strategies, which we call universal comparison strategies, are found to have intriguing connections with the problem of quantifying the distinguishability of a set of quantum states and also with unresolved conjectures in linear algebra. We finally investigate a potential generalisation of unambiguous state comparison, which we term unambiguous overlap filtering.

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, a conjecture on triangulation is confirmed. The relation between the piecewise linear algebraic curve and four-color conjecture is also presented. By Morgan-Scott triangulation, we will show the instability of Bezout number of piecewise algebraic curves. By using the combinatorial optimization method, an upper bound of the Bezout number defined as the maximum finite number of intersection points of two piecewise algebraic curves is presented.

CRITICAL POINTS OF MASTER FUNCTIONS AND FLAG VARIETIES

We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac-Moody algebra and prove the conjecture for algebras $sl_{N+1}, so_{2N+1}$, and $sp_{2N}$. We show that populations associated with a collection of integral dominant $sl_{N+1}$-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗ C^* -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

Splitting homomorphisms and the Geometrization Conjecture

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

Computation of Invariants for Reductive Groups

We will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group. The algorithm basically consists of a single Gröbner basis computation. We will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants.

Properness, strictness, and nilness in jordan systems

The basic theme of this paper, suggested by Ottmar Loos, is to show that certain sets S ofelements in a Jordan system J form an ideal by showing that S = J∩R([Jtilde]) is the Amitsur shrinkage of some well-knownl radical R on some extension [Jtilde] of J. The Jacobson radical Rad(J) and the degenerate radical Deg(J) have elemental characterizations as (respectively) the properly quasi-invertible elements and the m-finite elements. Frojn these two characterizations we show that: (1) the strictly properly nilpotent elements coincide with the strictly properly quasi-invertible elements and form the ideal J∩Rad(J[T]) (2) the strictly m-finite elements coincide with the m-finite elements and form the ideal Deg:(J) (3) the m-bounded elements form an ideal J ∩ Deg(Seq(J)) (Seq the algebra of sequences); and (4) the strictly m-bounded elements coincide with the strictly properly nilpotence-bounded elements and form the ideal J ∩ Deg(Seq(J[t])). We show that all these constructions are stable under structural pairs, a useful generalization of the concept of structural transformation. The question of whether the properly nilpotent elements form an ideal, and if so whether this is the nil radical, is an open question intimately related to the Kothe Conjecture for associative algebras.

Monoid gradings on algebras and the cartan determinant conjecture

In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A &lt; ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.

A geometric construction of the Iwahori–Hecke algebra for unramified groups

This paper concerns constructing the Iwahori-Hecke algebra for certain p-adic groups. It can be regarded as a natural extension of the ideas laid down in [KL2]; the objective (and conclusion) of that paper was the proof of Deligne-Langlands conjecture for Hecke algebras arising from split p-adic groups (with connected centers). The key observation of that paper was the geometric construction of the Iwahori-Hecke algebra for a split p-adic group.

The Kazhdan–Lusztig conjecture for 𝒲 algebras

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of 𝒲 algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, which constructs the 𝒲 algebras from affine Kac–Moody (KM) algebras and in a similar fashion 𝒲 modules from KM modules. Assuming certain properties of this functor, the 𝒲 characters are subsequently derived from the Kazhdan–Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan–Lusztig polynomials give the multiplicities in the characters.

A note on the stable equivalence conjecture

In this paper, algebras are finite dimensional over an algebraically closed field k, and modules are k-finite dimensional left modules. We prove the stable equivalence conjecture for algebras stably equivalent to algebras A with the following conditions: basic, connected and selfinjective; rad3A = 0 but rad2A ¬ 0; and the separated quiver Q3 A of the quiver QA of A consisting of more than two connected components.

Discrete Magnetic Laplacian

We consider a 2-dimensional discrete operator which we call the Discrete Magnetic Laplacian (DML); it is an analogue of the magnetic Schrödinger operator. It follows from well known arguments that DML has the same spectrum (as a subset inR) as the Almost Mathieu operator (AM). They also have the same Integrated Density of States (IDS) which is known to be continuous. We show that DML is an element in a II1-factor and its IDS can be expressed through the trace in the II1-factor. It follows that DML never has anyL 2-eigenfunctions (i.e. has no point spectrum). Then we formulate a natural algebraic conjecture which implies that the spectrum of DML (hence the spectrum of AM) is a Cantor set.