Modular Quantizations of Lie Algebras of Cartan Type 𝐻 via Drinfel’d Twists

Abstract

Abstract. We construct explicit Drinfel'd twists for the Lie algebras of generalized Cartan type H in characteristic 0 and also obtain the corresponding quantizations and their integral forms. By using modular reduction and base changes, we derive certain quantizations of the restricted universal enveloping algebra u(H(2n; 1)) of the restricted Hamiltonian algebra H(2n; 1) in prime characteristic p. These quantizations are new non-pointed Hopf algebras of prime-power dimension p p 2n −1 and contain the well-known Radford algebras as Hopf subalgebras. As a by-product we also obtain some Jordanian quanti-This paper is a continuation of [16] and [17] in which modular quantizations of Lie algebras of Cartan types W and S were studied. In the present paper we consider the same questions, both for the Lie algebras of generalized Cartan type H in characteristic 0 (for the definition, see [22]) and for the restricted Hamiltonian algebras H(2n; 1) in the modular case (for the definition, see [25] and [26]).For the convenience of the reader we review some previous related work. In

Pointed Hopf algebras of dimension $p^2q$ in characteristic $p$

“…where λ 1 ∈ I 0,1 , λ 2 , λ 3 ∈ k. To show that dim H = pq 2 by the Diamond Lemma, it suffices to verify the following overlaps: g q−1 (gx) = (g q−1 g)x, g q−1 (gy) = (g q−1 g)y, (27) g(xx p−1 ) = (gx)x p−1 , g(yy p−1 ) = (gy)y p−1 , (28) g(xy) = (gx)y, (29) x(yy p−1 ) = (xy)y p−1 , x p−1 (xy) = (x p−1 x)y, (30) are resolvable with the order y < x < h < g. The verification of (29) amounts to λ 3 = 0. The verification of (30) amounts to λ p 2 − λ 1 λ 2 = 0.…”

“…One need more efficient methods to study the connected graded braided Hopf algebras R to complete the classification of finite-dimensional Hopf algebras over an algebraically closed filed. It should be mentioned that N. Hu, X. Wang and Z. Tong constructed examples of braided Hopf algebras that are not Nichols algebras when deriving certain modular quantizations of the restricted universal enveloping algebras of the restricted modular simple Lie algebras of Cartan type by making modular reductions including modulo p and modulo p-restrictedness reduction, see [14,15,30,29]; V. C. Nguyen and X. Wang studied connected graded braided Hopf algebras of dimension p 2 that are not Nichols algebras by means of the Hochschild cohomology of coalgebras [21].…”

On non-connected pointed Hopf algebras of dimension 16 in characteristic 2

“…Let p, q, r be distinct prime numbers and char k = p. G. Henderson classified cocommutative connected Hopf algebras of dimension less than or equal to p 3 [13]; X. Wang classified connected Hopf algebras of dimension p 2 [34] and pointed ones with L. Wang [33]; V. C. Nguyen, L. Wang and X. Wang determined connected Hopf algebras of dimension p 3 [20,21]; Nguyen-Wang [22] studied the classification of non-connected pointed Hopf algebras of dimension p 3 and classified coradically graded ones; motivated by [27,22], the author gave a complete classification of pointed Hopf algebras of dimension pq, pqr, p 2 q, 2q 2 , 4p and pointed Hopf algebras of dimension pq 2 whose diagrams are Nichols algebras. It should be mentioned that S. Scherotzke classified finite-dimensional pointed Hopf algebras whose infinitesimal braidings are one-dimensional and the diagrams are Nichols algebras [26]; N. Hu, X. Wang and Z. Tong constructed many examples of pointed Hopf algebras of dimension p n for some n ∈ N via quantizations of the restricted universal enveloping algebras of the restricted modular simple Lie algebras of Cartan type, see [14,15,30,29]; C. Cibils, A. Lauve and S. Witherspoon constructed several examples of finite-dimensional pointed Hopf algebras whose diagrams are Nichols algebras of Jordan type [10]; N. Andruskiewitsch, et al constructed some examples of finite-dimensional coradically graded pointed Hopf algebras whose diagram are Nichols algebras of nondiagonal type [1], which extends the work in [10]. Until now, it is still an open question to give a complete classification of non-connected pointed Hopf algebras of dimension p 3 or pointed ones of dimension pq 2 whose diagrams are not Nichols algebras for odd prime numbers p, q.…”