# Quantization of the Super-BMS3 Algebra

Authors: Yu Yang; Xing Tao Wang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this article, we study quantization of super-BMS3 algebra W. We quantize W by the Drinfel’d twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.


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## Body

## 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-BMS3 algebra was introduced in [19, 20]. In [19], the author applies the construction to three-dimensional asymptotically flat N=1 supergravity, whose algebra of surface charges has been shown to realize the centrally extended super-BMS3 algebra. In this article, we study quantization of centerless super-BMS3 algebra W.The centerless super-BMS3 algebra W is an infinite-dimensional Lie superalgebra over ℂ with basis Lm,Pm,Qp∣m∈ℤ,p∈ε+ℤ (ε=0 or 1/2) and satisfying the following relations:
(1)Lm,Ln=m−nLm+n,Lm,Pn=m−nPm+n,Lm,Qp=m2−pQm+p,Qp,Qq=Pp+q,Pm,Pn=Pm,Qp=0,for any m,n∈ℤ and p,q∈ε+ℤ(ε=0 or 1/2). The fermionic generators Qp are labeled by (half-)integers in the case of (anti)periodic boundary conditions for the gravitino [19]. Clearly, the ℤ2-graded W is defined by W=W0¯⊕W1¯, where W0¯=Lm,Pm∣m∈ℤ and W1¯=Qp∣p∈ε+ℤ. W contains the Witt algebra. The Lie super-bialgebra structures of W have been determined in [21]. The non weight modules of W have been studied in [20].In this article, we study quantization of centerless super-BMS3 algebra W. In Section 2, we use the general method of quantization by Drinfel’d twist to quantize explicitly the Lie bialgebra structures on W and obtain a family of noncommutative and noncocommutative Hopf superalgebras. The main result of the article is stated in Theorem 18.


## 2. Quantization ofW

Theorem 1 (see [21]).
Every Lie super-bialgebra structure onW is a triangular coboundary Lie super-bialgebra.Proof.
The super-BMS3 algebra W in this article is the centerless case of that. By [21], we can deduce that every Lie super-bialgebra structure on W is a triangular coboundary Lie super-bialgebra.Definition 2 (see [11]).
LetW be a Lie superalgebra containing linearly independent elements a and b satisfying a,b=kb with a=b=0¯ and 0≠k∈ℂ. Then, we set r=a⊗b−b⊗a and define a linear map δ=δr:L⟶L⊗L by requiring that
(2)δrx=−1rxx·r=−1rxx,a⊗b+−1axa⊗x,b−x,b⊗a−−1bxb⊗x,afor all x∈W. Then, δr equips W with the structure of a triangular coboundary Lie super-bialgebra.Definition 3 (see [11]).
A superalgebra (H,μ,ι) is a superspace H equipped with a unit ι:ℂ⟶H, an associative product μ:H⊗H⟶H respecting the grading, and the identity element 1∈H0¯. A Hopf superalgebra (H,μ,ι,Δ,S,ε) is a superalgebra (H,μ,ι) equipped with a coproduct Δ:H⟶H⊗H, a counit ε:H⟶ℂ, and an antipode S:H⟶H, satisfying certain compatible conditions. Note that the antipode S satisfies Sxy=−1xySySx, ∀x,y∈H.Definition 4 (see [12, 15]).
For any elementx of a unital R-algebra (R is a ring) and a∈R, r,k∈ℕ+, we set
(3)xar=x+ax+a+1⋯x+a+r−1,xar=x+ax+a−1⋯x+a−r+1,ar=aa−1⋯a−r+1r!,ark=aa−k⋯a−r−1kr!.
In particular, we setx0r=xr, x0r=xr, and xa0=1, xa0=1.Lemma 5 (see [12, 15]).
