2018/08/12 by Gateva-Ivanova, Tatiana · 1 citation
#16S36 #16S37 #16W50 #81R50 (Primary) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1808.03938
We study noninvolutive set-theoretic solutions (X,r) of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid S(X,r), the quadratic Yang-Baxter algebra A= A(k, X, r) over a field k and its Koszul dual, A!. More generally, we continue our systematic study of nondegenerate quadratic sets (X,r) and the associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions (X,r). It contains the special class of self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets (X,r) of order n≥ 3 which satisfy "the minimality condition M", that is dimk A2 =2n-1. Examples are some simple racks of prime order p. Finally, we discuss general extensions of solutions and introduce the notion of "a generalized strong twisted union of braided sets". We prove that if (Z,r) is a non-degenerate 2-cancellative braided set splitting as Z = X\natural∗ Y, then its braided monoid SZ is a generalized strong twisted union SZ= SX\natural∗ SY of the braided monoids SX and SY. Moreover, if (Z,r) is injective then its braided group GZ=G(Z,r) also splits as GZ= GX\natural∗ GY of the associated braided groups of X and Y. We propose a construction of a generalized strong twisted union Z = X\natural∗ Y of braided sets (X,rX), and (Y, rY), where the map r has high, explicitly prescribed order.