Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them

Let A be a finite dimensional Hopf algebra, D(A) = A ⊗A its Drinfel’d double and H(A) = A#A its Heisenberg double. The relation between D(A) and H(A) has been found by J.-H. Lu in [24] (see also [33], p. 196): the multiplication of H(A) may be obtained by twisting the multiplication of D(A) by a certain left 2-cocycle which in turn is obtained from the R-matrix of D(A). It was also obtained in [24] that H(A) becomes a left D(A)-module algebra under a certain action of D(A) on H(A) (formula (35) in [24]). All these may be obtained alternatively using a more direct approach, which also shows that the above mentioned action of D(A) on H(A) is manifestly the left regular action of D(A) on D(A) (by identifying H(A) and D(A) as linear spaces). The general setting is the following: if (H,R) is a quasitriangular bialgebra and we define a new multiplication on H by f · g = ∑ (R ⇀ g)(R ⇀ f), then H with this new multiplication (denoted in what follows by H R) becomes a left H-module algebra under the left regular action ⇀ (this is well-known, see also [3], [29], [18], [10] for some more general versions in terms of Drinfel’d twists). Although very simple, this construction may have some nice applications, for instance H R may be noncommutative even if H was cocommutative−it was discovered recently in [49], [50] that an important algebra arising in noncommutative string theory is an example of this type; this discovery has also been applied to noncommutative quantum field theory in [36]. And, if A is a finite dimensional Hopf algebra and H = D(A), then H R is just H(A). For reasons to be discussed below, we were not satisfied with the description of H R as a left H-module algebra and we were led to consider also the right regular action of H on H R. It turns out that H ∗ R is a right H -module algebra (that is, the right action satisfies a “reversed Leibniz rule”), so H R is an algebra in the tensor category of H−H-bimodules (we say that it is an H−H-bimodule algebra). If we endow this category with the braiding given by multiplying to the left by R21 (as