On boundary superalgebras

On boundary superalgebrasAnastasia Doikou Citation: J. Math. Phys. 51, 043509 (2010); doi: 10.1063/1.3359005 View online: http://dx.doi.org/10.1063/1.3359005 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v51/i4 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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On boundary superalgebrasAnastasia Doikoua�

Department of Engineering Sciences, University of Patras, GR-26500 Patras, Greece

�Received 19 November 2009; accepted 17 February 2010; published online 15 April 2010�

We examine the symmetry breaking of superalgebras due to the presence of appro-priate integrable boundary conditions. We investigate the boundary breaking sym-metry associated with both reflection algebras and twisted super-Yangians. Weextract the generators of the resulting boundary symmetry as well as we provideexplicit expressions of the associated Casimir operators. © 2010 American Instituteof Physics. �doi:10.1063/1.3359005�

I. INTRODUCTION

Symmetry breaking processes are of the most fundamental concepts in physics. It was shownin a series of earlier works �see, e.g., Refs. 1–9�, within the context of quantum integrability, thatthe presence of suitable boundary conditions may break a symmetry down without spoiling theintegrability of the system. It was also shown in Ref. 10 that the breaking symmetry mechanismdue to the presence of integrable boundary conditions may be utilized to provide certain centrallyextended algebras. Here, we shall investigate in detail the resulting boundary symmetries in thecontext of quantum integrable models associated with various superalgebras.

More precisely, the main aim of the present work is the study of the algebraic structuresunderlying quantum integrable systems associated with certain superalgebras, Y�gl�m �n�� andUq�gl�m �n��, once nontrivial boundary conditions are implemented. In this investigation we shallfocus on the relevant algebraic content, and our primary objectives will be the study of the relatedexact symmetries as well as the construction of the relevant Casimir operators. The existence ofcentrally extended superalgebras emerging from these boundary algebras is one of the mainmotivations for the present investigation and will be discussed in full detail in a forthcoming workgiven that is a separate significant topic.

We shall focus here on the super-Yangian Y�gl�m �n�� and its q-deformed counterpart theUq�gl�m �n�� algebra. It is necessary to first introduce some useful notation associated with supe-ralgebras. Consider the m+n dimensional column vectors ei, with 1 at position i and zero every-where else, and the �m+n�� m+n� eij matrices: �eij�kl=�ik� jl. Then define the grades

�ei� = �i�, �eij� = �i� + �j� . �1.1�

The tensor product is also graded as

�Aij � Akl��Amn � Apq� = �− 1��k�+�l��m�+�n��AijAmn � AklApq. �1.2�

Define also the transposition as

AT = �i,j=1

m+n

�− 1��i��j�+�j�eji � Aij, A = �i,j=1

m+n

eij � Aij , �1.3�

and the supertrace as

a�Electronic mail: [email protected].

JOURNAL OF MATHEMATICAL PHYSICS 51, 043509 �2010�

51, 043509-10022-2488/2010/51�4�/043509/16/$30.00 © 2010 American Institute of Physics


http://dx.doi.org/10.1063/1.3359005

http://dx.doi.org/10.1063/1.3359005

str A = �i

�− 1��i�Aii. �1.4�

It will also be convenient for our purposes here to define the supertransposition as

At = V−1ATV , �1.5�

where the matrix V will be defined later in the text when appropriate. Also it is convenient forwhat follows to introduce the distinguished and symmetric grading, corresponding apparently tothe distinguished and symmetric Dynkin diagrams. In the distinguished grading, we define

�i� = �0, 1 � i � m

1, m + 1 � i � m + n.� �1.6�

In the gl�m �2k� we also define the symmetric grading as

�i� = �0, 1 � i � k,m + k + 1 � i � m + 2k

1, k + 1 � i � m + k.� �1.7�

II. THE SUPER-YANGIAN Y„gl„m n……

Let us first introduce the basic algebraic objects associated with the Yangian Y�gl�m �n��. TheR matrix solution of the Yang–Baxter equation11 associated with Y�gl�m �n�� is12–15

