# A Matrix Lie Superalgebra and Its Applications

Authors: Jingwei Han; Jing Yu; Jingsong He
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

A matrix Lie superalgebra is established. As its applications, multicomponent super
Ablowitz-Kaup-Newell-Segur (AKNS) equations and multicomponent super Dirac equations
are constructed. By making use of supertrace identity, their super-Hamiltonian structures
are presented, respectively.


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## Body

## 1. Introduction

It is well-known that by making use of the trace identity, many integrable equations can be written as the Hamiltonian forms, which was first proposed by Tu in [1]. The key ideas include the following three aspects. Firstly, from the finite-dimensional Lie algebra
(1)G={E1,…,Ed},
Tu constructs a loop Lie algebra
(2)G~={E1(n),…,Ed(n)},
where Ei(n)=Ei⊗λn=Eiλn. Secondly, an iso-spectral problem (λtn=0)
(3)ψx=U(u,λ)ψ
is considered, where λ is a spectral parameter and u is a potential. By solving the adjoint representation equation
(4)Vx=[U,V]
and zero curvature equations
(5)Utn-Vx(n)+[U,V(n)]=0,V(n)=(λnV)+,
where plus denotes the choice of the nonnegative power of λ, Tu obtains integrable equations. Lastly, by using the trace identity
(6)δδu〈V,∂U∂λ〉=(λ-γ∂∂λλγ)〈V,∂U∂u〉,
where γ is a undetermined constant, the obtained integrable equations can be written as the Hamiltonian form. Some well-known integrable Hamiltonian equations have been obtained by this method, such as AKNS equations, Kaup-Newll (KN) equations, and Wadati-Konno-Ichikawa (WKI) equations.This method for constructing Hamiltonian structures of integrable equations was extended to superintegrable equations by Hu in [2], where the supertrace identity
(7)δδu∫‍str(V∂U∂λ)dx=(λ-γ∂∂λλγ)str(∂U∂uV)
was first proposed by Hu in [3] and proved by Ma et al. in [4]. Similarly, many integrable super-Hamiltonian equations have also been constructed, such as super AKNS equations, super Dirac equations [4], super coupled Korteweg-de Vries (cKdV) equations [5, 6], and super KN equations [7, 8].It is a valuable generalization from one component to multicomponent in soliton equations because multicomponent soliton equations possess more complex structure and become more extensive than one-component ones. In physics, multicomponent integrable system is widely applied. For example, mixed N-coupled nonlinear Schrödinger equations (NLS), two-component Bose-Einstein condensates (BECs), and coupled Schrödinger equation are, respectively, discussed in [9–11], and they obtain many new results for these multicomponent equations. In mathematics, the inverse scattering method provides us a powerful tool to find multicomponent extensions of one-component soliton equations, like multicomponent AKNS equations [12, 13], multicomponent NLS equation [14], and so on [15–18]. Moreover, from the point of mathematics, we find that multicomponent generalizations mainly include the following aspects:(1)
symmetric space [14, 19, 20],(2)
matrix algebra [13, 21–23],(3)
soliton hierarchy associated with a matrix pseudodifferential operator [24, 25].Based on results from the above analysis, some questions are listed as follows.(1)
Do multicomponent superintegrable equations exist?(2)
If they exist, how can the multicomponent superintegrable equations be constructed?(3)
By making use of the supertrace identity (7), can multicomponent superintegrable equations be written as the super-Hamiltonian form?The purpose of this paper is to answer these questions. The paper is organized as follows. In the next section, we construct a matrix Lie superalgebraA(2m-1,m-1) [26]. As its applications, multicomponent super AKNS equations and multicomponent super Dirac equations are, respectively, constructed in Sections 3 and 4. Their super bi-Hamiltonian forms are also, respectively, constructed in these sections. Some conclusions and discussions are listed in the last section.


