Tubular Algebras Research Articles

On τ-tilting finite simply connected algebras

We show that $\tau$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $\tau$-tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of $\tau$-tilting finite sincere simply connected algebras.

Decidability of theories of modules over tubular algebras

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra (over a recursively given field) is tame if and only its common theory of modules is decidable. Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. These are the first examples of non-domestic algebras which have been shown to have decidable theory of modules.

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.

Computational Classification of Tubular Algebras

The effective method (based on Theorem 5.3) of classifying tubular algebras by the Cartan matrices of tilting sheaves over weighted projective lines with all indecomposable direct summands in some finite “fundamental domain” , by the reduction to the two elementary problems of discrete mathematics having algorithmic solutions is presented in details (see Problem A and B). The software package CART_TUB being an implementation of this method yields the precise classification of all up to isomorphism tubular algebras of a fixed tubular type p, by creating the complete lists of their Cartan matrices, and furnish their tilting realizations. In particular, the number of isomorphism classes of tubular algebras of the type p is determined (Theorem 2.3).

Dimension Vectors of Indecomposable Objects for Nilpotent Operators of Degree 6 with One Invariant Subspace

Formulas for the dimension vectors of all objects M in the category mathcal {S}(tilde {6}) of nilpotent operators with nilpotency degree bounded by 6, acting on finite dimensional vector spaces with invariant subspaces in a graded sense, are given (Theorem 2.3). For this purpose we realize a tubular algebra Λ, controlling the category mathcal {S}(tilde {6}), as an endomorphism algebra of a suitable tilting bundle over a weighted projective line of type (2,3,6) (Theorem 3.6). Using this description and a concept of mono-epi type, the interval multiplicity vector of an object in mathcal {S}(tilde {6}) is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector.

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

Modules with irrational slope over tubular algebras

Let A be a tubular algebra and let r be a positive irrational. Let D r be the definable subcategory of A-modules of slope r. Then the width of the lattice of pp formulas for D r is ∞. It follows that if A is countable, then there is a superdecomposable pure-injective module of slope r.

Description of the derived categories of tubular algebras in terms of dimension vectors

In this paper, we give a description of the derived category of a tubular algebra by calculating the dimension vectors of the objects in it.

A realization of elliptic Lie algebras of type [formula omitted] by the Ringel–Hall approach

This article is to study relations between an elliptic Lie algebra g of type F 4 ( 2 , 2 ) and the F-fixed point algebra A F of a tubular algebra A of type T ( 3 , 3 , 3 ) under a Frobenius morphism F. Using the explicit structure of the root category of the F-fixed point algebra A F , we prove that the elliptic Lie algebra g of type F 4 ( 2 , 2 ) is isomorphic to the Ringel–Hall Lie algebra of the root category of the F-fixed point algebra A F .

Representation Type Of One Point Extensions Of Iterated Tubular Algebras

The purpose of this work is to show that if \Gamma a strongly simply connected semi-regular iterated tubular algebra and M is an indecomposable \Gamma-module then \Gamma[M] is tame if and only if q_{\Gamma[M]} is weakly non negative.

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.

Semicanonical bases and preprojective algebras

We study the multiplicative properties of the dual of Lusztig's semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztig's nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ″ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components Z ′ and Z ″ is again an irreducible component. It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type A n with n ⩽ 4 . Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type A 5 . We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type ( 6 , 3 , 2 ) and by the corresponding elliptic root system of type E 8 ( 1 , 1 ) .

Elliptic Lie algebras and tubular algebras

This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito–Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel–Ringel, we prove that the elliptic Lie algebra of type D 4 ( 1 , 1 ) , E 6 ( 1 , 1 ) , E 7 ( 1 , 1 ) or E 8 ( 1 , 1 ) is isomorphic to the Ringel–Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel–Malkin–Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac–Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.

Indecomposable parabolic bundles

We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz’s middle convolution operation.

Cartan matrices of selfinjective algebras of tubular type

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras \( \hat B\)/G of the repetitive algebras \( \hat B\) of tubular algebras B with respect to the actions of admissible groups G of automorphisms of \( \hat B\). The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.

On the trivial extensions of tubular algebras

On the trivial extensions of tubular algebras

One-point extensions and derived equivalence

Work of the first author with de la Peña [M. Barot, J.A. de la Peña, Proc. Amer. Math. Soc. 127 (1999) 647–655], concerned with the class of algebras derived equivalent to a tubular algebra, raised the question whether a derived equivalence between two algebras can be extended to one-point extensions. The present paper yields a positive answer.

Varieties of modules over tubular algebras

We classify the irreducible components of varieties of modules over tubular algebras. Our results are stated in terms of root combinatorics. They can be applied to understand the varieties of modules over the preprojective algebras of Dynkin type ${\mathb

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety modA(d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in modA(d) is nonsingular if and only if ExtA2(M,M)=0.

Derived-tame Tree Algebras

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form χA is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of χA. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.