R�� = � + iP , �2.1�

where P is the superpermutation operator defined as

P = �i,j

�− 1��j�eij � eji. �2.2�

Also define

R12�� ª R12t1 �� − i��, R21�� ª R12

t2 �− � − i��, and R12�� = R21�� ,

� = − � − i� and � =n − m

2. �2.3�

The R matrix may be written as

R12�� = � + iQ12, �2.4�

where Q is a projector satisfying

Q2 = 2�Q, PQ = QP = Q . �2.5�

Consider also the L-operator expressed as

L�� = � + iP, P = �a,b

eab � Pab �2.6�

with Pab�gl�m �n�. L is a solution of the equation

R12��1 − �2�L1��1�L2��2� = L2��2�L1��1�R12��1 − �2� �2.7�

with R being the matrix above �2.1�. The algebra defined by �2.7� is equipped with a co product:let L��=�i,jeij � lij��,

043509-2 Anastasia Doikou J. Math. Phys. 51, 043509 �2010�


��L�� = L02��L01�� ⇒ ��lil�� = �j

ljl�� lij�� . �2.8�

Define also the opposite coproduct. Let � be the “shift operator” � :V1 � V2�V2 � V1,

�� = � � � �2.9�

in particular,

��L�� = L01��L02�� ⇒ ��lil�� = �j

�− 1��i�+�j��j�+�l��lij�� ljl�� . �2.10�

The L coproducts are derived by iteration as

��L� = �id � ��L−1��, ��L� = �id � ��L−1��. �2.11�

Let us now define the supercommutator as

�A,B�� = AB − �− 1��A��B�AB . �2.12�

It is easy to show from �2.7� that Pab satisfies the gl�m �n� algebra, which reads as

�Pij,Pkl�� = 0, k � j, i � l ,

�Pij,Pki�� = �− 1��i�Pkj, k � j ,

�Pij,P jl�� = − �− 1��i��j�+�l��+�j��l�Pil, i � l ,

�Pij,P ji�� = �− 1��i��P j j − Pii� . �2.13�

A. The reflection algebra

This section serves mostly as a warm up, although some alternative proofs for the symmetriesare provided, and explicit expressions of quadratic Casimir operators are also given. Consider nowthe situation of a boundary integrable system described by the reflection equation,16,17 which alsoprovides the exchange relations of the underlying algebra, i.e., the reflection algebra

R12��1 − �2�K1��1�R21��1 + �2�K2��2� = K2��2�R12��1 + �2�K1��1�R21��1 − �2� . �2.14�

As shown in Ref. 17, a tensorial type representation of the reflection algebra is given by

T�� = T��K��T�� , �2.15�

where we define

T�� = T−1�− �� and T�� = ��N+1��L� = L0N��, . . . ,L01�� , �2.16�

where K is a c-number solution of the reflection equation.The associated transfer matrix is defined as

t�� = strK+��T�� 2.17�

where K+ is also a solution of the reflection equation, and

�t��,t�� = 0. �2.18�

043509-3 Boundary superalgebras J. Math. Phys. 51, 043509 �2010�


In the special case where K+=K= I, the transfer matrix enjoys the full gl�m �n� symmetry. Herewe shall provide an explicit proof based on the linear relations satisfied by the algebra coproductsand the matrix T. The proof of this statement goes as follows: let us first recall the coproduct ofthe gl�m �n� elements

��Pab� = I � Pab + Pab � I . �2.19�

Define also the following representation � :gl�m �n��End�Cn+m� such that ��Pab�= Pab.The N+1 coproduct satisfies the following commutation relations with the monodromy ma-

trix:

�� id�N��N+1��Pab�T�� = T�� id�N��N+1��Pab� . �2.20�

The latter relations may be written in a more straightforward form as

�Pab � I + I � ��N��Pab��T�� = T��Pab � I + I � ��N��Pab�� . �2.21�

It is also clear that T−1�−�� also satisfies relation �2.21�, so for K= I, it is quite straightforward toshow that �recall also that Pab= �−1��b�eab�

��− 1��b�eab � I + I � ��N��Pab��T�� = T��− 1��b�eab � I + I � ��N��Pab�� . �2.22�

If we now express T��=�i,jeij � Tij��, then the latter relations become

�j

�− 1��b�eaj � Tbj�� + �i,j

�− 1��a�+�b��i�+�j��eij � ��N��Pab�Tij��

= �i

�− 1��b�+��a�+�b��a�+�i��eib � Tia�� + �i,j

eij � Tij��N��Pab� . �2.23�

We are, however, interested in the supertrace over the auxiliary space, so we are dealing basicallywith the diagonal terms of the above equation, hence we obtain the following exchange relations:

�Tii��,��N��Pab�� = 0, i � a, i � b ,

�Taa��,��N��Pab�� = �− 1��b�Tba��, �Tbb��,��N��Pab�� = − �− 1��a�Tba�� . �2.24�

By taking now the supertrace �we are considering K+= I�, we have

��i

�− 1��i�Tii��,��N��Pab� = ��− 1��a�Taa�� + �− 1��b�Tbb,��N��Pab�� = ¯ = 0

⇒ �t��,��N��Pab�� = 0 �2.25�

and, consequently,

�t��,gl�m�n�� = 0. �2.26�

Recall that here we are focusing on the distinguished Dynkin diagram �1.6�. Consider now thenontrivial situation where the K matrix has the following diagonal form �see also Ref. 4�:

K�� = diag�1, . . . 1m1

,− 1, . . . − 1m2+n2

,1, . . . ,1n1

� �2.27�

such that m=m1+m2 and n=n1+n2. In general, any solution4 may be written in the form K��= i+�E, E2= I. E may be diagonalized into �2.27� and that is why we make this convenient choicefor the K matrix �2.27�, we also choose, for simplicity, =0.

Let us first extract the nonlocal charges for any generic K matrix of the form



K�� = K +1

�1 +

1

�22 + O� 1

�3� �2.28�

�see also Ref. 18 for a brief discussion on the symmetry�. From the asymptotic behavior of thedynamical T, we then obtain

T�� → � � K +i

�Q�0� −

1

�2Q�1� + ¯ . �2.29�

The first order quantity provides the generators of the remaining boundary symmetry,

Q�0� = ��N��P�K + K��N��P� + 1, �2.30�

and for the special choice of K matrix �2.27�, we conclude

Q�0� = �i,j=1

m1

eij � ��N��Pij� + �i,j=m+n2

m+n

eij � ��N��Pij� − �i,j=1m1+1

m+n2

eij � ��N��Pij�

+ �i=1

m1

�j=m+n2+1

m+n

eij � ��N��Pij� + �i=m+n2+1

m+n

�j=1

m1

eij � ��N��Pij� . �2.31�

More precisely, the elements

Pij, i, j � �1,m1� � �m + n2 + 1,m + n� form the gl�m1�n1� ,

Pij, i, j � �m1 + 1,m + n2� form the gl�m2�n2� . �2.32�

These are exactly the generators that, in the fundamental representation, commute with the Kmatrix �2.27�.

That is, the gl�m �n� symmetry breaks down to gl�m1 �n1� � gl�m2 �n2�. Since the K matrixcommutes with all the above generators following the procedure above, we can show relations�2.21� but only with the generators �2.32� and finally

�t��,gl�m1�n1� � gl�m2�n2�� = 0. �2.33�

We shall focus now for simplicity on the “one-particle” representation N=1. The trace of thesecond order quantity provides the quadratic quadratic Casimir associated with the gl�m �n� in thecase where K=1. When K is of the diagonal form �2.27� the Casimir �see also Refs. 19 and 20� isassociated with gl�m1 �n1� � gl�m2 �n2�.

More specifically, for K� I �set N=1�,

Q�1� = 2P2 and C = str Q�1� = 2 �i,j=1

m+n

�− 1��j�PijP ji, �2.34�

and for K given by the diagonal matrix �2.28�,

Q�1� = PKP + KP2 − iP1 − i1P − 2 and C = str Q�1�. �2.35�

For the special choice of K matrix �2.27�, we have



C = �i=1

m1 ��j=1

m1

�− 1��j�PijP ji + �i=m+n2

m+n

�− 1��j�PijP ji� + �j=1

m1 ��i=1

m1

�− 1��j�PijP ji + �i=m+n2

m+n

�− 1��j�PijP ji�− �

i,j=m1+1

m+n2

�− 1��j�PijP ji. �2.36�

Higher Casimir operators may be extracted by considering the higher order terms in the expansionof the transfer matrix in powers of 1 /�. More precisely, let us focus on the N=1 case and see moreprecisely how one obtains the higher Casimir operators from the expansion of the transfer matrixt��=�k=1