## 2. A Matrix Lie Superalgebra

Let us start with the following linear spaceG={e1,e2,e3,e4,e5}:
(8)e1=(EOOO-EOOOO),e2=(OEOOOOOOO),e3=(OOOEOOOOO),e4=(OOEOOOO-EO),e5=(OOOOOEEOO),[e1,e2}=2e2,[e1,e3}=-2e3,[e2,e3}=e1,[e1,e4}=e4,[e1,e5}=-e5,[e2,e4}=[e3,e5}=0,[e2,e5}=e4,[e3,e4}=e5,[e4,e4}=-2e2,[e4,e5}=e1,[e5,e5}=2e3,
where E is a m×m unit matrix, O is a m×m zero matrix, G0={e1,e2,e3} is even, G1={e4,e5} is odd, [a,b}=ab-(-1)p(a)p(b)ba is the super Lie bracket, and p(f) denotes the parity of the arbitrary element f.It is easy to prove that the linear spaceG is a matrix Lie superalgebra A(2m-1,m-1). The corresponding loop superalgebra G~ is presented as
(9)ei(n)=eiλn,[e1(m),e2(n)}=2e2(m+n),[e1(m),e3(n)}=-2e3(m+n),[e2(m),e3(n)}=e1(m+n),[e1(m),e4(n)}=e4(m+n),[e1(m),e5(n)}=-e5(m+n),[e2(m),e4(n)}=[e3(m),e5(n)}=0,[e2(m),e5(n)}=e4(m+n),[e3(m),e4(n)}=e5(m+n),[e4(m),e4(n)}=-2e2(m+n),[e4(m),e5(n)}=e1(m+n),[e5(m),e5(n)}=2e3(m+n),degei(n)=n,(i=1,2,3,4,5).


## 3. Multicomponent Super AKNS Hierarchy and Its Super-Hamiltonian Structure

In this section, we will derive multicomponent super AKNS hierarchy by the above matrix Lie superalgebra, and further, its super-Hamiltonian structure will be constructed.Let us consider the following spectral problem:(10)ϕx=U(u,λ)ϕ,U=-e1(1)+ve2(0)+we3(0)+αe4(0)+βe5(0),
which can be written in the following matrix form:
(11)ϕx=U(u,λ)ϕ=(-λEvαwλEββ-α0)ϕ,
where v=diag(v1,…,vm), w=diag(w1,…,wm), α=diag(α1,…,αm), β=diag(β1,…,βm), u=(v,w,α,β)T, p(λ)=p(vj)=p(wj)=0, andp(αj)=p(βj)=1 (1≤j≤m). Note that U(u,λ)∈A(2m-1,m-1). Taking
(12)V=(abρc-aδδ-ρ0),
where a=diag(a1,…,am),b=diag(b1,…,bm),c=diag(c1,…,cm),ρ=diag(ρ1,…,ρm), andδ=diag(δ1,…,δm), the adjoint representation equation (4) gives
(13)ak,x=vkck-wkbk+αkδk+βkρk,bk,x=-2λbk-2vkak-2αkρk,ck,x=2λck+2wkak+2βkδk,ρk,x=-λρk+vkδk-αkak-βkbk,δk,x=λδk+wkρk-αkck+βkak,
where 1≤k≤m. Let ak=∑j≥0ak(j)λ-j, bk=∑j≥0bk(j)λ-j, ck=∑j≥0ck(j)λ-j, ρk=∑j≥0ρk(j)λ-j, and δk=∑j≥0δk(j)λ-j, and then (13) becomes
(14)bk(0)=ck(0)=ρk(0)=δk(0)=0,ak,x(j)=vkck(j)-wkbk(j)+αkδk(j)+βkρk(j),j≥0,bk,x(j)=-2bk(j+1)-2vkak(j)-2αkρk(j),j≥0,ck,x(j)=2ck(j+1)+2wkak(j)+2βkδk(j),j≥0,ρk,x(j)=-ρk(j+1)+vkδk(j)-αkak(j)-βkbk(j),j≥0,δk,x(j)=δk(j+1)+wkρk(j)-αkck(j)+βkak(j),j≥0,
where 1≤k≤m. Moreover, we find that (14) can be written as the following recursive form:
(15)(c1(j+1)⋱cm(j+1)b1(j+1)⋱bm(j+1)2δ1(j+1)⋱2δm(j+1)-2ρ1(j+1)⋱-2ρm(j+1))=ℒ(c1(j)⋱cm(j)b1(j)⋱bm(j)2δ1(j)⋱2δm(j)-2ρ1(j)⋱-2ρm(j)),
where the recursive operator is given by
(16)ℒ=(ℒ11ℒ12ℒ13ℒ14ℒ21ℒ22ℒ23ℒ24ℒ31ℒ32ℒ33ℒ34ℒ41ℒ42ℒ43ℒ44),
where
(17)ℒ11=diag(12∂-w1∂-1v1,…,12∂-wm∂-1vm12),ℒ12=diag(w1∂-1w1,…,wm∂-1wm),ℒ13=diag(-12β1-12w1∂-1α1,…,hhhhhh-12βm-12wm∂-1αm),ℒ14=diag(12w1∂-1β1,…,12wm∂-1βm),ℒ21=diag(-v1∂-1v1,…,-vm∂-1vm),ℒ22=diag(-12∂+v1∂-1w1,…,-12∂+vm∂-1wm),ℒ23=diag(-12v1∂-1α1,…,-12vm∂-1αm),ℒ24=diag(12α1+12v1∂-1β1,…,12αm+12vm∂-1βm),ℒ31=diag(2α1-2β1∂-1v1,…,2αm-2βm∂-1vm),ℒ32=diag(2β1∂-1w1,…,2βm∂-1wm),ℒ33=diag(∂-β1∂-1α1,…,∂-βm∂-1αm),ℒ34=diag(w1+β1∂-1β1,…,wm+βm∂-1βm),ℒ41=diag(2α1∂-1v1,…,2αm∂-1vm),ℒ42=diag(2β1-2α1∂-1w1,…,2βm-2αm∂-1wm),ℒ43=diag(-v1+α1∂-1α1,…,-vm+αm∂-1αm),ℒ44=diag(-∂-α1∂-1β1,…,-∂-αm∂-1βm).After a direct calculation, we obtain thatak,x(0)=0. Choosing ak(0)=-1(1≤k≤m) and constant of integration to be zero, the first few terms are listed as follows:
(18)ak(1)=0,bk(1)=vk,ck(1)=wk,ρk(1)=αk,δk(1)=βk,ak(2)=12vkwk+αkβk,bk(2)=-12vk,x,ck(2)=12wk,x,ρk(2)=-αk,x,δk(2)=βk,x,
