Quantum Groupoids for Physics
A weak Hopf algebra A will have an algebra (B,.) and a dual algebra (Y,*), both of which are finite dimensional, ie. sums of matrix algebras. Given as input a simple Lie group L and a positive integer k, there will be a family of weak Hopf algebras associated to (L,k). Note that k is the level (affine rep theory), related to the deformation parameter q of quantum groups. The families consist of three types of algebra, Ak, Dk and Ek, associated to graphs.
The algebras (B,.) and (Y,*) are associative and compatible in the sense that the coproduct is a homomorphism. B is the direct sum of spaces Hn of a certain class of paths on the graph G of length n. The product turns out to be a vertical composition of basic >-< diagrams. In the case of SU(2) for k=1, the graph is the two node Dynkin diagram A2. A vertex (two leafed tree) in H0 is labelled by 0 or 1 at the inputs. There are two paths in H1, depending on the orientation of the edge. And for this example, that's it. Dually, one has triangles instead of basic 3-valent vertices, with labelled edges. All these diagrams form a basis for vector spaces over C.
Similarly, the algebra (Y,*) deals with horizontal diagram composition. Both algebras have units, but it is not true that the coproduct on 1 gives 1 x 1. Also, the counit is not a homomorphism. This is the reason for the term 'weak'. Given B and Y, there are two character theories, A and O respectively. O is a bimodule over A and a lot of the physical motivation for this type of weak Hopf algebra comes from the fact that the bimodule structure is given by certain matrices W(xy), where W(00) is a so-called modular invariant. The graph G is a Z+ module over A, and also a Z+ module over O.
The paths with fixed source or target generate two spaces Ds and Dt which appear when one looks at the coproduct on 1. The antipode switches labels on the >-< diagram and may have a coefficient, but for SU(2) this is 1. The natural pairing given by the basis elements acts on the matrix units to give a labelled tetrahedron. These are the Ocneanu cells.
Now let's define creation and annihilation operators. The latter takes paths of length n to paths of length n-2 by removing backtracks. One must also insert a coefficient which is the square root of the ratio of the quantum numbers associated to the vertices at the source and target of the backtracking piece. To compute these quantum numbers on the graph, take the graph's adjacency matrix, compute the eigenvalues/eigenvectors, normalise with respect to the leading eigenvector, and the scaled eigenvalues are the quantum numbers.
For example, for the Dynkin diagram for A3, there is a 4x4 matrix and the labels are (1,r,r,1) where r is the Golden ratio.
Taking En to be the number operator here and scaling by the highest eigenvalue b one recovers the relations for the Temperley-Lieb-Jones algebra. Cool stuff. The restriction of self-adjointness forces b to take on a certain range of well-known values.