For any elementx of a unital ℂ-algebra and a∈ℂ, r,s,t∈ℕ+, one has
(4)xas+t=xasxa+st,xas+t=xasxa+st,xas=xa−s+1s,∑s+t=r−1ts!t!xasxbt=a−br=a−b⋯a−b−r+1r!,∑s+t=r−1ts!t!xasxb−st=a−b+r−1r=a−b⋯a−b+r−1r!.Definition 6 (see [1]).
Let (H,μ,ι,Δ0,S0,ε) be a Hopf superalgebra. A Drinfel’d twist F on H is an invertible element in H⊗H such that
(5)F⊗1Δ0⊗IdF=1⊗FId⊗Δ0F,ε⊗IdF=1⊗1=Id⊗εF.Lemma 7 (see [22]).
Let (H,μ,ι,Δ0,S0,ε) be a Hopf superalgebra and F be a Drinfel’d twist on H. Then, w=μId⊗S0F is invertible in H with w−1=μId⊗S0F−1. Moreover, we denote Δ:H⟶H⊗H and S:H⟶H by
(6)Δx=FΔ0xF−1,Sx=wS0xw−1,∀x∈H.
Then, (H,μ,ι,Δ,S,ε) is a new Hopf superalgebra, which is called twisting of H by the Drinfel’d twist F.Lemma 8 (see [12, 15]).
For any elementsx,y in an associative algebra, p∈ℕ, one has
(7)xyp=∑k=0p−1kpkyp−kadykx.Definition 9.
LetUW be the universal enveloping algebra of W and (UW,μ,ι,Δ0,S0,ε) be the standard Hopf algebra structure on UW. Then, the coproduct Δ0, the antipode S0, and the counit ε are defined by
(8)Δ0X=X⊗1+1⊗X,S0X=−X,εX=0,∀X∈W.
In particular,Δ01=1⊗1 and S01=ε1=1.Lemma 10.
LetX=−1/mL0 and Y=Pm (m∈ℤ∗); we have X,Y=Y; then, X and Y generate a two-dimensional nonabelian subalgebra of W.Proof.
For anym∈ℤ∗, by L0,Pm=−mPm, we can get X,Y=Y. Then, we set r=X⊗Y−Y⊗X. By Definition 2, δr equips W with the structure of a triangular coboundary Lie super-bialgebra.Lemma 11.
For anya∈ℂ, n∈ℤ, p∈ε+ℤ, m∈ℤ∗, and r∈ℕ+, we have
(9)LnXar=Xa−n/mrLn,LnXar=Xa−n/mrLn,(10)PnXar=Xa−n/mrPn,PnXar=Xa−n/mrPn,(11)QpXar=Xa−n/mrQp,QpXar=Xa−n/mrQp.Proof.
We only prove that (9), (10), and (11) can be obtained similarly. We prove (9) by induction on r. It is true for the case of r=1. Assume the case of r is also true; then, we consider the case of r+1; we have
(12)LnXar+1=LnXarX+a+r=Xa−n/mrLnX+a+r,(13)X,Ln=−1mL0,Ln.
By (12) and (13), we have
(14)Xa−n/mrLnX+a+r=Xa−n/mrXLn−nmLn+a+rLn=Xa−n/mrX+a−nm+rLn=Xa−n/mr+1Ln.
Therefore, we deduce thatLnXar+1=Xa−n/mr+1Ln, which means that LnXar=Xa−n/mrLn is true. The proof of LnXar=Xa−n/mrLn is similar.Lemma 12.
For anya∈ℂ, s,r∈ℕ+, we have
(15)YsXar=Xa−srYs,YsXar=Xa−srYs.Proof.
The case ofs=r=1 is clear. If s=1, we prove (15) by induction on r. We have
(16)YXar+1=YXarX+a+r=Xa−1rYX+a+r=Xa−1rXY−Y+a+rY=Xa−1rX+a+r−1Y=Xa−1r+1Y,which means that
(17)YXar=Xa−1rY.
Suppose thatYsXar=Xa−srYs. By (17), we have
(18)Ys+1Xar=YXa−srYs=Xa−s+1rYYs=Xa−s+1rYs+1.