2N t�k−1� /�k. Recall the N=1 representation of the reflection algebra

T�� = L��kL�� = �1 +i

�P�k�1 +

i

�P −

1

�2P2 −

i

�3P3 +

1

�4P4. . .�

= k +i

��Pk + kP� −

1

�2 �PkP + kP2� −1

�2 �PkP2 + kP3� ¯ , �2.37�

where k is diagonal, then

t�k−1� � �a,b

�PabkbbPbak−1 + kaaPaa

k � . �2.38�

All t�k�’s are the higher Casimir quantities and, by construction, they commute with each other andthey commute as shown earlier with the exact symmetry of the system. Depending on the rank ofthe considered algebra, the expansion of t�� should truncate at some point; note that expressions�2.37� and �2.38� are generic and hold for any gl�m �n�. The spectra of all Casimir operatorsassociated with a specific algebra may be derived via the Bethe ansatz methodology. In particular,the spectrum for generic representations of supersymmetric algebras is known �see, e.g., Ref. 21�.By appropriately expanding the eigenvalues in powers of 1 /�, we may identify the spectrum ofeach one of the relevant Casimir operators.

B. The twisted super-Yangian

Note that we focus here on the gl�m �2k� case and the symmetric Dynkin diagram �1.7�. Let usfirst define some basic notation useful or our purposes here. Consider the matrix

V = �i

f ieii, where i = m + 2k − i + 1. �2.39�

More precisely, we shall consider here the following antidiagonal matrix:

V = antidiag�1, . . . ,1m+k

,− 1, . . . − 1k

� . �2.40�

The twisted super-Yangian defined in Refs. 22–24 �for more details on the physical meaningof reflection algebra and twisted Yangian, see Refs. 25 and 4�,

R12��1 − �2�K1��1�R12��1 − �2�K2��2� = K2��2�R12��1 − �2�K1��1�R12��1 − �2� , �2.41�

the matrix R12 is defined in �2.3�.Define also

L0n�� = L0nt0 �− � − i�� 1 +

i

�P0n, where P0n = � − P0n

t0 . �2.42�

Consider now the generic tensorial representation of the twisted super-Yangian,



T�� = T��K��Tt0�− � − i�� , �2.43�

where K is a c-number solution of the twisted Yangian �2.41�.As in Sec. II A, express the above tensor representation in powers of 1 /�,

T�� = 1 +i

��Q�0� + N�� −

1

�2Q�1� + ¯ , �2.44�

where

Q�0� = ��N��P� − ��N��Pt0� . �2.45�

It is clear that the elements Qab form the osp�m �2n� algebra, and this corresponds essentially to afolding of the gl�m �2n� to osp�m �2n�. Such a folding occurs in the corresponding symmetricDynkin diagram. Henceforth, we shall consider the simplest solution K� I, although a full classi-fication is presented in Ref. 4. Based on the same logic as in the previous paragraph, we mayextract the corresponding exact symmetry. We have in this case

�� id�N��N+1��Qab�0��T�� = T�� id�N��N+1��Qab

�0�� , �2.46�

which leads to

�t��,osp�m�2n�� = 0 �2.47�

so the exact symmetry of the considered transfer matrix is indeed osp�m �2n�.The quadratic Casimir operator associated with osp�m �n� emerges from the supertrace of Q�1�

�N=1�,

Q�1� = PP, C = str Q�1� = �i,j

�− 1��j�PijP ji. �2.48�

III. THE Uq„gl„m n…… ALGEBRA

We come now to the q deformed situation. The R matrix associated with the Uq�gl�m �n��algebra is given by the following expressions:26

R�� = �i=1

m+n

ai��eii � eii + b�� i�j=1

m+n

eii � ejj + �i�j=1

m+n

cij��eij � eji, �3.1�

where we define

aj�� = sinh�� + i� − 2i��j��, b�� = sinh �, cij�� = sinh�i��esign�j−i��− 1��j�. �3.2�

Let us now introduce the super symmetric Lax operator associated with Uq�gl�m �n��,

L�� = e�L+ − e−�L− �3.3�

and L satisfies the fundamental algebraic relation �2.7� with the R matrix given in �3.1�. Theelements L satisfy27,28

R12 L1

+L+ = L2+L1

+R12 ,

R12 L1

−L− = L2−L1

−R12 ,

R12 L1

L2� = L2

�L1 R12

, �3.4�

where L are expressed as



L+ = �i�j

eij � lij+, L− = �

i�j

eij � lij− . �3.5�

Definitions of lij and li j

in terms of the Uq�gl�m �n�� algebra generators in the Chevalley–Serrebasis are given in the Appendix. The above equation �3.4� provides all the exchange relations ofthe Uq�gl�m �n�� algebra. This is, in fact, the so-called FRT realization of the Uq�gl�m �n�� algebra�see, e.g., Ref. 27�.