where 1≤k≤m.Then, let us consider the spectral problem (11) with the following auxiliary spectral problem:
(19)ϕtn=V(n)ϕ,V(n)=(λnV)+.
The compatibility conditions of (11) and (19), that is, the zero curvature equations (5) give an infinite hierarchy of nonlinear partial differential equations:
(20)vk,tn=-2bk(n+1),1≤k≤m,wk,tn=2ck(n+1),1≤k≤m,αk,tn=-ρk(n+1),1≤k≤m,βk,tn=δk(n+1),1≤k≤m,
or
(21)vtn=-2b(n+1),wtn=2c(n+1),αtn=-ρ(n+1),βtn=δ(n+1),
where a(j)=diag(a1(j),…,am(j)),  b(j)=diag(b1(j),…,bm(j)), c(j)=diag(c1(j),…,cm(j)), ρ(j)=diag(ρ1(j),…,ρm(j)), δ(j)=diag(δ1(j),…,δm(j)), with j≥0. When m=1, (21) is equivalent to the super AKNS soliton hierarchy [27–30], and thus (21) is called multicomponent super AKNS hierarchy.In what follows, a super-Hamiltonian structure of (21) is derived by means of the supertrace identity (7). To this end, the following quantities are needed:
(22)str(V∂U∂λ)=-2tra,str(∂U∂vV)=c,str(∂U∂wV)=b,str(∂U∂αV)=2δ,str(∂U∂βV)=-2ρ.
Thus, the supertrace identity (7) gives the following equality:
(23)(δδv1⋱δδvmδδw1⋱δδwmδδα1⋱δδαmδδβ1⋱δδβm)∫‍-2∑k=1m‍∑j≥0‍ak(j)λ-jdx=(λ-γ∂∂λλγ)(∑j≥0c1(j)λ-j⋱∑j≥0‍cm(j)λ-j∑j≥0‍b1(j)λ-j⋱∑j≥0‍bm(j)λ-j2∑j≥0‍δ1(j)λ-j⋱2∑j≥0‍δm(j)λ-j-2∑j≥0‍ρ1(j)λ-j⋱-2∑j≥0‍ρm(j)λ-j).
Equating the coefficients of λ-j-2 on two sides of the above equality, we have
(24)(δδv1⋱δδvmδδw1⋱δδwmδδα1⋱δδαmδδβ1⋱δδβm)∫‍-2∑k=1m‍ak(j+2)dx=(γ-j-1)(c1(j+1)⋱cm(j+1)b1(j+1)⋱bm(j+1)2δ1(j+1)⋱2δm(j+1)-2ρ1(j+1)⋱-2ρm(j+1)).
By taking j=0, we obtain that the constant γ=0. Thus, we have
(25)(c1(j+1)⋱cm(j+1)b1(j+1)⋱bm(j+1)2δ1(j+1)⋱2δm(j+1)-2ρ1(j+1)⋱-2ρm(j+1))=(δδv1⋱δδvmδδw1⋱δδwmδδα1⋱δδαmδδβ1⋱δδβm)Hj,
where Hj=∫‍(2/(j+1))∑k=1mak(j+2)dx.Therefore, (21) can be written in the following super bi-Hamiltonian form:
(26)(v1⋱vmw1⋱wmα1⋱αmβ1⋱βm)tn=J(c1(n+1)⋱cm(n+1)b1(n+1)⋱bm(n+1)2δ1(n+1)⋱2δm(n+1)-2ρ1(n+1)⋱-2ρm(n+1))=JδδuHn=JℒδδuHn-1,
where J=(0-2E002E000000(1/2)E00(1/2)E0) is the supersymplectic operator.Example 1.
Letm=2,n=2 in (21), and we have
(27)v1,t2=-12v1,xx+v12w1+2v1α1β1-2α1α1,x,v2,t2=-12v2,xx+v22w2+2v2α2β2-2α2α2,x,w1,t2=12w1,xx-v1w12-2w1α1β1-2β1β1,x,w2,t2=12w2,xx-v2w22-2w2α2β2-2β2β2,x,α1,t2=-α1,xx-v1β1,x+12v1w1α1-12v1,xβ1,α2,t2=-α2,xx-v2β2,x+12v2w2α2-12v2,xβ2,β1,t2=β1,xx+w1α1,x+12w1,xα1-12v1w1β1,β2,t2=β2,xx+w2α2,x+12w2,xα2-12v2w2β2,
which is the first nonlinear two-component super AKNS equations.


