Related papers: Non-associative Loops for Holger Bech Nielsen
We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras…
We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative…
Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed…
This is a very brief but selfcontained review of the concept of quantum group symmetries and their anomalies. Remarkably, general constructions can be very simply illustrated on the standard harmonic oscillator which is shown to possess a…
For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional…
A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over…
$p$-Adic compactifications of geometric loop and diffeomorphism groups of compact manifolds on finite-dimensional spaces over non-Archimedean fields are investigated. Weakened topology is introduced. The structure of newly constructed…
We provide new bounds for the divisibility function of the free group F_2 and construct short laws for the symmetric groups Sym(n). The construction is random and relies on the classification of the finite simple groups. We also give bounds…
We provide a simplified characterization of entanglement in physical systems which are symmetric under the action of subgroups of the symmetric group acting on the party labels. Sets of entanglements are inherently equal, lying in the same…
In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of…
We compute an explicit representation of the (topological) automorphism group or a particular Toeplitz subshift. The automorphism group is a (non-finitely generated) subgroup of rational numbers under addition and the shift map corresponds…
Our primary purpose is to isolate the abstract, mathematical properties of circuits -- both classical Boolean circuits and quantum circuits -- that are essential for their computational interpretation. A secondary purpose is to clarify the…
We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an…
We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of…
We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element centralisers are all soluble, and a reduction theorem for the…
We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its elements such that the set of all elements of $M$ that associate with $a$ and $b$ does not form a subloop. This is also an example of a nonassociative Moufang loop…
The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer…
Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…
Various kinds of infinitary operations satisfying forms of associativity have been considered in the literature by various authors, including A. Tarski, C. Karp, J. H. Conway, D. Krob, N. Bedon, and C. Rispal. Applications include the…
Finite groups with very few character values are characterized. The following is the main result of this article: a finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a…