Related papers: Division-Alebra/Poncare-Conjecture correspondence
Automata learning is a popular technique for inferring minimal automata through membership and equivalence queries. In this paper, we generalise learning to the theory of coalgebras. The approach relies on the use of logical formulas as…
By using MacMahon partition analysis technique, the Poincar\'e series for the algebras of invariants of the ternary, quaternary and quinary forms of small orders are calculated.
We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the…
The deformations of the Galilei algebra and their associated noncommutative Newtonian spacetimes are investigated. This is done by analyzing the possible nonrelativistic limits of an eleven generator (pseudo)extended \kap-Poincar\'e algebra…
Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic…
The theory of integrals is used to analyse the structure of Hopf algebroids, introduced in math.QA/0302325. We prove that the total algebra of the Hopf algebroid is a separable extension of the base algebra if and only if it is a…
Systems of cooperation and interaction are usually studied in the context of real or complex vector spaces. Additional insight, however, is gained when such systems are represented in vector spaces with multiplicative structures, i.e., in…
Useful relations describing arbitrary parameters of given quantum systems can be derived from simple physical constraints imposed on the vectors in the corresponding Hilbert space. This is well known and it usually proceeds by partitioning…
We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius…
Generalised observables (POM observables) are necessary for representing all possible measurements on a quantum system. Useful algebraic operations such as addition and multiplication are defined for these observables, recovering many…
Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to prove the existence of enveloping actions, i.e., every…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…
By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known…
Motivated by the Poincare conjecture, we study properties of digital n-dimensional spheres and disks, which are digital models of their continuous counterparts. We introduce homeomorphic transformations of digital manifolds, which retain…
We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference…
Unified graded differential algebra, generated by $\kappa$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $\kappa$-Poincar\'e-Hopf algebra. For time- and…
Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint…
We show that some associative algebras whose product splits up into the sum of several operations and are free, in a certain sense, with respect to these operations, admit a Hopf algebra structure. We show that the operad of dendriform…