Tree Algebras Research Articles

On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra A is a 2-cluster category of Dynkin type A. Here we show that, if A has an admissible action by the group G with two elements, then the stable Cohen-Macaulay category of the skew group algebra AG is a 2-cluster category of Dynkin type D. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible G-action on the path algebra of type A the resulting skew group algebra is of type D. Moreover, we provide a geometric model of the syzygy category of AG in terms of a punctured polygon P with a checkerboard pattern in its interior, such that the 2-arcs in P correspond to indecomposable syzygies in AG and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type A and D respectively. We also provide examples of types E6,E7, and E8.

Broué's abelian defect group conjecture for blocks with cyclic hyperfocal subgroups

AbstractIn this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

A Quantization of the Loday-Ronco Hopf Algebra

We propose a quantization algebra of the Loday-Ronco Hopf algebra k[Y∞]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k[Y^\\infty ]$$\\end{document}, based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra k[Y∞]h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k[Y^\\infty ]_h$$\\end{document} is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion ATopRech\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}^h_{\ ext {TopRec}}$$\\end{document} is a subalgebra of a quotient algebra ARegh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_{\ ext {Reg}}^h$$\\end{document} obtained from k[Y∞]h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k[Y^\\infty ]_h$$\\end{document} that nevertheless doesn’t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of ATopRech\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}^h_{\ ext {TopRec}}$$\\end{document} in low degree.

The Power-Set Construction for Tree Algebras

We study power-set operations on classes of trees and tree algebras. Our main result consists of a distributive law between the tree monad and the upwards-closed power-set monad, in the case where all trees are assumed to be linear. For non-linear ones, we prove that such a distributive law does not exist.

The Ext-algebra of the Brauer tree algebra associated to a line

The Ext-algebra of the Brauer tree algebra associated to a line

A Geometric Model for Syzygies Over 2-Calabi–Yau Tilted Algebras II

Abstract In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $B$, we construct a polygon $\mathcal {S}$ with a checkerboard pattern in its interior, which defines a category $\text {Diag}(\mathcal {S})$. The indecomposable objects of $\text {Diag}(\mathcal {S})$ are the 2-diagonals in $\mathcal {S}$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $\text {Diag}(\mathcal {S})$ is equivalent to the stable syzygy category of the algebra $B$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $\mathbb {A}$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $\mathcal {S}$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver.

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

On the nilpotency index of the radical of a module category

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of QA are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type An, Dn and E6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras.

Behavioural theory of reflective algorithms I: Reflective sequential algorithms

We develop a behavioural theory of reflective sequential algorithms (RSAs), i.e. sequential algorithms that can modify their own behaviour. The theory comprises a set of language-independent postulates defining the class of RSAs, an abstract machine model, and the proof that all RSAs are captured by this machine model. As in Gurevich's behavioural theory for sequential algorithms RSAs are sequential-time, bounded parallel algorithms, where the bound depends on the algorithm only and not on the input. Different from the class of sequential algorithms every state of an RSA includes a representation of the algorithm in that state, thus enabling linguistic reflection. Bounded exploration is preserved using terms as values. The model of reflective sequential abstract state machines (rsASMs) extends sequential ASMs using extended states that include an updatable representation of the main ASM rule to be executed by the machine in that state. Updates to the representation of ASM signatures and rules are realised by means of a sophisticated tree algebra.

Affine Completeness of Some Free Binary Algebras

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial function. We show that the algebra of (possibly empty) binary trees whose leaves are labeled by letters of an alphabet containing at least one letter, and the free monoid on an alphabet containing at least two letters are affine complete.

The Singularity and Cosingularity Categories of C∗BG for Groups with Cyclic Sylow p-subgroups

We construct a differential graded algebra (DGA) modelling certain A_{infty } algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H_{*}{Omega } BG{^{^{wedge }}_p}. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.

Blocks of profinite groups with cyclic defect group

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group Z p $\mathbb {Z}_{p}$ is of star type.

The algebra of binary trees is affine complete

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial.

The differential graded stable category of a self-injective algebra

Let A be a finite-dimensional, self-injective algebra, graded in non-positive degree. We define A -dgstab, the differential graded stable category of A , to be the Verdier quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab / Ω ( 1 ) , reducing computations in the dg-stable category to those in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.

Affine completeness of the algebra of full binary trees

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of full binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative 1–affine complete algebra. As far as we know, it is the first example of such an algebra.

Mutations and Pointing for Brauer Tree Algebras

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the classical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices.In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)n−1, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.

Simplicial complexes and tilting theory for Brauer tree algebras

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f-vector of the simplicial complexes of Brauer tree algebras only depends on the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support τ-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.

Hopf algebras of planar binary trees: an operated algebra approach

Parallel to operated algebras built on top of planar rooted trees via the grafting operator $B^+$, we introduce and study $\vee$-algebras and more generally $\vee_\Omega$-algebras based on planar binary trees. Involving an analogy of the Hochschild 1-cocycle condition, cocycle $\vee_\Omega$-bialgebras (resp.~$\vee_\Omega$-Hopf algebras) are also introduced and their free objects are constructed via decorated planar binary trees. As a special case, the well-known Loday-Ronco Hopf algebra $H_{LR}$ is a free cocycle $\vee$-Hopf algebra. By means of admissible cuts, a combinatorial description of the coproduct $\Delta_{LR(\Omega)}$ on decorated planar binary trees is given, as in the Connes-Kreimer Hopf algebra by admissible cuts.