Magnetic Monopole and Quantum Groups

Zaczynamy od omówienia modelu Wu-Yanga monopolu magnetycznego w języku geometrii różniczkowej (rozwłóknienie Hopfa). Następnie wprowadzamy pojęcie grup kwantowych i wyznaczamy bazy liniowe, ślady (cykliczne 0-kocykle) oraz nieprzywiedlne *-reprezentacje kwantowej grupy SU(2) i zdegenerowanych kwantowych sfer Podlesia. Na koniec dyskutujemy nieprzemienne modele monopolu magnetycznego w terminach rozszerzeń Hopfa-Galois (kwantowe rozwłóknienie Hopfa).We begin with a discussion of the Wu-Yang model of a magnetic monopole in the language of differential geometry (Hopf fibration). Next, we introduce the concept of quantum groups and we designate line bases, traces (cyclic 0-cocycles) and irreducible *-representations of the quantum SU(2) group and the degenerate quantum Podleś spheres. Finally, we discuss non-commutative models of a magnetic monopole in terms of Hopf-Galois extensions (quantum Hopf fibration)

Structure algebras of finite set-theoretic solutions of the Yang--Baxter equation

Algebras related to finite bijective or idempotent left non-degenerate solutions $(X,r)$ of the Yang--Baxter equation have been intensively studied. These are the monoid algebras $K[M(X,r)]$ and $K[A(X,r)]$, over a field $K$, of its structure monoid $M(X,r)$ and left derived structure monoid $A(X,r)$, which have quadratic defining relations. In this paper we deal with arbitrary finite left non-degenerate solutions $(X,r)$. Via divisibility by generators, i.e., the elements of $X$, we construct an ideal chain in $M(X,r)$ that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras $K[M(X,r)]$ and $K[A(X,r)]$ are left or right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution $(X,r)$ are proven, which extends known results on bijective non-degenerate solutions. Furthermore, we describe the cancellative congruences of $A(X,r)$ and $M(X,r)$ as well as the prime spectrum of $K[A(X,r)]$. This then leads to an explicit formula for the Gelfand--Kirillov dimension of $K[M(X,r)]$ in terms of the number of orbits in $X$ under actions of certain finite monoids derived from $(X,r)$. It is also shown that the former coincides with the classical Krull dimension of $K[M(X,r)]$ in case the algebra $K[M(X,r)]$ is left or right Noetherian. Finally, we obtain the first structural results for a class of finite degenerate solutions $(X,r)$ of the form $r(x,y)=(\lambda_x(y),\rho(y))$ by showing that structure algebras of such solution are always right Noetherian.Comment: 35 page

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang–Baxter equation

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang–Baxter equation.Fil: Jespers, Eric. Vrije Unviversiteit Brussel; BélgicaFil: Kubat, Łukasz. Vrije Unviversiteit Brussel; Bélgica. Uniwersytet Warszawski; ArgentinaFil: Van Antwerpen, Arne. Vrije Unviversiteit Brussel; BélgicaFil: Vendramin, Claudio Leandro. Vrije Unviversiteit Brussel; Bélgica. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin