Niveau: Supérieur, Doctorat, Bac+8
Under revision for the Transactions of the American Mathematical Society Preprint version available at YANG–BAXTER DEFORMATIONS AND RACK COHOMOLOGY MICHAEL EISERMANN Abstract. In his study of quantum groups, Drinfeld suggested to consider set-theoretic solutions of the Yang–Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group, or more generally every rack Q provides such a Yang–Baxter operator cQ : x ? y 7? y ? xy. In this article we study deformations of cQ within the space of Yang–Baxter opera- tors. Over a complete ring these are classified by Yang–Baxter cohomology. We show that the general Yang–Baxter cohomology complex of cQ homotopy- retracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of cQ, including the modular case which had previously been left in suspense, by establishing that every deformation of cQ is gauge equivalent to a quasi-diagonal one. In a quasi-diagonal deformation only behaviourally equivalent elements of Q interact; if all elements of Q are behaviourally distinct, then the Yang–Baxter cohomology of cQ collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Ger- stenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory. 1. Introduction and statement of results 1.1. Motivation and background.
- diagonal deformation
- group
- quantum group
- deformation theory
- term rack
- theoretic solution
- yang–baxter cohomology
- diagonal means diagonal
- called behaviourally