## 4. MultiComponent Super Dirac Hierarchy and Its Super-Hamiltonian Structure

As another application of the matrix Lie superalgebra, multicomponent super Dirac hierarchy will be constructed, and further, its super-Hamiltonian structure will also be obtained.Let us consider the following spectral problem:(28)ϕx=U(u~,λ)ϕ,U=e2(1)-e3(1)+re1(0)+se2(0)+se3(0)+αe4(0)+βe5(0),
which can be written in the following matrix form:
(29)ϕx=U(u~,λ)ϕ=(rλE+sα-λE+s-rββ-α0)ϕ,
where r=diag(r1,…,rm), s=diag(s1,…,sm), α=diag(α1,…,αm), β=diag(β1,…,βm), u~=(r,s,α,β)T,  p(λ)=p(ri)=p(si)=0, and p(αi)=p(βi)=1  (1≤i≤m). U(u~,λ)∈A(2m-1,m-1). Solving the adjoint representation equation (4), where
(30)V=(CA+BρA-B-Cδδ-ρ0),
with A=diag(A1,…,Am), B=diag(B1,…,Bm), C=diag(C1,…,Cm), ρ=diag(ρ1,…,ρm), and  δ=diag(δ1,…,δm), we have
(31)Ak,x=2rkBk-2λCk-αkρk+βkδk,Bk,x=2rkAk-2skCk-αkρk-βkδk,Ck,x=2λAk-2skBk+αkδk+βkρk,ρk,x=λδk+rkρk+skδk-Ckαk-βkAk-βkBk,δk,x=-λρk+skρk-rkδk-αkAk+αkBk+βkCk,
where 1≤k≤m. Let Ak=∑j≥0Ak(j)λ-j, Bk=∑j≥0Bk(j)λ-j, Ck=∑j≥0Ck(j)λ-j, ρk=∑j≥0ρk(j)λ-j, and δk=∑j≥0δk(j)λ-j, and then we have
(32)Ak(0)=Ck(0)=ρk(0)=δk(0)=0,Ak,x(j)=-2Ck(j+1)+2rkBk(j)-αkρk(j)+βkδk(j),Bk,x(j)=2rkAk(j)-2skCk(j)-αkρk(j)-βkδk(j),Ck,x(j)=2Ak(j+1)-2skBk(j)+αkδk(j)+βkρk(j),ρk,x(j)=δk(j+1)+rkρk(j)+skδk(j)-αkCk(j)-βkAk(j)-βkBk(j),δk,x(j)=-ρk(j+1)+skρk(j)-rkδk(j)-αkAk(j)+αkBk(j)+βkCk(j),
where j≥0,1≤k≤m, which can be written as a recursive form:
(33)(C1(j+1)⋱Cm(j+1)A1(j+1)⋱Am(j+1)δ1(j+1)⋱δm(j+1)-ρ1(j+1)⋱-ρm(j+1))=L(C1(j)⋱Cm(j)A1(j)⋱Am(j)δ1(j)⋱δm(j)-ρ1(j)⋱-ρm(j)),
where the recursive operator is given by
(34)L=(L11L12L13L14L21L22L23L24L31L32L33L34L41L42L43L44),
with
(35)L11=diag(-2r1∂-1s1,…,-2rm∂-1sm),L12=diag(-12∂+2r1∂-1r1,…,-12∂+2rm∂-1rm)L13=diag(12β1-r1∂-1β1,…,12βm-rm∂-1βm12),L14=diag(12α1+r1∂-1α1,…,12αm+rm∂-1αm12),L21=diag(12∂-2s1∂-1s1,…,12∂-2sm∂-1sm12),L22=diag(2s1∂-1r1,…,2sm∂-1rm),L23=diag(-12α1-s1∂-1β1,…,-12αm-sm∂-1βm),L24=diag(12β1+s1∂-1α1,…,12βm+sm∂-1αm12),L31=diag(α1-2β1∂-1s1,…,αm-2βm∂-1sm),L32=diag(β1+2β1∂-1r1,…,βm+2βm∂-1rm),L33=diag(-s1-β1∂-1β1,…,-sm-βm∂-1βm),L34=diag(-∂+r1+β1∂-1α1,…,-∂+rm+βm∂-1αm),L41=diag(-β1+2α1∂-1s1,…,-βm+2αm∂-1sm),L42=diag(α1-2α1∂-1r1,…,αm-2αm∂-1rm),L43=diag(∂+r1+α1∂-1β1,…,∂+rm+αm∂-1βm),L44=diag(s1-α1∂-1α1,…,sm-αm∂-1αm).It is easy to verify thatBk,x(0)=0(1≤k≤m). Choosing Bk(0)=1 and constant of integration to be zero, the first few terms can be worked out as follows:
(36)Ak(1)=sk,Bk(1)=0,Ck(1)=rk,ρk(1)=αk,δk(1)=βk,Ak(2)=12rk,x,Bk(2)=12(rk2+sk2)+αkβk,Ck(2)=-12sk,x,ρk(2)=-βk,x,δk(2)=αk,x.In what follows, the spectral problem (29) is associated with the auxiliary spectral problem:
(37)ϕtn=V(n)ϕ,
where(38)V(n)=∑j=1n‍(C1(j)A1(j)+B1(j)ρ1(j)⋱⋱⋱Cm(j)Am(j)+Bm(j)ρm(j)A1(j)-B1(j)-C1(j)δ1(j)⋱⋱⋱Am(j)-Bm(j)-Cm(j)δm(j)δ1(j)-ρ1(j)0⋱⋱⋱δm(j)-ρm(j)0)λn-j.The compatibility condition between (29) and (37) gives the following super nonlinear soliton hierarchy:
