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This paper contributes to the proof of the conjecture posed in arXiv:1606.02521, stating that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming…
Suppose G is a finite group, such that |G| = 27p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G;S).
Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…
Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other…
The Cayley-Dickson loop Q_n is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers,…
We show how to construct an explicit Hamilton cycle in the directed Cayley graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >... k). The existence of such cycles was shown by Jackson (Discrete Mathematics, 149 (1996)…
We prove functional equations for multiple Dirichlet series defined by a collection of five geometric axioms. We find functional equations of two types: one modeled on the functional equations of Dirichlet $L$-functions, and another modeled…
Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems…
For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we extend some well-known theorems of the Hamiltonicity of…
As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…
In this paper, we initiate the study of nondiagonal finite quasi-quantum groups over finite abelian groups. We mainly study the Nichols algebras in the twisted Yetter-Drinfeld module category $_{\k G}^{\k G}\mathcal{YD}^\Phi$ with $\Phi$ a…
Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, powers of cycles, Cayley graphs on some non-abelian groups, and…
Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…
It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show…
The classical Cayley-Hamilton identities are generalized to quantum matrix algebras of the GL(m|n) type.
We introduce certain directed multigraphs with extra structure, called Weyl graphs, which model quotients of Tits buildings by type-preserving chamber-free group actions. Their advantage over complexes of groups, which are often used for…
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the…
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the…
Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are…
We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free…