Science AI Journal
Mathematics · Research topic

Open research questions in Algebraic structures and combinatorial models

84 unresolved questions extracted from the limitations and future-work sections of 285 Algebraic structures and combinatorial models papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • A technical challenge is the complexity of the Auslander-Reiten quiver and exceptional tubes. Another challenge is the need to leverage prior work on Brauer graph algebras and representation theory.

    Universal deformation rings of a special class of modules over generalized Brauer tree algebras · 2026 · DOI

  • The paper suggests that future research could focus on generalizing the results to higher rank cluster algebras. The paper suggests that future research could explore the applications of the quantum-weighted bijection in algebraic geometry and representation theory.

    Broken lines and compatible pairs for rank 2 quantum cluster algebras · 2026 · DOI

  • The lack of a combinatorial proof of the coincidence of the greedy basis and the theta basis in rank 2. The need for a quantum-weighted bijection between compatible pairs and broken lines.

    Broken lines and compatible pairs for rank 2 quantum cluster algebras · 2026 · DOI

  • Future research directions include the study of other types of association schemes. The application of the results to the design of experiments and the analysis of data.

    On the Terwilliger Algebras of Quasi-Thin Schurian Association Schemes · 2026 · DOI

  • The lack of a full characterization of triply-transitive quasi-thin association schemes. The need to study the properties of quasi-thin Schurian association schemes and their Terwilliger algebras.

    On the Terwilliger Algebras of Quasi-Thin Schurian Association Schemes · 2026 · DOI

  • To further study the properties of band modules and their connection to string algebras. To explore the implications of the paper's results for the study of representation types of finite-dimensional associative algebras.

    Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words · 2026 · DOI

  • The paper identifies a gap in the understanding of the connection between string algebras and band bricks. The paper identifies a need for a new characterization of band bricks over string algebras.

    Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words · 2026 · DOI

  • There is a lack of understanding of the q-discrete Painlevé equations and their higher-order analogues. There is a need to construct new discrete integrable equations.

    Co-primeness preserving higher dimensional extension of $q$-discrete Painlevé I, II equations · 2026 · DOI

  • The gap is the lack of understanding of the number of full exceptional collections modulo spherical twists. The paper identifies the need for a recursive formula and categorical interpretation to calculate this number.

    The number of full exceptional collections modulo spherical twists for extended Dynkin quivers · 2026 · DOI

  • The obstruction in the simultaneous spectral-correction reading. The lack of a compact and structurally explicit form of the Laurent expansion.

    Structural identities, mode-level reformulation, and a three-term alternative for α−1 in the SO(3,3) matrix model on P3 · 2026 · DOI

  • Further study of color Lie algebras and their applications - Investigation of other types of graded Casimir elements

    Graded Casimir Elements and Central Extensions of Color Lie Algebras · 2026 · DOI

  • The need for a method to construct graded Casimir elements - The need for a understanding of graded central extensions of loop algebras

    Graded Casimir Elements and Central Extensions of Color Lie Algebras · 2026 · DOI

  • Abstract One of the fundamental open problems in the field of tensors is the border Comon’s conjecture : given a symmetric tensor $$F\in (\mathbb {C}^n)^{\otimes d}$$ <mml:math xmlns:mml="http://www.

    Symmetrization maps and minimal border rank Comon’s conjecture · 2026 · DOI

  • Further study of the properties of distributive Ockham algebras is needed. The results of the paper can be used to study the properties of other algebraic systems motivated by non-classical propositional logics.

    Topological duality for distributive Ockham algebras · 1983 · DOI

  • The study of distributive Ockham algebras is an ongoing area of research. There is a need for a new approach to the study of distributive Ockham algebras, using a canonical representation by homming into a maximal subdirectly irreducible algebra.

    Topological duality for distributive Ockham algebras · 1983 · DOI

  • Lemma 6.8 establishes isomorphisms ΨG(V) ≅ OΩ ⊗K V and ΨH(W) ≅ DDr(OPn−1 ⊗K W) for representations with trivial Gm action, but does not address the case where Gm acts non-trivially or provide a unified treatment of the relationship between Lubin-Tate bundles and Drinfeld bundles for higher-level representations.

    THE CATEGORIES OF LUBIN-TATE AND DRINFELD BUNDLES · 2026 · DOI

  • The functors ΨG and ΨH are defined as direct limits over levels m via Lm ⊗K −, but the paper does not characterize the image of these functors or determine which Lubin-Tate bundles in VectConG_LT(Ω) arise from smooth representations in Repsm_K(G) and Repsm_K(H).

    THE CATEGORIES OF LUBIN-TATE AND DRINFELD BUNDLES · 2026 · DOI

  • Future research could involve the study of braided Z2-crossed tensor categories for odd groups. Future research could involve the application of the construction to other areas of physics.