A. The reflection algebra

Our main objective here is to extract the exact symmetry of the open transfer matrix associ-

ated with Uq�gl�m �n��. We shall focus in this section on the distinguished Dynkin diagram. Theopen transfer matrix is given by �2.17�, and from now, on we consider for our purposes here theleft boundary to be the trivial solution

K+ = M = �k=1

m+n

qn+m−2k+1q−2�k�+4�i=1k �i�ekk. �3.6�

The elements one extracts from the asymptotic expansion of T by keeping the leading contribu-tion, Tab

, are the boundary nonlocal charges, which form the boundary superalgebra with ex-change relations dictated by

R12 T1

R21 T2

= T2 R21

T1 R12

. �3.7�

In general, it may be shown that the boundary superalgebra is an exact symmetry of the doublerow transfer matrix. Indeed, by introducing the element

� = str�eabT � = �− 1��a�Tba

, �3.8�

it is quite straightforward to show along the lines described in Ref. 8 that

�� ,t�� = 0 ⇒ �Tab ,t�� = 0. �3.9�

Let

T = �ab

eab � Tab , T = �

ab

eab � Tab and Tab

= ��N��lab �, Tab

= ��N��lab � , �3.10�

see also the Appendix for the definitions of lij and li j

.The boundary nonlocal charges emanating from T are of the explicit form,

Tad = �

b,c�− 1��a�+�b��b�+�d��Tab

Kbc+ Tcd

. �3.11�

It is clear that different choices of K matrix lead to different nonlocal charges and, consequently,to different symmetries. Also, the quadratic Casimir operators are obtained by

C = str0�M0T0 � = �

a,b,c�− 1��b�MaaTab

Kbc Tca

. �3.12�

Explicit expressions for the quadratic Casimir operators will be given below for particular simpleexamples. Note that higher Casimir operators may be extracted from the higher order terms in theexpansion of the transfer matrix in powers of e 2�.

1. Diagonal reflection matrices

We shall distinguish two main cases of diagonal matrices, and we shall examine the corre-sponding exact symmetry. First consider the simplest boundary conditions described by K� I, we



shall explicitly show that the associated symmetry is the Uq�gl�m �n��. One could just notice thatthe resulted T form essentially the Uq�gl�m �n��, but this is not quite obvious. Then via �3.9� onecan show the exact symmetry of the transfer matrix. Let us, however, here follow a more straight-forward and clean approach regarding the symmetry, that is, we shall show that the open transfermatrix commutes with each one of the algebra generators. Our proofs hold for generic nontrivialintegrable boundary conditions as will be more transparent subsequently. In fact, in this and theprevious section we prepare somehow the general algebraic setting so that we may extract thesymmetry for generic integrable boundary conditions.

Let us now set the basic algebraic machinery necessary for the proofs that follow. Our aimnow is to show that the double row transfer matrix with trivial boundary conditions, that is, K= I , K�L�=M, enjoys the full Uq�gl�m �n�� symmetry. We shall show, in particular, that the opentransfer matrix commutes with each one of the generators of Uq�gl�m �n��. Let us outline theproof; first it is quite easy to show from the form of the coproduct for �i and following the logicdescribed in Sec. III A �see Eq. �2.21�� that

��N��i�,t�� = 0. �3.13�

We may now show the commutation between the transfer matrix and the other elements of thesuperalgebra. Introduce the representation � :Uq�gl�m �n��End�CN�,

��ei� = �i, ��f i� = �i, ��qhi/2� = ki. �3.14�

Consider also the following more convenient notation:

��N��q hi/2� � �KNi � 1, ��N��ei� � EN

i , ��N��f i� � FNi . �3.15�

We shall now make use of the generic relations, which clearly T satisfies due to the particularchoice of boundary conditions �see also Ref. 7�,

�� id�N��N+1��Y�T�� = T�� id�N��N+1��Y�, Y � Uq�gl�m�n�� . �3.16�

Let us restrict our proof to ei, although the same logic follows for proving the commutation ofthe transfer matrix with f i. In addition to the algebra exchange relations, presented in the Appen-dix, we shall need for our proof the following relations:

Mei = qaiieiM, Mfi = q−aiif iM ,

�k0i KN

i � 1T0�� = T0��k0i KN

i � 1. �3.17�

It is convenient to rewrite the above relation for ei as follows:

�k0i EN

i + �0i �KN

i �−1�T0�� = T0��k0i EN

i + �0i �KN

i �−1� . . .

