Cohomology of Frobenius Algebras and the Yang-Baxter Equation

arXiv:0801.2567v1 [math.QA] 16 Jan 2008 Cohomology of Frobenius Algebras and the Yang-Baxter Equation J. Scott Carter∗ University of South Alabama Alissa S. Crans Loyola Marymount University Mohamed Elhamdadi University of South Florida Enver Karadayi University of South Florida Masahico Saito† University of South Florida November 21, 2018 Dedicated to the memory of Xiao-Song Lin Abstract A cohomology theory for multiplications and comultiplications of Frobenius algebras is de- veloped in low dimensions in analogy with Hochschild cohomology of bialgebras based on defor- mation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using mul- tiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain de- formations of R-matrices thus obtained. 1 Introduction Frobenius algebras are interesting to topologists as well as algebraists for numerous reasons includ- ing the following. First, 2-dimensional topological quantum field theories are formulated in terms of commutative Frobenius algebras (see [13]). Second, a Frobenius algebra structure exists on any finite-dimensional Hopf algebra with a left integral defined in the dual space. These Hopf algebras have found applications in topology through Kuperberg’s invariant [14, 15], the Henning invariant [11, 17], and the theory of quantum groups from which the post-Jones invariants arise. Third, there is a 2-dimensional Frobenius algebra that underlies Khovanov’s cohomology theory [12]. See also [1]. Our interest herein is to extend the cohomology theories defined in [3, 4] to Frobenius algebras and thereby construct new solutions to the Yang-Baxter equation (YBE). We expect that there are connections among these cohomology theories that extend beyond their formal definitions. Furthermore, we anticipate topological, categorical, and/or physical applications because of the diagrammatic nature of the theory. ∗Supported in part by NSF Grant DMS #0603926. †Supported in part by NSF Grant DMS #0603876. 1 The 2-cocycle conditions of Hochschild cohomology of algebras and bialgebras can be interpreted via deformations of algebras [8]. In other words, a map satisfying the associativity condition can be deformed to obtain a new associative map in a larger vector space using 2-cocycles. The same interpretation can be applied to quandle cohomology theory [2, 5, 6]. A quandle is a set equipped with a self-distributive binary operation satisfying a few additional conditions that correspond to the properties that conjugation in a group enjoys. Quandles have been used in knot theory extensively (see [2] and references therein for more aspects of quandles). Quandles and related structures can be used to construct set-theoretic solutions (called R-matrices) to the Yang-Baxter equation (see, for example, [10] and its references). From this point of view, combined with the deformation 2-cocycle interpretation, a quandle 2-cocycle can be regarded as giving a cocycle deformation of an R-matrix. Thus we extend this idea to other algebraic constructions of R-matrices and construct new R-matrices from old via 2-cocycle deformations. In [3, 4], new R-matrices were constructed via 2-cocycle deformations in two other algebraic contexts. Specifically, in [3], self-distributivity was revisited from the point of view of coalgebra categories, thereby unifying Lie algebras and quandles in these categories. Cohomology theories of Lie algebras and quandles were given via a single definition, and deformations of R-matrices were constructed. In [4], the adjoint map of Hopf algebras, which corresponds to the group conjuga- tion map, was studied from the same viewpoint. A cohomology theory was constructed based on equalities satisfied by the adjoint map that are sufficient for it to satisfy the YBE. In this paper, we present an analog for Frobenius algebras according to the following organiza- tion. After a brief review of necessary materials in Section 2, a cohomology theory for Frobenius algebras is constructed in Section 3 via deformation theory. Then Yang-Baxter solutions are con- structed by skein methods in Section 4, followed by deformations of R-matrices by 2-cocycles. The reader should be aware that the composition of the maps is read in the standard way from right to left (gf)(x) = g(f(x)) in text and from bottom to top in the diagrams. In this way, when reading from left to right one can draw from top to bottom and when reading a diagram from top to bottom, one can display the maps from left to right. The argument of a function (or input object from a category) is found at the bottom of the diagram. 2 Preliminaries A Frobenius algebra is an (associative) algebra (with multiplication µ : A ⊗A →A and unit η : k →A) over a field k with a nondegenerate associative pairing β : A ⊗A →k. Throughout this paper all algebras are finite-dimensional unless specifically stated otherwise. The pairing

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