Open research questions in Advanced Topics in Algebra
104 gap statements mined from Advanced Topics in Algebra papers in our 4.5M-paper local library, which holds 918 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
Further study of the properties of integro-derivation Dzhumadildaev algebras is needed. Further study of the derivations of these algebras is needed. Further study of the applications of these algebras is needed.
There is a lack of understanding of infinite-dimensional simple conservative algebras. There is a lack of examples of non-(anti)commutative conservative simple algebras.
Further study of the properties of skew left braces is needed. The construction of new solutions with specific properties is an open problem. The application of the approach to other areas of mathematics and physics is a potential direction for future research.
One-Generator Skew Braces and Indecomposable Set-Theoretic Solutions to the Yang-Baxter Equation · 2026 · DOIA comprehensive description of all solutions of the Yang-Baxter equation remains currently beyond reach. The study of simple set-theoretic solutions is an open problem. The paper identifies a gap in the understanding of the simplicity of set-theoretic solutions.
One-Generator Skew Braces and Indecomposable Set-Theoretic Solutions to the Yang-Baxter Equation · 2026 · DOIThe study is limited to the analysis of flat quasi-Frobenius Lie superalgebras. The use of supersymmetric methods may not be applicable to all types of Lie superalgebras.
The study of Lie superalgebras is an important area of research in mathematics and physics. Prior work has focused on the algebraic properties of these structures, leaving a gap in the understanding of their geometric properties.
The 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 · DOIThe lack of study on transposed Poisson conformal superalgebras. The need to derive a family of identities forced by the transposed conformal super-Leibniz rule. The need to determine all compatible transposed Poisson conformal superalgebra structures on Lie conformal superalgebras of rank (1+1).
On transposed Poisson conformal superalgebras · 2026The gap in prior work is the lack of a complete classification of 2-generated primitive axial algebras of Monster type. The paper identifies the need to deal with certain parameters, subalgebras, axets, and axial dimensions.
The Classification of the 2-generated Primitive Axial Algebras of Monster Type · 2026The 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.
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.
There is a need to extend this result to more general classes of algebras, such as generalized block-triangular algebras.
The lack of a general description of the algebra of invariants for the dihedral group. The need for a computation of the Hilbert series for specific cases.
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 lack of a unified notion of Lie superalgebra over a field of characteristic 2. The need for a study on mixed characteristic deformation theory of Lie superalgebras.
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 paper identifies a gap in the understanding of the structural properties of involutive Yang-Baxter groups. The study aims to contribute to the theory of quantum groups.
Identifying sufficient conditions for (R-IUUI). Obtaining a more complete characterization of its solutions.
The need to generalize algebras with brackets to the conformal setting - The lack of a framework for constructing dendriform-CAWB…
Further study of the applications of Lie-Yamaguti algebras and their extensions. Investigation of the relationships between Lie-Yamaguti algebras and other areas of mathematics.
Most-cited papers in Advanced Topics in Algebra
- States on Pseudo MV-Algebras · Studia Logica · 2001 · 97 citations
- Perfect MV-algebras are categorically equivalent to abelianl-groups · Studia Logica · 1994 · 87 citations
- Simple derivation of general Fierz-type identities · American Journal of Physics · 2005 · 69 citations
- Varieties of Monadic Heyting Algebras. Part I · Studia Logica · 1998 · 60 citations
- Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory · Communications in Mathematical Physics · 2024 · 47 citations
- Free Algebras in Varieties of Glivenko MTL-algebras Satisfying the Equation 2(x2) = (2x)2 · Studia Logica · 2006 · 43 citations
- Pretabular varieties of modal algebras · Studia Logica · 1980 · 43 citations
- Quantum MV algebras · Studia Logica · 1996 · 35 citations
- Dynamic algebras: Examples, constructions, applications · Studia Logica · 1991 · 32 citations
- Equality Algebras · Studia Logica · 2012 · 31 citations
Most recent work
- Generic graded contractions of Lie algebras · Journal of Algebra · 2026
- Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra · Journal of Algebra · 2026
- Geometry of Yang-Baxter Matrix Equations over Finite Fields · Experimental Mathematics · 2026
- Nijenhuis deformations of Poisson algebras and F -manifold algebras · Communications in Algebra · 2026
- Classification of simple commutative algebras in the Delannoy category · Mathematische Zeitschrift · 2026
- On the renormalization and quantization of topological–holomorphic field theories · Advances in Mathematics · 2026
- Lie algebraic invariants in quantum linear optics · Quantum · 2026
- On commuting integer matrices · Transactions of the American Mathematical Society · 2026
- Representations of Hamiltonian Lie algebras · Journal of Algebra · 2026
- Quantum superalgebras and the free-fermionic Yang-Baxter equation · Journal of Algebra · 2026
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