⇒ ENi M0T0�� + q�ii/2�0

i M0T0��k0i �−1�KN

i �−1

= �k0i �−1M0T0��k0

i ENi + �k0

i �−1M0T0��0i �KN

i �−1, �3.18�

where the subscript 0 denotes the auxiliary space, whereas the N coproducts leave exclusively onthe quantum space, recall also that �i�eii+1. By focusing only on the diagonal contributions overthe auxiliary space, after some algebraic manipulations, and bearing in mind �3.18�, we end up tothe following exchange relations �recall M is diagonal�:



�ENi ,MaaTaa�� = 0, a � i, i + 1,

�ENi ,MiiTii�� = − �iMi+1i+1Ti+1i��KN

i �−1,

�ENi ,Mi+1i+1Ti+1i+1�� = �− 1��i�+�i+1��iMi+1i+1Ti+1i��KN

i �−1, �3.19�

where �i is a scalar depending on aii and the grading. Then based on the above relations, we have

�ENi ,�

a=1

N

�− 1��a�MaaTaa�� = 0 ⇒ �ENi ,t�� = 0. �3.20�

Similarly, one may show the commutativity of the transfer matrix with FNi , hence

��N��x�,t�� = 0, x � Uq�gl�m�n�� , �3.21�

and this concludes our proof on the exact symmetry of the double row transfer matrix for theparticular choice of boundary conditions. It is clear that the proof above can be easily applied tothe usual nonsupersymmetric deformed algebra. We have not seen, as far as we know, such anexplicit and elegant proof elsewhere not even in the nonsupersymmetric case. In Ref. 7 the proofis explicit, but rather tedious, whereas in Ref. 2, one has to realize that the emanating nonlocalcharges form the Uq�gl�N��, and this is not quite obvious. Note that Tab

are quadratic combina-tions of the algebra generators and, in fact, the corresponding supertrace provides the associatedCasimir, which again gives a hint about the associated exact symmetry. In the case of trivialboundary conditions �K� I�, there is a discussion on the symmetry in Ref. 29 but is restricted onlyin this particular case, whereas our proof holds for generic nontrivial integrable boundary condi-tion �see Sec. III A 2�.

Let us now give explicit expressions of the quadratic q-Casimir for the simplest case �see alsoRef. 30�, that is, the Uq�gl�1 �1�� situation. In general, the quadratic Casimir operator is given by�3.12�, but now Kab

=�ab,

C+ � ��N��q2�1� − ��N��q2�2� − �q − q−1�2��N��q��1+�2�/2��N��f1��N��q��1+�2�/2��N��e1�

C− � ��N��q−�1� − ��N��− q2�2� + �q − q−1�2��N��e1��N��q−��1+�2�/2��N��f1��N��q−��1+�2�/2� .

�3.22�

Consider diagonal nontrivial solutions of the reflection equation �see also a brief discussion onthe symmetry in Ref. 21�,

K�� = diag�a��, . . . ,a�� ,

�

b��, . . . ,b��N−�

� . �3.23�

In this case, it is clear that the K matrix satisfies �recall �3.14��

��x�,K�� = 0, x � Uq�gl�m − ��n�� Uq�gl�� if � bosoni,

x � Uq�gl�m�� Uq�gl�n − �� if � = m + � fermionic �3.24�

and, consequently,



�� id�N��N+1��x�T�� = T�� id�N��N+1��x� ,

x � Uq�gl�m − ��n�� Uq�gl�� if � bosonic,

x � Uq�gl�m�� Uq�gl�n − �� if � = m + � fermionic. �3.25�

Thus, following the logic described in the case where K� I, we show that

�t��,Uq�gl��n�� Uq�gl�m − �� if � bosonic,

�t��,Uq�gl�m�� Uq�gl�n − �� if � = m + � fermionic. �3.26�

Consider the simplest nontrivial cases, that is, the Uq�gl�2 �1�� and Uq�gl�2 �2�� algebras withdiagonal K matrices �3.23� with �=2. According to the preceding discussion, the Uq�gl�2 �1�� andUq�gl�2 �2�� symmetries break down to Uq�gl�2�� u�1� and Uq�gl�2�� Uq�gl�2��, respectively. Itis worth presenting the associated Casimir operators for the tow particular examples. Consider firstthe Uq�gl�2 �1�� case, then from the �→ asymptotic behavior of the open transfer matrix weobtain

