Tree Algebras Research Articles

Spherical objects of the multiplicity free Brauer tree algebra with two edges

We explicitly describe all spherical objects, as well as tilting objects, of the multiplicity free Brauer tree algebra with two edges.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

FOLDINGS AND TWO-SIDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS

In this note, for a Brauer tree algebra $A$ and a star-shaped Brauer tree algebra $B$ which is derived equivalent to $A$, we give operations on the two-sided tilting complex $D_T$ of $A\otimes B^{op}$-modules constructed in [3] which is isomorphic to the Rickard tree-to-star complex $T$ constructed in [5] in $D^b(A)$, and we show that the operations on $D_T$ correspond to operations called $foldings$ on the Rickard tree-to-star complex $T$ given in [7].

Two-sided tilting complexes for Brauer tree algebras

In this note, we construct two-sided tilting complexes corresponding to one-sided tilting complexes for Brauer tree algebras.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ,V) and we describe R(Λ,V) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.

Some remarks on the Lehmer conjecture

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

On Representation-Finite Gendo-Symmetric Biserial Algebras

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

Representation type for block algebras of Hecke algebras of classical type

We find representation type of the cyclotomic quiver Hecke algebras RΛ(β) of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type in good characteristic. We conjecture that this statement should hold for Hecke algebras of exceptional type.

Polynomial functors and combinatorial Dyson–Schwinger equations

We present a general abstract framework for combinatorial Dyson–Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson–Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild 1-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structures. Precisely, for any finitary polynomial endofunctor P defined over groupoids, the system of combinatorial Dyson–Schwinger equations X = 1 + P(X) has a universal solution, namely the groupoid of P-trees. The isoclasses of P-trees generate naturally a Connes–Kreimer-like bialgebra, in which the abstract Dyson–Schwinger equation can be internalised in terms of canonical B+-operators. The solution to this equation is a series (the Green function), which always enjoys a Faà di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Faà di Bruno bialgebra. Varying P yields different bialgebras, and cartesian natural transformations between various P yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson–Schwinger equations. Finally, all constructions can be pushed inside the classical Connes–Kreimer Hopf algebra of trees by the operation of taking core of P-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-Löf type theory (expounded elsewhere).

Derived Picard groups of selfinjective Nakayama algebras

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

Spectra of Tukey types of ultrafilters on Boolean algebras

Extending recent investigations on the structure of Tukey types of ultrafilters on \({\mathcal{P}(\omega)}\) to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.

Hamiltonian B-series and a Lie algebra of non-rooted trees

The conditions for a B-series to be a Hamiltonian vector field imply that it may be given as a series indexed by free trees, i.e. trees without root. At present, the pre-Lie structure of rooted trees plays an important role in the study of numerical methods for ordinary differential equations, as does the associated Lie bracket on rooted trees obtained by antisymmetrization. We give a substitute for this Lie bracket defined on free trees that reflects the Lie bracket of Hamiltonian B-series, and illustrate an application of this to the backward error analysis of symplectic numerical integrators.

Hopf algebras and topological recursion

We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293–309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347–452).

Two-Term Tilting Complexes over Brauer Tree Algebras

In this paper, all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described, using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.

Offer semantics: Achieving compositionality, flattening and full expressiveness for the glue operators in BIP

Based on a concise but comprehensive overview of some fundamental properties required from component-based frameworks, namely compositionality, incrementality, flattening, modularity and expressiveness, we review three modifications of the semantics of glue operators in the Behaviour–Interaction–Priority (BIP) framework. We provide theoretical results and examples illustrating the degree to which the three semantics meet these requirements. In particular, we show that the most recent semantics, based on the offer predicate is the only one that satisfies all of them.The classical and offer semantics are not comparable: there are systems that can be assembled in the classical semantics, but not in the offer one. We present a strict characterisation of the behaviour hierarchy determining the conditions, under which systems in the classical semantics can be transposed into the offer semantics: directly, with minor modifications, by introducing a new type of synchronisation or cannot be transposed at all.The offer semantics allows us to extend the algebras, which are used to model glue operators in BIP, to encompass priorities. This extension uses the Algebra of Causal Interaction Trees, T(P), as a pivot: existing transformations automatically provide the extensions for the Algebra of Connectors. We then extend the axiomatisation of T(P), since the equivalence induced by the new operational semantics is weaker than that induced by the interaction semantics. This extension leads to canonical normal forms for all structures and to a simplification of the algorithm for the synthesis of connectors from Boolean coordination constraints.

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant {\kappa}(A) for A and count the number of isomorphism classes of Leavitt path algebras with {\kappa}(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.

Two-Term Partial Tilting Complexes Over Brauer Tree Algebras

In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are computed.

Tree algebra of sofic tree languages

We consider the languages of finite trees called tree-shift languages which are factorial extensible tree languages. These languages are sets of factors of sub-shifts of infinite trees. We give effective syntactic characterizations of two classes of regular tree-shift languages: the finite type tree languages and the tree languages which are almost of finite type. Each class corresponds to a class of subshifts of trees which is invariant by conjugacy. For this goal, we define a tree algebra which is finer than the classical syntactic tree algebra based on contexts. This allows us to capture the notion of constant tree which is essential in the framework of tree-shift languages.

Multisorted Tree Algebra

This paper introduces basic concepts describing a hierarchical algebraic structure called multisorted tree algebra. This structure is constructed by placing multisorted algebra at the bottom of a hierarchy and placing at other intermediate nodes the aggregation of algebras placed at their immediate subordinate nodes. These constructions are different from the one of subalgebras, homomorphic images and product algebras used to characterize varieties in universal algebra theory. The resulting hierarchical algebraic structures cannot be easily classified in common universal algebra varieties. The aggregation method and the fundamental properties of the aggregated algebras have been presented with an illustrative example. Multisorted tree algebras spans multisorted algebra concepts and can be used as modelling framework for building hierarchical abstract data types for information processing in organizations.