Open research questions in Advanced Topics in Algebra
67 unresolved questions extracted from the limitations and future-work sections of 396 Advanced Topics in Algebra papers in our library. Each links back to the study that raised it.
What the literature leaves open
The paper identifies a gap in the understanding of weak generating sets for classical W-algebras. The paper aims to fill this gap by finding weak generating sets for classical W-algebras.
The paper suggests further study of Hom-M-dendriform algebras and their connections to other algebraic structures. The paper proposes the investigation of Hom-M-dendriform algebras in the context of Hom-pre-Malcev algebras.
The paper identifies a gap in the prior work on the commutativity of the matrices Lk and Lk. The paper discusses the need for a correct proof of the commutativity of the matrices Lk and Lk.
The classification of solvable Lie algebras is not yet complete. The problem of classifying algebraic objects is a central topic in algebra.
The paper identifies a gap in the study of Hom-Malcev algebras. The paper recognizes the need for a dendriform version of Hom-Malcev algebras.
No discussion is provided regarding the computational feasibility of verifying the conditions in Proposition 5.12 for large or complex skew left braces.
One-Generator Skew Braces and Indecomposable Set-Theoretic Solutions to the Yang-Baxter Equation · 2026 · DOINo explicit examples or concrete calculations are provided in the excerpt demonstrating how to apply the isomorphism criterion in Proposition 5.12 to specific pairs of skew left braces.
One-Generator Skew Braces and Indecomposable Set-Theoretic Solutions to the Yang-Baxter Equation · 2026 · DOISection 3.7.1 introduces integro-differential forms Φˡ for m=1 (one odd coordinate) by constructing a K-graded Ω-module with specific relations on generators ˆxˡ. The extension of this construction to superdomains with m>1 odd coordinates and its compatibility with the module structure on Σ(U) for general flat quasi-Frobenius Lie superalgebras is not addressed.
The proof that the canonical volume form v̂U is coordinate-independent (Section 3.6.1) uses the Berezinian of the Jacobian matrix with non-standard block format, but the paper does not discuss how this coordinate-independence generalizes to non-trivial flat quasi-Frobenius Lie superalgebra structures beyond the standard superomain case.
Further study of the algebraic structure of the image of a polynomial on an algebra. Investigation of the implications of the results for Ring Theory and the theory of polynomial identities with involution.
Upper Triangular Matrices with Superinvolution: Identities and Images of Multilinear Polynomials · 2026 · DOIThe problem of studying the algebraic structure of the image of a polynomial on an algebra is important in Ring Theory. Prior work has not addressed the case of upper triangular matrices with superinvolution.
Upper Triangular Matrices with Superinvolution: Identities and Images of Multilinear Polynomials · 2026 · DOIIdentifying sufficient conditions for (R-IUUI). Obtaining a more complete characterization of its solutions.
The need for a deeper understanding of distributive equations for k-generated implications. The lack of necessary and sufficient conditions for these equations.
The lack of a comprehensive theory of quasitriangular and factorizable Poisson bialgebras. The need for a one-to-one correspondence between factorizable Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weight.
The paper identifies a gap in the literature regarding the study of partial actions of Lie algebras on nonassociative algebras. It notes the lack of a comprehensive theory of Lie inverse semialgebras.
Further study of the MS model in the Granik basis is needed to fully understand its implications for particle physics. The application of noncommutative geometry and Hopf algebras to other areas of physics may provide new insights.
The relativistic quantum mechanics of the MS model in the Granik basis has not been fully explored. The implications of the MS model for particle physics are not well understood.
Future research can focus on extending the results to other types of matrices. The study of linear preserver problems on Toeplitz matrices can be generalized to other fields. Further research can explore the applications of the paper's findings in signal processing and other fields.
There is a gap in the literature regarding the characterization of linear preservers on Toeplitz matrices. The paper identifies the need for a characterization of linear preservers of rank one matrices and the determinant on Toeplitz matrices.
The classical Albert-Muckenhoupt-Shoda theorem only applies to full matrix algebras. There is a need to extend this result to more general classes of algebras, such as generalized block-triangular algebras.
The need to generalize algebras with brackets to the conformal setting - The lack of a framework for constructing dendriform-CAWB structures
The gap is the lack of understanding of the core of inner ideals of 4-dimensional real Lie algebras with 2-dimensional derived algebras. The gap is the need for a proof that the core of each inner ideal of such a Lie algebra is zero.
The Core of Inner Ideals of Four Dimensional Real Lie Algebra With Tow Dimensional Derived · 2026 · DOIThe understanding of simple commutative algebras in the Delannoy category is limited. Previous results have been limited to interpolation categories.
The paper identifies a gap in the study of the connections between Hopf algebras and Frobenius algebras. The paper identifies a gap in the study of Hopf categories and Hopf opcategories.