C+ = q2��N��q�1+�2��q��N��q�1−�2� + q−1��N��q−�1+�2� + �q − q−1�2��N��f1��N��e1�� ,

C− = q2��N��q−2�3� . �3.27�

Notice that all �i , i=1,2 ,3 elements commute with the transfer matrix and also the parenthesis inthe first line of �3.27� is essentially the typical Uq�sl2� quadratic Casimir. C− is basically a �u�1��type quantity. It is thus clear that C+ is the quadratic Casimir associated with Uq�gl2�; indeed, inthis case the Uq�gl�2 �1�� symmetry breaks down to Uq�gl�2�� u�1�. In general, the implementa-tion of boundary conditions described by the diagonal matrix �3.23� with �=m obviously breaksthe super symmetry to Uq�gl�m�� Uq�gl�n��, so, in fact, the superalgebra reduces to two nonsupersymmetric quantum algebras. Also, C+ is the Casimir associated with Uq�gl�m��, whereas C−

is the Casimir associated with Uq�gl�n��.This will become more transparent when examining the Uq�gl�2 �2�� case with boundary

conditions described by K �3.23� and �=2. The associated Casimir operators are then given by

C+ = q2��N��q�1+�2��q��N��q�1−�2� + q−1��N��q−�1+�2� + �q − q−1�2��N��f1��N��e1�� ,

C− = q2��N��q−�3−�4��q��N��q�4−�3� + q−1��N��q−�4+�3� + �q − q−1�2��N��f3��N��e3�� .

�3.28�

As expected in this case, since now the symmetry is broken to Uq�gl2� � Uq�gl2�, C+ Casimir isassociated with the one Uq�gl2� symmetry, whereas C− is associated with the other Uq�gl2�.

2. Nondiagonal reflection matrices

Let us finally consider nondiagonal reflection matrices. A new class of nondiagonal reflectionmatrices associated with Uq�glm �n� was recently derived in Ref. 31. Specifically, first define theconjugate index a such that �a�= �a� and, more specifically,

a = 2k + m + 1 − a, symmetric diagram,

a = m + 1 − a, a bosonic, a = 2m + n + 1 − a, a fermionic, distinguished diagram.

�3.29�

Then the nondiagonal matrices read as follows.



Symmetric Dynkin diagram.

Kaa�� = e2� cosh i�m − cosh 2i��, Kaa�� = e−2� cosh i�m − cosh 2i�� ,

Kaa�� = ica sinh 2�, Kaa�� = ica sinh 2�, 1 � a � L ,

Kaa�� = Kaa�� = cosh�2� + im�� − cosh 2i��, Kaa�� = Kaa�� = 0, L � a �m + 2k

2,

1 � L �m + 2k

2,

KAA = cosh�2� + im�� − cosh 2i��, A =m + 2k + 1

2if m odd. �3.30�

Distinguished Dynkin diagram.Bosonic.


Kaa�� = ica sinh 2�, Kaa�� = ica sinh 2�, 1 � a � L ,

Kaa�� = Kaa�� = cosh�2� + im�� − cosh 2i��, Kaa�� = Kaa�� = 0, L � a �m

21 � L �

m

2,

KAA = cosh�2� + im�� − cosh 2i��, A =m + 1

2if m odd and Kaa�� = Kaa�� = cosh�2�

+ im�� − cosh 2i��, Kaa�� = Kaa�� = 0, a � m, �3.31�

Fermionic.


Kaa�� = ica sinh 2�, Kaa�� = ica sinh 2�, m + 1 � a � L ,

Kaa�� = Kaa�� = cosh�2� + im�� − cosh 2i��, Kaa�� = Kaa�� = 0, L � a � m +n

2, m + 1

� L � m +n

2,

KAA = cosh�2� + im�� − cosh 2i��, A = m +n + 1

2if n odd and Kaa�� = Kaa��

= cosh�2� + im�� − cosh 2i��, Kaa�� = Kaa�� = 0, a � m, �3.32�

where m and � are free boundary parameters.Note that for the distinguished Dynkin diagram we have purely bosonic or purely fermionic

reflection matrices and not mixed. Let us first focus on the solution with the minimal number ofnonzero entries, and let us consider the symmetric case. The elements Tij

, i , j� 2, . . . ,2k+m



−1� form essentially the Uq�gl�m �2�k−1�� algebra, and it can be shown, based on the logicdescribed in the Sec. I, that the transfer matrix commutes with Uq�gl�m �2�k−1�� plus the ele-ments T1j

�T j1 �TNj

�T jN �N=m+2k�. Let us now deal with the generic situation described by

the solutions presented above �3.30�–�3.32�. First define U=TAj �T jA

�TAj

�T

jA

=0, A

� 1, . . . ,L�. Then it is shown for the generic nondiagonal K matrix.Symmetric Dynkin diagram.