(39)rk,tn=2Ak(n+1),sk,tn=-2Ck(n+1),αk,tn=δk(n+1),βk,tn=-ρk(n+1),
where 1≤k≤m. By denoting A(j)=diag(A1(j),…,Am(j)), B(j)=diag(B1(j),…,Bm(j)), C(j)=diag(C1(j),…,Cm(j)), ρ(j)=diag(ρ1(j),…,ρm(j)), and δ(j)=diag(δ1(j),…,δm(j)), with j≥0, and (39) can be written as follows:
(40)rtn=2A(n+1),stn=-2C(n+1),αtn=δ(n+1),βtn=-ρ(n+1).
Similarly, when m=1, (40) is equivalent to the super Dirac soliton hierarchy [4, 30], and thus (40) is called multicomponent super Dirac hierarchy.To derive super bi-Hamiltonian structure of (40), we need to use the supertrace identity (7). To this end, we firstly obtain the following equalities:
(41)str(V∂U∂λ)=-2trB,str(∂U∂sV)=2A,str(∂U∂rV)=2C,str(∂U∂αV)=2δ,str(∂U∂βV)=-2ρ.
Thus, the supertrace identity (7) becomes the following equality:
(42)(δδr1⋱δδrmδδs1⋱δδsmδδα1⋱δδαmδδβ1⋱δδβm)∫‍-∑k=1m‍∑j≥0‍Bk(j)λ-jdx=(λ-γ∂∂λλγ)(∑j≥0‍C1(j)λ-j⋱∑j≥0‍Cm(j)λ-j∑j≥0‍A1(j)λ-j⋱∑j≥0‍Am(j)λ-j∑j≥0‍δ1(j)λ-j⋱∑j≥0‍δm(j)λ-j-∑j≥0‍ρ1(j)λ-j⋱-∑j≥0‍ρm(j)λ-j).
Equating the coefficients of λ-j-2 in the above equality, we have
(43)(δδr1⋱δδrmδδs1⋱δδsmδδα1⋱δδαmδδβ1⋱δδβm)∫‍-∑k=1m‍Bk(j+2)dx=(γ-j-1)(C1(j+1)⋱Cm(j+1)A1(j+1)⋱Am(j+1)δ1(j+1)⋱δm(j+1)-ρ1(j+1)⋱-ρm(j+1)).
Let j=0, and we have γ=0. Thus, we obtain
(44)(C1(j+1)⋱Cm(j+1)A1(j+1)⋱Am(j+1)δ1(j+1)⋱δm(j+1)-ρ1(j+1)⋱-ρm(j+1))=(δδr1⋱δδrmδδs1⋱δδsmδδα1⋱δδαmδδβ1⋱δδβm)H~j,H~j=∫‍1j+1∑k=1m‍Bk(j+2)dx.
Therefore, (39) or (40) can be written in the following super bi-Hamiltonian structure:
(45)(r1⋱rms1⋱smα1⋱αmβ1⋱βm)tn=J~(C1(n+1)⋱Cm(n+1)A1(n+1)⋱Am(n+1)δ1(n+1)⋱δm(n+1)-ρ1(n+1)⋱-ρm(n+1))=J~δδu~H~n=J~Lδδu~H~n-1,
where J~=(02E00-2E00000E0000E) is the supersymplectic operator.Example 2.
Letm=2, n=2 in (45), and we obtain two-component super Dirac equations:
(46)r1,t2=-12s1,xx+r12s1+s13+2s1α1β1-α1α1,x+β1β1,x,r2,t2=-12s2,xx+r22s2+s23+2s2α2β2-α2α2,x+β2β2,x,s1,t2=12r1,xx-r13-r1s12-2r1α1β1-α1β1,x+α1,xβ1,s2,t2=12r2,xx-r23-r2s22-2r2α2β2-α2β2,x+α2,xβ2,α1,t2=-β1,xx+r1β1,x-s1α1,x-12α1s1,x+12β1r1,x+12r12β1+12s12β1,α2,t2=-β2,xx+r2β2,x-s2α2,x-12α2s2,x+12β2r2,x+12r22β2+12s22β2,β1,t2=α1,xx+s1β1,x+r1α1,x+12α1r1,x-12r12α1-12s12α1+12β1s1,x,β2,t2=α2,xx+s2β2,x+r2α2,x+12α2r2,x-12r22α2-12s22α2+12β2s2,x.


## 5. Conclusions and Discussions

Starting from the matrix Lie superalgebraA(2m-1,m-1), we have, respectively, constructed multicomponent super AKNS equations (21) and multicomponent super Dirac equations (40). By making use of the supertrace identity (7), (21), and (40) have been rewritten as integrable super-Hamiltonian forms (26) and (45), respectively. Moreover, we believe that many multicomponent superintegrable equations can also be constructed, which may be helpful to many physical and mathematical researchers in their future work.



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*Source: 416520-2013-12-19.xml*