Let $R=\oplus_{Γ\inΓ}R_γ$ be a $Γ$-graded $K$-algebra over a field $K$, where $Γ$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $Γ$-grading filtration $FR$ of $R$ and the $Γ$-filtration $FA$ induced by $FR$ for the quotient $K$-algebra $A=R/I$, we show that there is a $Γ$-graded $K$-algebra isomorphism $G(A)\cong \bar{A}=R/<\hbox{\bf HT} (I)>$, where $G(A)$ is the associated $Γ$-graded $K$-algebra of $A$ defined by $FA$, and $<\hbox{\bf HT}(I)>$ is the $Γ$-graded ideal of $R$ generated by the set of head terms of $I$. In the case that $Γ$ is an ordered monoid with a well-ordering, this result enables us to lift many nice structural properties of $\bar{A}$ to $A$ theoretically, and the natural connection with Gröbner basis theory leads to effective realization lifting information from the associated monomial algebras in both commutative and noncommutative cases.
A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of nonzero weight on the algebra of polynomials in several variables without constant term.
In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.
We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show this result holds for the class of \emph{automaton algebras}; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz.
Let A be a connected graded noncommutative monomial algebra. We associate to A a finite graph Γ(A) called the CPS graph of A. Finiteness properties of the Yoneda algebra Ext_A(k,k) including Noetherianity, finite GK dimension, and finite generation are characterized in terms of Γ(A). We show these properties, notably finite generation, can be checked by means of a terminating algorithm.
We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.
In recent times, there has been a lot of active research on monomial groups in two different directions. While group theorists are interested in the study of their normal subgroups and Hall subgroups, the interest of group ring theorists lie in the structure of their rational group algebras due to varied applications. The purpose of this paper is to bind the two threads together. Revisiting Dade's celebrated embedding theorem which states that a finite solvable group can be embedded inside some monomial group, it is proved here that the embedding is indeed done inside some generalized strongly…
A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, it is the…
Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent: QGr(A) denotes the quotient category of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules; QGr(kQ) has a similar definition. The proof makes use of an algebra homomorphism A--->kQ that may be of independent interest.
In this paper, we construct a pre-Lie structure on the free Lie algebra L(E) generated by a set E, giving an explicit presentation of L(E) as the quotient of the free pre-Lie algebra generated by E, by some ideal I. The main result in this paper is a description of Gröbner bases in terms of trees.
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.
We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent paper. Our construction goes through a construction of finitely generated just-infinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type. We also discuss the connection between entropy of algebras and their homomorphic images, as well as the deg…
Let $A$ be a monomial associative finite dimensional algebra over a field $\Bbbk$ of characteristic zero. It is well known that the Hochschild cohomology of $A$ can be computed using Bardzell's complex $B(A)$. The aim of this article is to describe an explict $L_\infty$-structure on $B(A)$ that induces a weak equivalence of $L_\infty$-algebras between $B(A)$ and the Hochschild complex $C(A)$ of $A$. This allows us to describe the Maurer-Cartan equation in terms of elements of degree $2$ in $B(A)$. Finally, we make concrete computations when $A$ is a truncated algebra, and we prove that Bardzel…
In one of our former papers {\it Endomorphisms of the measure algebra of commutative hypergroups arXiv:2204.07499 we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.
We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution and Bardzell's resolution; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $HH^*(A)$ and the second one has been shown to be an efficient tool for computations of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A=k Q/I$ a monomial algebra such that $dim_k \ e_i A e_j = 1…
We consider the bar complex of a monomial non-unital associative algebra $A=k \langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes $B_w$, defined for any fixed monomial $w=x_1...x_n \in A$. We give a simple argument, showing that the homology of this subcomplex is at most one-dimensional, and describe the place where the nontrivial homology appears. It has a very simple expression in terms of the length of the generalized Dyck path associated to a given monomial in $w \in A$. The operadic analogue of the question about dichotomy in homology is considered. It is shown that…
This article sets out to understand the categories $\QGr A$ where $A$ is either a monomial algebra or a path algebra of finite Gelfand-Kirillov dimension. The principle questions are: 1) What is the structure of the point modules up to isomorphism in $\QGr A$? 2) When is $\QGr A \equiv \QGr A'$? These two questions turn out to be intimately related. It is shown that up to isomorphism in $\QGr A$, there are only finitely many point modules and these give all the simple objects in the category. Then, a finite quiver $E_A$, which can be constructed from the algebra $A$ rather simply, is associate…
We classify indecomposable non-projective Gorenstein-projective modules over a monomial algebra via the notion of perfect paths. We apply this classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.