The proof ofYsXar=Xa−srYs is similar.Lemma 13.
For anym∈ℤ∗, n∈ℤ, p∈ε+ℤ, and r∈ℕ+, we have
(19)LnYr=YrLn+rYr−1n−mPm+n,(20)PnYr=YrPn,(21)QpYr=YrQp.Proof.
We only prove (19) and (21). The proof of (20) is similar.
(22)LnYr=∑k=0r−1krkYr−kadYkLn=YrLn+rYr−1n−mPm+n,QpYr=∑k=0r−1krkYr−kadYkQp=YrQp.Definition 14 (see [12, 15]).
Fora∈ℂ, set
(23)Fa=∑r=0∞−1rr!Xar⊗Yrtr,Fa=∑r=0∞1r!Xar⊗Yrtr,(24)Ua=μS0⊗IdFa,Va=μId⊗S0Fa.
In particular, we setF=F0, F=F0, u=u0, and v=v0. Since S0Xar=−1rX−ar and S0Yr=−1rYr, we have
(25)Ua=∑r=0∞−1rr!X−arYrtrVa=∑r=0∞1r!XarYrtr.Lemma 15.
Fora,b∈ℂ, we have FaFb=1⊗1−Yta−b and VaUb=1−Yt−a−b; Fa, Fa, Ua, and Va are invertible elements with Fa=Fa−1 and Ua=Va−1.Proof.
By (23) and Lemma 5, we have
(26)FaFb=∑r,s=0∞−1rr!s!XarXbs⊗YrYstrts=∑k=0∞−1k∑r+s=k−1sr!s!XarXbs⊗Yktk=∑k=0∞−1ka−bk⊗Yktk=1⊗1−Yta−b.
From (15), (25), and Lemma 5, we obtain
(27)VaUb=∑r,s=0∞−1rr!s!XarYrX−bsYstr+s=∑k=0∞−1k∑r+s=k−1sr!s!XarX−b−rsYktk=∑k=0∞a+b+k−1kYktk=1−Yt−a−b.
Then, we can deduce thatFa=Fa−1, Ua=Va−1, F=F−1, and U=V−1.Lemma 16.
For anym∈ℤ∗, n∈ℤ, p∈ε+ℤ, and a∈ℂ, we have
(28)Ln⊗1Fa=Fa−n/mLn⊗1,(29)Pn⊗1Fa=Fa−n/mPn⊗1,(30)Qp⊗1Fa=Fa−n/mQp⊗1,(31)1⊗LnFa=Fa1⊗Ln+n−mFa+1Xa1⊗Pm+nt,(32)1⊗PnFa=Fa1⊗Pn,(33)1⊗QpFa=Fa1⊗Qp.Proof.
We only prove (28), (31), and (32); the proof of other equations is similar.By (9) and (23), we have
(34)Ln⊗1Fa=∑r=0∞1r!LnXar⊗Yrtr=∑r=0∞1r!Xa−n/mrLn⊗Yrtr=Fa−n/mLn⊗1.Theproof of (29)–(30) is similar to (28).By (19) and (23), we have
(35)1⊗LnFa=∑r=0∞1r!Xar⊗YrLn+rYr−1n−mPm+ntr=∑r=0∞1r!Xar⊗YrLntr+n−m∑r=1∞1r−1!Xar⊗Yr−1Pm+ntr=∑r=0∞1r!Xar⊗Yrtr1⊗Ln+n−m∑r=0∞1r!Xar+1⊗YrPm+ntr+1=Fa1⊗Ln+n−m∑r=0∞1r!Xa1Xa+1r⊗YrPm+ntr·t=Fa1⊗Ln+n−m∑r=0∞1r!Xa+1r⊗YrtrXa1⊗Pm+nt=Fa1⊗Ln+n−mFa+1Xa1⊗Pm+nt.By (21) and (25), we have
(36)1⊗QpFa=1⊗Qp∑r=0∞1r!Xar⊗Yrtr=∑r=0∞1r!Xar⊗QpYrtr=∑r=0∞1r!Xar⊗YrQptr=Fa1⊗Qp.Lemma 17.
