Abstract
In this article, we study quantization of super-BM algebra . We quantize by the Drinfel’d twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.
1. Introduction
Lie (super-)bialgebras as well as their quantizations provide important tools in searching for solutions of quantum Yang-Baxter equations and in producing new quantum groups [1, 2]. The notion of Lie bialgebras was introduced by Drinfel’d in 1983 [1, 3]. In 1992, the problem that whether there exists a general approach to quantize all Lie (super-)bialgebras was posed by Drinfel’d in [4]. Later, a positive answer was given by Etingof and Kazhdan in [5], but they did not present a uniform method to realize quantizations for all Lie (super-)bialgebras. Since then, the study of quantizations of Lie (super-)bialgebras has attracted more and more attention. A growing number of people studied the structure theory of Lie (super-)bialgebras, such as [6–11].
The “quantum group” appeared in the work of Drinfel’d as a deformation of the universal enveloping algebra of a Lie algebra in the category of Hopf algebras. In the theory of Hopf algebras and quantum groups, there exist two standard methods to yield new bialgebras from old ones. Twisting the product by a 2-cocycle but keeping the coproduct unchanged is one way; using a Drinfel’d twist element to twist the coproduct but preserving the product is the other approach. Constructing quantizations of Lie bialgebras is an important approach to producing new quantum groups [1, 2, 12].
As an application of quantum groups, quantizations of Lie (super-)bialgebra structures were intensively investigated. Recently, some authors have considered the quantization of several algebras, such as [11–18]. These algebras are all centerless. The case with center is similar.
In order to study the precise boundary conditions for the gauge field describing the theory, the super-BM algebra was introduced in [19, 20]. In [19], the author applies the construction to three-dimensional asymptotically flat supergravity, whose algebra of surface charges has been shown to realize the centrally extended super-BM algebra. In this article, we study quantization of centerless super-BM algebra .
The centerless super-BM algebra is an infinite-dimensional Lie superalgebra over with basis ( or ) and satisfying the following relations: for any and ( or ). The fermionic generators are labeled by (half-)integers in the case of (anti)periodic boundary conditions for the gravitino [19]. Clearly, the -graded is defined by , where and . contains the Witt algebra. The Lie super-bialgebra structures of have been determined in [21]. The non weight modules of have been studied in [20].
In this article, we study quantization of centerless super-BM algebra . In Section , we use the general method of quantization by Drinfel’d twist to quantize explicitly the Lie bialgebra structures on and obtain a family of noncommutative and noncocommutative Hopf superalgebras. The main result of the article is stated in Theorem 18.
2. Quantization of
Theorem 1 (see [21]). Every Lie super-bialgebra structure on is a triangular coboundary Lie super-bialgebra.
Proof. The super-BM algebra in this article is the centerless case of that. By [21], we can deduce that every Lie super-bialgebra structure on is a triangular coboundary Lie super-bialgebra.
Definition 2 (see [11]). Let be a Lie superalgebra containing linearly independent elements and satisfying with and . Then, we set and define a linear map by requiring that for all . Then, equips with the structure of a triangular coboundary Lie super-bialgebra.
Definition 3 (see [11]). A superalgebra () is a superspace equipped with a unit , an associative product respecting the grading, and the identity element . A Hopf superalgebra () is a superalgebra () equipped with a coproduct , a counit , and an antipode , satisfying certain compatible conditions. Note that the antipode satisfies , .
Definition 4 (see [12, 15]). For any element of a unital -algebra ( is a ring) and , , we set In particular, we set , , and , .
Lemma 5 (see [12, 15]). For any element of a unital -algebra and , , one has
Definition 6 (see [1]). Let () be a Hopf superalgebra. A Drinfel’d twist on is an invertible element in such that
Lemma 7 (see [22]). Let () be a Hopf superalgebra and be a Drinfel’d twist on . Then, is invertible in with . Moreover, we denote and by Then, () is a new Hopf superalgebra, which is called twisting of by the Drinfel’d twist .
Lemma 8 (see [12, 15]). For any elements in an associative algebra, , one has
Definition 9. Let be the universal enveloping algebra of and () be the standard Hopf algebra structure on . Then, the coproduct , the antipode , and the counit are defined by In particular, and .
Lemma 10. Let and (); we have ; then, and generate a two-dimensional nonabelian subalgebra of .
Proof. For any , by , we can get . Then, we set . By Definition 2, equips with the structure of a triangular coboundary Lie super-bialgebra.
Lemma 11. For any , , , , and , we have
Proof. We only prove that (9), (10), and (11) can be obtained similarly. We prove (9) by induction on . It is true for the case of . Assume the case of is also true; then, we consider the case of ; we have By (12) and (13), we have Therefore, we deduce that , which means that is true. The proof of is similar.
Lemma 12. For any , , we have
Proof. The case of is clear. If , we prove (15) by induction on . We have which means that Suppose that . By (17), we have The proof of is similar.
Lemma 13. For any , , , and , we have
Proof. We only prove (19) and (21). The proof of (20) is similar.
Definition 14 (see [12, 15]). For , set In particular, we set , , , and . Since and , we have
Lemma 15. For , we have and ; , , , and are invertible elements with and .
Proof. By (23) and Lemma 5, we have From (15), (25), and Lemma 5, we obtain Then, we can deduce that , , , and .
Lemma 16. For any , , , and , we have
Proof. We only prove (28), (31), and (32); the proof of other equations is similar.
The proof of (29)–(30) is similar to (28).
Lemma 17. For any , , , and , we have
Proof. We only prove (37) and (39); the case of (38) is similar. By (9), (11), (15), (19), and (24), we have
Theorem 18. With the choice of two distinguished elements and () such that in , there exists a structure of noncommutative algebra and noncocommutative Hopf algebra structure () on , such that , which preserve the product and counit of ; the coproduct and antipode are defined by where , , and .
Proof. We only prove (41), (43), (44), and (46); the cases of (42) and (45) are similar. By Definition 14 and Lemmas 15–17, we have
Remark 19. We can use the method in this paper to study the quantization of other Lie (super-)algebras in the future.
Remark 20. The case with center is similar, because the center element can be exchanged. Namely, .
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Conflicts of Interest
The authors declare that they have no conflicts of interest.