NON-TRIVIALLY ASSOCIATED TENSOR CATEGORIES AND LEFT COSET REPRESENTATIVES*) E.J. BEGGS Dept. of Mathematics, University of Wales Swansea, Wales SA2 8PP Received 24 August 2000 In this talk I will motivate and describe a generalisation of the group doublecross prod- uct construction, involving a set of coset representatives for the left action of a subgroup on a group. From this data a non-trivially associated tensor category can be made. I shall briefly mention the corresponding double construction, which gives a non-trivially associated braided tensor category, containing a braided Hopf algebra. 1 Introduction I shall begin with a brief description of the group doublecross formalism for solving integrable 1+1 dimensional classical field theories [1]. Take a space-time S, and a group X with two subgroups G and M so that X = GM = MG. Here X = G ~ M is called the doublecross product of G and M [2], and we shall give all example of this later. This is similar to the cross product G x M, except that the subgroups may not commute with each other. We now consider a function a : S --* M, which we shall call the 'classical vacuum' map. The group G may be called the classical phase space. For a given classical solution ¢0 E G, we perform a factorisation for all space-time positions x E S: a(x)¢0 = ¢(x) b(x), ¢(x) eG, b(x) eU. The solution to the field theory, the fields at the point x E 5', is encoded into b(x) and ¢(x). We can recover a linear system from the factorisation by differentiating it along a space-time vector v to give av¢o = ¢~b + ebv (we use subscripts for differentiation). This can be rearranged to give ev = ava-l¢ - ebvb -1. By using meromorphic and analytic loop groups this method can construct all the soliton solutions of the principal chiral and sine-Gordon models, and has been used to find new charges for single solitons in the affine Toda equation [3]. We can now form a semidirect product algebra C(G) :>~CM combining the observables C(G) (functions on the phase space) and the group algebra CM (containing symmetries of the system such as space and time translation). This algebra can also be assigned a coproduct structure, making it into a Hopf algebra [2]. Now we might note that there is no physical reason why G and M should actually be groups, in fact it is possible that M not being a group might correspond to curved *) Presented at the 9th Colloquimn "Qumltmn Groups mad Integrable Systems", Prague, 22-24 June 2000. Czechoslovak Journal of Physics, Vol. 50 (2000), No. 11 1195