Modular quantizations of Lie algebras of Cartan type K via Drinfeld twists of Jordanian type

Abstract

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra u(K(2n+1; 1)) for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new Hopf algebras of noncommutative and noncocommutative and with dimension p p 2n+1 +1 (if 2n + 4 0 (modp)) or p p 2n+1 (if 2n + 4 ≡ 0 (modp)) over a truncated p-polynomials ring, which also contain the well-known Radford algebras as Hopf subalgebras. Some open questions are proposed. This paper is a continuation of [14,15,28] in which quantizations of Cartan type Lie algebras of types W, S and H were studied. In the present paper, we continue to treat the same questions both for the generalized Cartan type K Lie algebras in characteristic 0 (for the definition, see [24]) and for the restricted simple contact algebra K(2n+1; 1) of Cartan type K in the modular case (for the definition, see [25], [26]). This work, together with papers [14, 15, 28], completes the work of this kinds of modular quantizations of Jordanian type for the Cartan type Lie algebras over a field K with char (K) ≥ 7.We survey some previous related work. After the works on quantum groups which were introduced by Drinfel'd [4] and Jimbo [16], Drinfeld in [5] raised the question of the existence of a universal quantization for Lie bialgebras. Etingof-Kazhdan gave a positive answer to this question in [7,8], where the Lie bialgebras they considered including finiteand infinite-dimensional ones are those defined by generalized Cartan matrices. Enriquez-Halbout showed that any coboundary Lie bialgebra, in principle, can be quantized via a certain Etingof-Kazhdan quantization functor [6], and Geer [10] further extended Etingof-Kazhdan's work from Lie bialgebras to the setting of Lie superbialgebras. After the work [7,8], it is natural to consider the quantizations of Lie algebras of Cartan type that are 1991 Mathematics Subject Classification. Primary 17B37, 17B62; Secondary 17B50.

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

“…If ν 2 = 0 = ν 4 and µ 1 = 0 = µ 2 , then H is isomorphic to one of the Hopf algebras described in ( 28)- (29).…”

“…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.…”

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].…”

Matching realization of U q (sl n+1) of higher rank in the quantum Weyl algebra W q (2n)

“…This is a hard question aimed by the first author when he was a Humbolt research fellow almost twenty years ago. On the other hand, fortunately, a series work on the mudular quantizations of the modular simple restricted Lie algebras of Cartan type have been done in recent years via the Jordanian twists and modular reductions, and these provide us with some new non-pointed Hopf algebras of prime-power dimensions defined over a field of positive characteristic, see [6], [7], [14]. This will be significant if one considers the Kaplansky's 10 questions proposed in early 1975 which are related to classifying some finite-dimensional Hopf algebras in some sense, and compares with the seminal work on classifying finite-dimensional complex pointed Hopf algebras due to Andruskiewitsch-Schneider [1] in 2010.…”