Connected (graded) Hopf algebras

$\mathfrak{R}$-matrix for quantum superalgebra $\mathfrak{sl}(2|1)$ at roots of unity and its application to centralizer algebras

“…We mention the following papers in which are used methods close to us [17], [14], [16]. The case when the quantization parameter is the root of unity has been less studied (see [13], [1]), but has recently been in the focus of attention, along with the case of quantum superalgebras (see [9], [1], [22], [2], [8], [23], [24]).…”

“…These results can be applied to supercase. Some classes of Hopf superalgebras were investigated in [3], [4], [8], [12], [24]. The number of articles devoted to applications in modern mathematical physics is huge.…”

Unimodular graded Poisson Hopf algebras

“…When working with graded Hopf algebras it is frequently useful to make use of the adjustment permitted by the following lemma, a slight improvement of [BGZ,Lemma 2.1(2)].…”

“…The possible Hopf structures in this case are as described in the following theorem. Details concerning part (i) can be found at [BGZ,Theorem 0.1]. Its non-trivial content is due to Serre for (1) =⇒ ( 3), (see e.g.…”

“…Example 1.9. Let H be the connected Hopf algebra of GK-dimension 5 constructed in [BGZ,Theorem 5.6]. That is, H = k â, b, ĉ, ẑ, ŵ with relations given by setting all commutators of the generators equal to 0, except for [â, b] = ĉ and [ẑ, ŵ] = 1 3 ĉ3 .…”