Quiver Algebras Research Articles

Quantisation via Branes and Minimal Resolution

The ‘brane quantisation’ is a quantisation procedure developed by Gukov and Witten (Adv Theor Math Phys 13(5):1445–1518, 2009). We implement this idea by combining it with the tilting theory and the minimal resolutions. This way, we can realistically compute the deformation quantisation on the space of observables acting on the Hilbert space. We apply this procedure to certain quantisation problems in the context of generalised Kähler structure on P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Our approach differs from and complements that of Bischoff and Gualtieri (Commun Math Phys 391(2):357–400, 2022). We also benefitted from an important technical tool: a combinatorial criterion for the Maurer–Cartan equation, developed by Barmeier and Wang (Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001).

Leavitt path algebras of quantum quivers

Leavitt path algebras of quantum quivers

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

On clusters and exceptional sets

In this paper, we first study clusters in type [Formula: see text] by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type [Formula: see text]. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type [Formula: see text] quivers with straight orientation and using this to count the number of families of exceptional sets of type [Formula: see text] with straight orientation.

One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has aradical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremumbeing finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2 (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case.

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

Hochschild homology dimension and a class of Hochschild extension algebras of truncated quiver algebras

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite.

Non-perturbative Symmetries of Little Strings and Affine Quiver Algebras

We consider Little String Theories (LSTs) that are engineered by N parallel M5-branes probing a transverse ℤM geometry. By exploiting a dual description in terms of F-theory compactified on a toric Calabi-Yau threefold XN,M, we establish numerous symmetries that leave the BPS partition function \U0001d4b5N,M invariant. They furthemore act in a non-perturbative fashion from the point of view of the low energy quiver gauge theory associated with the LST. We present different group theoretical organisations of these symmetries, thereby generalising the results of [1] to the case of generic M ≥ 1. We also provide a Mathematica package that allows to represent them in terms of matrices that act linearly on the Kähler parameters of XN,M. From the perspective of dual realisations of the LSTs the symmetries found here act in highly nontrivial ways: as an example, we consider a formulation of \U0001d4b5N,M in terms of correlation functions of a vertex operator algebra, whose commutation relations are governed by an affine quiver algebra. We show the impact of the symmetry transformations on the latter and discuss invariance of \U0001d4b5N,M from this perspective for concrete examples.

Quipu Quivers and Nakayama Algebras with Almost Separate Relations

A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow. In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with almost separate relations. As an application, we include a list of the derived equivalence classes of all Nakayama algebras of length ≤8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le 8$$\\end{document} with almost separate relations.

Generators for K-theoretic Hall algebras of quivers with potential

Generators for K-theoretic Hall algebras of quivers with potential

Curves in the disc, the type $B$ braid group, and a type $B$ zigzag algebra

We construct a finite-dimensional quiver algebra from the non-simply laced type B Dynkin diagram, which we call type B zigzag algebra. This leads to a faithful categorical action of the type B Artin (braid) group {\mathcal A}(B) , acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of {\mathcal A}(B) , viewed as mapping class group of the punctured disc – hence, our exposition can be seen as a type B analogue of Khovanov–Seidel's work.

Exact Borel subalgebras of path algebras of quivers of Dynkin type [formula omitted

Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on {1,…,n}, we compute the quiver and relations for the Ext-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with n vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of König and we show that there is always a regular exact Borel subalgebra containing the idempotents e1,…,en and find a minimal generating set for it. For a quiver Q and a deconcatenation Q=Q1⊔Q2 of Q at a sink or source v, we describe the Ext-algebra of standard modules over KQ, up to an isomorphism of associative algebras, in terms of that over KQ1 and KQ2. Moreover, we determine necessary and sufficient conditions for KQ to admit a regular exact Borel subalgebra, provided that KQ1 and KQ2 do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.

FAITHFULNESS OF THE 2-BRAID GROUP VIA ZIGZAG ALGEBRA IN TYPE B

AbstractWe show that a certain category of bimodules over a finite-dimensional quiver algebra known as a type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier’s conjecture on the faithfulness of the 2-braid groups for type B.

On cohomological and K-theoretical Hall algebras of symmetric quivers

We give a brief review of the cohomological Hall algebra CoHA H and the K-theoretical Hall algebra KHA R associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) R→Hˆσ˜ where Hˆσ˜ is a Zhang twist of the completion of H. Moreover, we establish the equivalence of categories of “locally finite” graded modules H-Modlf≃RQ-Modlf. Examples of locally finite Hˆ-, resp. RQ-modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.

BPS cohomology for rank 2 degree 0 Higgs bundles (and more)

We give a formula comparing the E-series of the moduli stacks of rank 2 degree 0 semistable Higgs bundles in genus g≥2 to intersection E-polynomials of its coarse moduli space. A parallel formula holds in various 2-Calabi–Yau settings, for example for sheaves on K3 surfaces, or preprojective algebras of g-loop quivers. As a consequence we provide evidence for a conjecture of Davison on the BPS cohomology of Higgs bundles, which has implications for non-abelian Hodge theory for stacks. We apply the formula to cohomological χ-independence tests for BPS cohomology of Higgs bundles and K3 surfaces.

Dimer Algebras, Ghor Algebras, and Cyclic Contractions

A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}-modules of maximal dimension and give an explicit description of the center of Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.

Frobenius-Perron theory of the bound quiver algebras containing loops

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example.

On the number of tilting modules over a class of Auslander algebras

Let [Formula: see text] be a radical square zero algebra of a Dynkin quiver and let [Formula: see text] be the Auslander algebra of [Formula: see text]. Then the number of tilting right [Formula: see text]-modules is [Formula: see text] if [Formula: see text] is of [Formula: see text] type for [Formula: see text]. Otherwise, the number of tilting right [Formula: see text]-modules is [Formula: see text] if [Formula: see text] is either of [Formula: see text] type for [Formula: see text] or of [Formula: see text] type for [Formula: see text].

Categorical properties of generalized σ-derivations on modules

Let S be an algebra over a commutative ring R, and let σ be an automorphism on S. In this paper, we investigate the notion of generalized σ-derivations on modules, which is an extension of generalized derivations on modules introduced by Nakajima. Namely, we study homological properties of generalized σ-derivations. Also, we equip the category of functors that send S/R-modules to R-modules of generalized σ-derivations with a tensor product. We show this category is semi-monoidal. As an application, we characterize when a generalized derivation on a path algebra of an acyclic quiver is a generalized inner derivation.

Tame hereditary path algebras and amenability

AbstractIn this note we revisit the notion of amenable representation type introduced by Gábor Elek. We show that tame hereditary path algebras of quivers of extended Dynkin type over any field are of amenable type. This verifies a conjecture of Elek, which draws similarities to the tame‐wild dichotomy, for another class of tame algebras. We also show that path algebras of wild acyclic quivers over finite fields are not amenable.