The complete algebraic structure of semisimple finite group algebra of a generalized strongly monomial group is provided. This work extends the work of Broche and del R{\'ı}o on strongly monomial groups. The theory is complimented by an algorithm and is illustrated with an example.
Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A = kQ/I. Let QGr A denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph Q' with all its arrows placed in degree 1 and an equivalence of categories QGr A = QGr kQ'. A result of Smith now implies that QGr A =…
We describe the Galois objects and biGalois groups of monomial nonsemisimple Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. These computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.
We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.
We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.
Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous Gröbner basis for the ideal $I=<{\cal G}>$ of $K<X>$ with respect to some monomial ordering $\prec$ on $K<X>$. It is proved that if the monomial algebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, where ${\bf LM}({\cal G})$ is the set of leading monomials of ${\cal G}$ with respect to $\prec$, then the $\mathbb{N}$-graded algebra $A=K<X>/I$ is semiprimitive (in…
We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either…
Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.
We classify $n$-hereditary monomial algebras in three natural contexts: First, we give a classification of the $n$-hereditary truncated path algebras. We show that they are exactly the $n$-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the $n$-hereditary quadratic monomial algebras. In the case $n=2$, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is …
Our published paper contains an incorrect statement of a result due to Artin and Zhang. This corrigendum gives the correct statement of their result and includes a new result that allows us to use their result to prove our main theorem. Thus the main theorem of our published paper is correct as stated but its proof must be modified.
We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Barth…
We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. For a stably hereditary algebra or a gentle one-cycle algebra, these considerations show that the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated if and only if the algebra is piecewise hereditary.
We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a mon…
We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.
In this article we study a class of orders called {\it monomial orders} in a central simple algebra over a non-Archimedean local field. Monomial orders are easily represented and they may be also viewed as a direct generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two.
Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension of $HH^1(TA)$ can be derived.
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline …
Rota-Baxter operators were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. The polynomial algebra $\mathbf{k}[x]$ plays a central role both in analysis and algebra. In this paper, we explicitly classified all monomial Rota-Baxter operators on $\mathbf{k}[x]$.
Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid S=< X_{1}, ..., X_{n} | R >. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac-Moody groups. We prove that all cluster monomials with g-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero-Chapoton description via quiver representations. In type A_1^{(1)}, we further show that elements of several canonical bases (generic, triangular, and…
The 3-dimensional Sklyanin algebras, S(a,b,c), over a field k, form a flat family parametrized by points (a,b,c) lying in P^2-D, the complement of a set D of 12 points in the projective plane, P^2. When (a,b,c) is in D the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are said to be "degenerate". Chelsea Walton showed the degenerate 3-dimensional Sklyanin algebras do not have the same properties as the non-degenerate ones. Here we prove that a degenerate Sklyanin algebra is isomorphic to the free algebra on u,v,w, modulo either the relations u^2=v^2=w^2=0 o…
We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to deduce some finiteness results about leading monomial ideals of any fixed ideal, namely: (1) the number of minimal leading monomial ideals with respect to total orderings is finite; (2) the number of leading monomial ideals with respect to degree orderings is finite; (3) the number of leading monomial ideals with respect to admissible orderings is finite under …
We prove that the basis of cluster monomials of a skew-symmetric cluster algebra A of finite type is the atomic basis of A. This means that an element of A is positive if and only if it has a non-negative expansion in the basis of cluster monomials. In particular cluster monomials are positive indecomposable, i.e. they cannot be written as a sum of positive elements.
Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincaré-Birkhoff-Witt type basis of $A$ in terms of its \emph{Lyndon atoms} $N$, but, in general, $N$ may be infinite. We prove that if $A$ has polynomial growth of degree $d$ then $A$ has global dimension $d$ and is standard finitely presented, with $d-1 \leq |W| \leq d(d-1)/2$. Furthermore, $A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\…
In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli spaces of objects. On the other hand, categories of higher homological dimension remain more mysterious from this point of view, with few general methods. In this paper, we will see how some of the methods used to study quivers can be extended to work for representations of any (noncommutative) monomial algebra with relations of length two. In particular, we…
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras) from the perspective of Laurent phenomenon. In this general framework, we firstly prove that each positive and d-vector-positive GLP algebra has the proper Laurent monomial property and thus its cluster monomials are linearly independent. Skew-symmetric cluster algebras are verified to be d-vector-positive, which gives the affirmation of Conjecture [conjd] …
We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $…
