Jordan forms for mutually annihilating nilpotent pairs

In this paper we completely characterize all possible pairs of Jordan canonical forms for mutually annihilating nilpotent pairs, i.e. pairs (A,B) of nilpotent matrices such that AB=BA=0.     

Auslander-Reiten correspondence for tilting pairs

Let A be an Artin algebra. A pair (C,T) of A-modules is tilting provided that C and T are (finitely generated) self-orthogonal, and . Particularly, T is a tilting module if and only if (A,T) is a tilting pair. In the note, we will extend the Auslander-Reiten correspondence for tilting modules to the context of tilting pairs.     

On the zero set of semi-invariants for regular modules over tame canonical algebras

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete intersections.     

Moving subvarieties by endomorphisms

Let K be an algebraically closed field of characteristic zero and let A be a semiabelian variety defined over K. Let End(A) be the ring of endomorphisms of A. Let X ⊂ A be a subvariety of smaller dimension. We show that does not equal A(K). In particular, we may take K to be countable.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

---

Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

A categorification of finite-dimensional irreducible representations of quantum \mathfraksl2{\mathfrak{sl}_2} and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.     

Semi-invariants of canonical algebras

We describe the algebras of semi-invariants on the varieties of regular representations of canonical algebras. In particular, we show that these algebras are polynomial algebras or complete intersections. Received: 29 March 1999     

The existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators

We prove the existence of Hall polynomials for $x^2$-bounded invariant subspaces of nilpotent linear operators.     

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

On the number of stable quiver representations over finite fields

We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.     

On module variety dimension for coverings and degenerations of algebras

Given a pair of G-covering functors F¹:R→R₁ and F⁰:R→R₀ such that F⁰ is a Galois covering, the inequality for all z,t, of the dimensions of the first kind module sets under some assumptions is proved (Theorem 2.2). The result is applied to show the equality of the module variety dimensions for some special degenerations of algebras. Certain consequences for preserving wild and tame representation types by G-covering functors are also presented (Theorems 2.4 and 3.1).     

Degenerations for indecomposable modules in almost cyclic coherent Auslander-Reiten components

We give a necessary and sufficient condition for the existence of degeneration M≤_degN for arbitrary modules M, N of the same dimension from the additive category of a generalized standard almost cyclic coherent component of the Auslander-Reiten quiver of finite-dimensional algebra.     

The subregular variety to the variety of special lattices

We recall the basic geometric properties of the full lattice variety, the projective variety parametrizing special lattices over Witt vectors which was introduced in Haboush (2005) [6]. It is an analog in unequal characteristic, of a certain Schubert variety in the affine Grassmannian for , and it is normal and a locally complete intersection (Haboush and Sano, submitted for publication [7], Sano (2004) [15]). In this paper, I prove that the complement of its smooth locus, the subregular variety in it, is also normal and a locally complete intersection. The result is analogous to the geometry of the subregular subvariety of the nilpotent cone.     

Wild hypersurfaces

Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type.     

Preservation of the injectivity of the Gauss map for general projections

Let X be a smooth irreducible quasi-projective variety of dimension n in P ^N with N ≥ 2n + 2. Let γ be its Gauss map, let be the embedding obtained from the general projection in P ^N and let γ′ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ(Q) ≠ γ(Q′) implies γ′(Q) ≠ γ′ (Q′). We prove that this property holds in the following cases: N≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n−1 and X does contain a linear (n−1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map. The author is affiliated with the University at Leuven as a research fellow. This paper belongs to the FWO-project G.0318.06.     

Triangulable locally nilpotent derivations in dimension three

In this paper we give an algorithm to recognize triangulable locally nilpotent derivations in dimension three. In case the given derivation is triangulable, our method produces a coordinate system in which it exhibits a triangular form.     

Geometry of the tetrahedron space

Let X^° be the space of all labeled tetrahedra in P³. In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285-312] we constructed a smooth symmetric compactification of X^°. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .     

Hadamard products of hypersurfaces

In this paper we characterize hypersurfaces for which their Hadamard product is still a hypersurface Then we study hypersurfaces and, more generally, varieties which are idempotent under Hadamard powers.