    Modular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> -crossed Tambara-Yamagami-like categories for even groups · 2026 · DOI

  • The lack of a general method for constructing braided Z2-crossed tensor categories for even groups. The need for a physically motivated approach to construct these categories.

    Modular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> -crossed Tambara-Yamagami-like categories for even groups · 2026 · DOI

  • There is a gap in the theory of homomorphisms and cv-polynomials between double Ore extensions. The absence of inclusions between the classes of all double Ore extensions of an algebra and of all two-step iterated Ore extensions of the same algebra is a problem.

    A View Toward Homomorphisms and CV-Polynomials Between Double Ore Extensions · 2026 · DOI

  • As we said above, it is a pending task to exemplify dcv-matrices consisting of polynomials of degree two. On the other hand, skew PBW extensions were introduced by Gallego and Lezama [16] with the aim of generalizing Ore extensions of injective type and PBW extensions defined by Bell and Goodearl [3]. Over the years, ring-theoretic, geo- metric and homological properties of the objects have been studied, and it has shown that these extensions generalize several families of noncommutative algebras (see [14, Chapter 2] and [35, Section 2] for more details). Suárez [44] defined a subclass of these extensions, the graded skew PBW extensions, and then, by using [7], in [18] he presented necessary the comparison carried out by Carvalho et al. and sufficient conditions for a graded trimmed double Ore extension to be a graded skew PBW extension, and in fact, they proved the Artin-Schelter regularity and the property of being skew Calabi-Yau for graded skew PBW extensions. Having in mind these works and the theory established in this paper, a question arises whether it is possible to develop the theory of homomorphisms and cv-polynomials for graded skew PBW extensions, and if so, compare it with the results obtained here and those corresponding in [7, 18, 20, 39, 45, 46, 47]. Last but not least, Lü et al. [32] proved that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra, and then showed that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension [31]. Related with this, Lou et al. [30] gave a definition of Poisson double extension - which may be considered as an analogue of double Ore extension -, and showed that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions. These works together Zambrano’s paper [49] where he gave a description of Poisson brackets on some families of skew PBW extensions, motivate us to ask ourselves about the relations between all these algebras through the notion of homomorphism and cv-polynomial developed here.

    A View Toward Homomorphisms and CV-Polynomials Between Double Ore Extensions · 2026 · DOI

  • To apply the non-unitary version of the Bisognano-Wichmann property to the study of non-unitary conformal field theories. To extend the results to more general classes of conformal field theories.

    The Bisognano-Wichmann Property for Non-unitary Wightman Conformal Field Theories · 2026 · DOI

  • The Bisognano-Wichmann property has not been established for non-unitary Wightman conformal field theories. The existing results are limited to unitary theories.

    The Bisognano-Wichmann Property for Non-unitary Wightman Conformal Field Theories · 2026 · DOI

  • The current understanding of the structural relations between coefficients is limited. The Laurent expansion is not in a compact and structurally explicit form.

    Structural identities, mode-level reformulation, and a three-term alternative for α−1 in the SO(3,3) matrix model on P3 · 2026 · DOI

  • The paper does not study all possible Argyres-Douglas theories. The method is limited to cases where the 3d TFT has a finite number of half-BPS monopole operators.

    3D TFTs and boundary VOAs from BPS spectra of (G, G′) Argyres-Douglas theories · 2026 · DOI

Most-cited papers in Algebraic structures and combinatorial models

Most recent work

Find a gap in your own Algebraic structures and combinatorial models sub-topic

This page shows what the Algebraic structures and combinatorial models literature already flags as unresolved. To narrow it to your specific question, run the guided finder — it searches the gap library on demand and checks candidates against 250M+ OpenAlex works.

Open the Research Gap Finder →

Related topics in Mathematics

Cognitive and developmental aspects of mathematical skills
398 open questions
Mathematics Education and Pedagogy
319 open questions
Statistics Education and Methodologies
146 open questions
Geometric Analysis and Curvature Flows
143 open questions
COVID-19 epidemiological studies
142 open questions
Advanced Causal Inference Techniques
140 open questions
Analytic Number Theory Research
118 open questions
Numerical methods for differential equations
113 open questions
84 open questions have been extracted from the limitations and future-work passages of 285 Algebraic structures and combinatorial models papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

Tools for your next paper

Gaps & Literature SearchFind open questions in any field — 120,000+ open gaps mined from our own library.Pre-CheckIs your paper ready? A Tier 1–5 acceptance probability in seconds.Journal FinderA ranked shortlist of target journals from a 17,500-venue index.Duplicate Publication CheckerCheck prior-publication & salami-slicing risk across 8 sources.Citation GeneratorFreeA DOI or title → APA, MLA, Vancouver, Chicago, BibTeX, RIS. No signup.Graphical Abstract MakerFreeTurn your findings into a colorblind-safe graphical abstract.

Compare the category — Honest roundups of the AI research tools, ours listed alongside the alternatives.

Command palette

Jump anywhere, run any action.