For anym∈ℤ∗, n∈ℤ, p∈ε+ℤ, and a∈ℂ, we have
(37)LnUa=Ua+n/mLn+m−nUa+n/mX−a−n/m1Pm+nt,(38)PnUa=Ua+n/mPn,(39)QpUa=Ua+n/mQp.Proof.
We only prove (37) and (39); the case of (38) is similar. By (9), (11), (15), (19), and (24), we have
(40)LnUa=Ln∑r=0∞−1rr!X−arYrtr=∑r=0∞−1rr!LnX−arYrtr=∑r=0∞−1rr!X−a−n/mrLnYrtr=∑r=0∞−1rr!X−a−n/mrYrLn+rYr−1n−mPm+ntr=∑r=0∞−1rr!X−a−n/mrYrLntr+∑r=1∞−1rr−1!X−a−n/mrYr−1n−mPm+ntr=Ua+n/mLn+∑r=0∞−1r+1r!X−a−n/mr+1Yrn−mPm+ntr+1=Ua+n/mLn+m−n∑r=0∞−1rr!X−a−n/mrX−a−n/m−r1YrPm+ntr·t=Ua+n/mLn+m−n∑r=0∞−1rr!X−a−n/mrYrX−a−n/m1Pm+ntr·t=Ua+n/mLn+m−nUa+n/mX−a−n/m1Pm+nt,QnUa=Qn∑r=0∞−1rr!X−arYrtr=∑r=0∞−1rr!QnX−arYrtr=∑r=0∞−1rr!X−a−n/mrQnYrtr=∑r=0∞−1rr!X−a−n/mrYrQntr=Ua+n/mQn.Theorem 18.
With the choice of two distinguished elementsX=−1/mL0 and Y=Pm (m∈ℤ∗) such that X,Y=Y in W, there exists a structure of noncommutative algebra and noncocommutative Hopf algebra structure (UWt,μ,ι,Δ,S,ε) on UWt, such that UWt/tUWt≅UW, which preserve the product and counit of UWt; the coproduct and antipode are defined by
(41)ΔLn=Ln⊗1−Ytn/m+1⊗Ln+n−mX1⊗1−Yt−1Pm+nt,(42)ΔPn=Pn⊗1−Ytn/m+1⊗Pn,(43)ΔQp=Qp⊗1−Ytn/m+1⊗Qp,(44)SLn=−1−Yt−n/mLn+m−nX−n/m1Pm+nt,(45)SPn=−1−Yt−n/mPn,(46)SQp=−1−Yt−n/mQp,where m∈ℤ∗, n∈ℤ, and p∈ε+ℤ.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
(47)ΔLn=FΔ0LnF−1=FLn⊗1+1⊗LnF−1=FLn⊗1F+F1⊗LnF=FF−n/mLn⊗1+FF1⊗Ln+n−mF1X1⊗Pm+nt=1⊗1−Ytn/mLn⊗1+1⊗Ln+n−mX1⊗1−Yt−1Pm+nt=Ln⊗1−Ytn/m+1⊗Ln+n−mX1⊗1−Yt−1Pm+nt,ΔQn=FΔ0QnF−1=FQn⊗1+1⊗QnF−1=FQn⊗1F+F1⊗QnF=FF−n/mQn⊗1+FF1⊗Qn=Qn⊗1−Ytn/m+1⊗Qn,SLn=−VLnU=−VUn/mLn+m−nUn/mX−n/m1Pm+nt=−VUn/mLn+m−nX−n/m1Pm+nt=−1−Yt−n/mLn+m−nX−n/m1Pm+nt,SQn=−VQnU=−VUn/mQn=−1−Yt−n/mQn.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 elementC can be exchanged. Namely, C,W=0.



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