6.1. Remaining questions. In this paper, we studied measuring coalgebras and comeasuring algebras between Frobenius algebras. As a main result we showed that the semi-Hopf category (i.e. the coalgebra enriched category) of universal measuring coalgebras between Frobenius algebras admits an invertible antipode. We discussed duality between measuring and comeasurings and computed several explicit examples. Based on these results and examples, we formulate a few remaining questions that require further investigations. (1) Recall that if ℓ is a field extension of k, then a k-algebra A is called a form of an ℓ-algebra R, if and only if R ∼= ℓ ⊗k A. So far, the only examples we have where CA,B is non-zero for some Frobenius k-algebras A and B, is when A and B are forms of the same ℓ-algebra for some field extension ℓ of k. Therefore, and in view of Proposition 5.6 above, we pose the following question: Is it true that CA,B is nonzero for some some Frobenius k-algebras A and B if and only if A and B are forms of the same ℓ-algebra for some field extension ℓ of k ? For the above to hold, it would be sufficient to see that there exists a base extension ℓ of k, such that ℓ ⊗k CA,B has a grouplike element, since such a grouplike element acts as a morphism of Frobenius algebras, which is therefore an isomorphism. The results we have proven so far (see Propositions 5.7, 5.10 and 5.9), already show that there are strong dimension restrictions for group algebras and matrix algebras in order to have existence of non-zero (co)measurings between them. Since both group algebras and matrix algebras are so-called special Frobenius, meaning that the composition of multiplication with comultiplicaton equals a scalar multiple of the identity, one could wonder whether it is possible to generalize the triviality conditions to this setting. (2) In all examples we computed, the universal measuring coalgebras and comeasuring algebras were finite dimensional. However, there is a priori no reason why this should always be the case. We therefore ask Do there exist Frobenius algebras A and B, whose universal measuring coalgebra CAB is infinite dimensional ? Furthermore, one could also wonder, if there is a formula to compute the dimension of the universal measuring coalgebra. Based on the very small examples we have, one already sees that the dimension of the universal acting Hopf algebra on a Frobenius algebra grows more rapidly than the dimension of the Frobenius algebra itself (the dimension of Ck[C2],k[C2] being 2 and the dimension of Ck[C3],k[C3] being 6, computer algebra computations showed that the dimension of CC[C4],C[C4] is at least 96). (3) One could wonder, if there exist other types of Ω-algebras such that the associated measuring semi-Hopf category is Hopf ? Or even stronger: what are the conditions on Ω and its Ω-algebras for this to be the case ? Based on Section 4.2 we expect that this could be the case for Ω’s that are “self-dual”. 6.2.
Most-cited papers in Advanced Topics in Algebra
- Maps preserving n-tuple A*B − B*A derivations on factor von Neumann algebras · Publications de l Institut Mathematique · 2023 · 11 citations
- An Algebraic Study of Tense Operators on Nelson Algebras · Studia Logica · 2020 · 7 citations
- On Sheffer stroke UP-algebras · Discussiones Mathematicae - General Algebra and Applications · 2021 · 7 citations
- On Sheffer stroke BE-algebras · Discussiones Mathematicae - General Algebra and Applications · 2022 · 5 citations
- Hilbert Algebras with Hilbert–Galois Connections · Studia Logica · 2022 · 4 citations
- On Heyting Algebras with Negative Tense Operators · Studia Logica · 2023 · 3 citations
- Filters, ideals and power of double Boolean algebras · Discussiones Mathematicae - General Algebra and Applications · 2024 · 3 citations
- Set-theoretical solutions for the Yang-Baxter equation in triangle algebras · Discussiones Mathematicae - General Algebra and Applications · 2023 · 3 citations
- Transposed Poisson superalgebra · Proceedings of the Estonian Academy of Sciences · 2024 · 3 citations
- A Levi–Civita Equation on Monoids, Two Ways · Annales Mathematicae Silesianae · 2022 · 2 citations
Most recent work
- Geometry of Yang-Baxter Matrix Equations over Finite Fields · Experimental Mathematics · 2026
- Nijenhuis deformations of Poisson algebras and <i>F</i> -manifold algebras · Communications in Algebra · 2026
- Integro-derivation Dzhumadildaev algebras: from the algebra of polynomials · Communications in Mathematics · 2026
- Identities related to left generalized derivations in near-rings · Asian-European Journal of Mathematics · 2026
- One-Generator Skew Braces and Indecomposable Set-Theoretic Solutions to the Yang-Baxter Equation · Results in Mathematics · 2026
- Classification of uniformly bounded simple Lie conformal algebras with upper bound one · Letters in Mathematical Physics · 2026
- Flat quasi-Frobenius Lie superalgebras · European Journal of Mathematics · 2026
- Simple n-Lie Poisson algebras · European Journal of Mathematics · 2026
- The algebraic and geometric classification of transposed δ-Poisson algebras · Journal of Algebra and Its Applications · 2026
- Characterizing Nilpotent Associative Algebras by Their Multiplier · Vietnam Journal of Mathematics · 2026
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