Degenerations and Stable Homomorphisms #

Let Λ be an algebra over an algebraically closed field. We compare the partial order ≤ _hom in the module category of Λ with a certain relation ≤ _stab in the stable module category of Λ. Both relations coincide if Λ is hereditary. Starting with any non-hereditary representation-finite algebra Λ, we construct a representation-finite algebra Λ′, obtained by a covering of the Auslander-Reiten quiver of Λ, such that for Λ′ both relations do not coincide.     

Degenerations of <Emphasis Type="Italic">k</Emphasis>[<Emphasis Type="Italic">t</Emphasis>]/(<Emphasis Type="Italic">t</Emphasis><Superscript>r</Superscript>)-Modules Giving Stratification of the Orbits

Let Λ be a finitely generated associative k-algebra where k is an algebraically closed field. For each natural number d, we have the variety of d-dimensional module structures on k^d given by the multiplication of the elements from a generating set of Λ. The general linear group Gl_d(k) acts on this variety by conjugation and the orbits under this action correspond to isomorphism classes of d-dimensional Λ-modules. For two d-dimensional Λ-modules M and N one says that M degenerates to N if the orbit corresponding to N is in the Zariski-closure of the orbit corresponding to M. Now in this situation the stabilizers of the elements in the orbit corresponding to N acts on the orbit corresponding to M. In this paper we characterize degenerations of k[t]/(t^r)-modules with the property that for each y in the orbit corresponding to N, there is an x_y in the orbit corresponding to M such that the orbit corresponding to M is the disjoint union of orbits of the x_y’s under the action of the stabilizer of y where y runs through the orbit corresponding to N. Presented by Idun ReitenMathematics Subject Classifications (2000) 14L30, 16G10.     

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(M_n), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(M_n) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(M_n).     

Homological Behavior of Auslander’s k-Gorenstein Rings

In this paper we mainly study the homological properties of dual modules over k-Gorenstein rings. For a right quasi k-Gorenstein ring Λ, we show that the right self-injective dimension of Λ is at most k if and only if each M?∈ mod Λ satisfying the condition that Ext $_{\Lambda}^i(M, \Lambda)=0$ for any 1?≤?i?≤?k is reflexive. For an ∞-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of Λ are identical. In addition, we show that if Λ is a left quasi ∞-Gorenstein ring and M?∈ mod Λ with gradeM finite, then Ext $_{\Lambda}^i($ Ext $_{\Lambda ^{op}}^i($ Ext $_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if i?≠gradeM. For a 2-Gorenstein ring Λ, we show that a non-zero proper left ideal I of Λ is reflexive if and only if Λ/I has no non-zero pseudo-null submodule.     

On the Multiplicity Problem and the Isomorphism Problem for the Four Subspace Algebra

Let Λ be the four subspace algebra. We show that for any Λ-module M there exists an algorithm (up to the problem of finding roots of the so-called characteristic polynomial of M) with relatively low polynomial complexity of determining multiplicities of all direct summands of M. Moreover, we give a fully algorithmic criterion for deciding if two Λ-modules M and N are isomorphic.     

Graph representation of projective resolutions

We generalize the concept — dimension tree and the related results for monomial algebras to a more general case — relations algebras Λ by bringing Gröbner basis into play. More precisely, we will describe the minimal projective resolution of a left Λ-module M as a rooted ‘weighted’ diagraph to be called the minimal resolution graph for M. Algorithms for computing such diagraphs and applications as well will be presented.     

Constructing Minimal Projective Resolutions

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.     

The Category of Firm Modules for Nonunital Monomial Algebras

Let k be a field and X a set and P be a set of words over X. Consider the free nonunital k-algebra over X generated by the nonempty words over X and let R be the quotient of this algebra modulo the ideal generated by the words in P. R is called a “nonunital monomial algebra”. A right R-module M is said to be “firm” if M? _R R → M given by m ? r? mr is an isomorphism. In this article we prove that if R is a nonunital monomial algebra, the category of firm modules is Grothendieck.     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.     

Extension closure of k-torsionfree modules

Let Λ be a left and right Noetherian ring. For a positive integer k, we give an equivalent condition that flat dimensions of the first k terms in the minimal injective resolution of Λ are less than or equal to k. In this case we show that the subcategory consisting of k-torsionfree modules is extension closed. Moreover we prove that for a Noetherian algebra every subcategory consisting of i-torsionfree modules is extension closed for any 1 ≤ i ≤ k if and only if every subcategory consisting of i-th syzygy modules is extension closed for any 1 ≤ i ≤ k. Our results generalize the main results in Auslander and Reiten [4].     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Mutation of Auslander Generators for Selfinjective Algebras

Let Λ be an artin algebra with representation dimension equal to three and M an Auslander generator of Λ. We show how, under certain assumptions, we can mutate M to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of M. We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.     

Finitely Presented Functors and Degenerations#

Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF]  [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M   _deg N even when M ⊕ X ≤  _deg N ⊕ X for some X for modules over Λ _q  = k〈 x,y〉/〈 x ²,y ²,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field.     

On the Hochschild (Co)Homology of Quantum Exterior Algebras

We compute the Hochschild cohomology and homology of the algebra Λ = k 〈x, y〉/(x ², xy + qyx, y ²) with coefficients in ₁ Λ_ψ for every degree preserving k-algebra automorphism ψ : Λ → Λ. As a result we obtain several interesting examples of the homological behavior of Λ as a bimodule.     

On the maximum cocliques of the rank 3 graph of 211 : M24

In this article, we consider the maximum cocliques of the 2¹¹: M₂₄ ‐graph Λ. We show that the maximum cocliques of size 24 of Λ can be obtained from two Hadamard matrices of size 24, and that there are exactly two maximum cocliques up to equivalence. We verify that the two nonisomorphic designs with parameters 5‐(24,9,6) can be constructed from the maximum cocliques of Λ, and that these designs are isomorphic to the support designs of minimum weights of the ternary extended quadratic residue and Pless symmetry [24,12,9] codes. Further, we give a new construction of Λ from these 5‐(24,9,6) designs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 323–332, 2009     

On Semigenerically Tame Algebras Over Perfect Fields

Given a semigenerically tame finite-dimensional algebra Λ over a (possibly finite) perfect field, we give, for each natural number d, parametrizations of the indecomposable Λ-modules with central endolength bounded by d, modulo finite scalar extensions, over polynomial algebras.     

ON ORDERED COMPLETELY SIMPLE SEMIGROUPS

We consider an ordered completely simple semigroup S = M(G; I, Λ; P) which is non-degenerate in the sense that G, I, Λ are non-trivial. We prove that if every element of S has a biggest inverse then S contains at least one of seven particular types of subsemigroup.