�t��,Uq�gl�m�2�k − L�� U� = 0 bosonic solution,

�t��,Uq�gl�m − 2l�� U� = 0 L = k + l fermionic solution. �3.33�

Distinguished Dynkin diagram.

�t��,Uq�gl�m − 2L�� Uq�gl�n�� U� = 0 bosonic solution,

�t��,Uq�gl�m�� Uq�gl�n − 2l�� U� = 0, L = m + l, fermionic solution. �3.34�

The generic Casimir is given by �3.12�, where now the only nonzero entries are given by Kaa andKaa

It is clear that different choices of nondiagonal reflection matrices lead to distinct preservingsymmetries. This may perhaps be utilized together with the contraction process presented in Ref.10 to offer an algebraic description regarding of the underlying algebra emerging in the AdS/CFTcontext. Recall that this type of contractions leads to centrally extended algebras, and as in knownin the context of AdS/CFT, we deal basically with a centrally extended gl�2 �2� algebra.32

The explicit form of the boundary non local charges Tab �3.11�, in addition to the existence ofsome familiar symmetry, is essential, and may be, for instance, utilized for deriving reflectionmatrices associated with higher representations of Uq�gl�m �n�� see, e.g., Ref. 33�. In fact, thelogic we follow here is rather twofold: on the one hand, we try to extract a familiar symmetryalgebra, if any, and following the process of Sec. III A 1 to derive the exact symmetry. On theother hand, for generic K that may break all familiar symmetries, we show via the reflectionequation that the boundary nonlocal charges form an algebra, and via �3.9� we show that providean extra symmetry for the open transfer matrix. Moreover, the knowledge of the explicit form ofthe boundary nonlocal charges is of great significance given that they may be used, as alreadymentioned, for deriving reflection matrices for arbitrary representations.33

B. The q twisted super-Yangian

To complete our analysis on the boundary supersymmetric algebras, we shall now brieflydiscuss the q twisted Yangian. A more detailed analysis together with the classification of thecorresponding c-number solutions will be pursued elsewhere. As in the case of the twisted Yan-gian, we focus on the symmetric Dynkin diagram �1.7� and introduce

V = �i

f ieii:VTV = M �3.35�

and define the supertransposition as in the rational case. Also define the following matrices:

R12�� ª R21t1 �− � − i��, R21�� ª R12

t2 �− � − i�� . �3.36�

Recall that in the isotropic case R12=R21.Then the q-twisted Yangian is defined by



R12��1 − �2�K1��1�R21��1 + �2�K2��2� = K2��2�R12��1 + �2�K1��1�R21��1 − �2� . �3.37�

The nonlocal charges are derived from the asymptotic behavior of the tensor representation ofthe q-twisted Yangian,

Tab = �

k

�− 1��a�+�k��k�+�b��Tak+ Tkb

+ . �3.38�

As in the nonsupersymmetric case, the nonlocal charges derived above satisfy exchange relationsof the type �see also, e.g., Ref. 9�

R12 T1

R21 T2

= T2 R12

T1 R21

, �3.39�

it turns out that they do not provide an exact symmetry of the transfer matrix,9 as opposed to theisotropic limit described in Sec. II B. Although in the context of discrete integrable models de-scribed by the double row transfer matrix the boundary nonlocal charges do not form an exactsymmetry, one may show that in the corresponding field theoretical context they do provide exactsymmetries �see, e.g., Ref. 3�. Such investigations regarding supersymmetric field theories will beleft, however, for future investigations. Note finally that the classification of solutions of the qtwisted Yangian for the Uq�gl�m �n�� is still an open question, which we hope to address in aforthcoming publication.

APPENDIX: MORE ON THE Uq„gl„m n……

We shall recall here some details regarding the Uq�gl�m �n�� algebra. The algebra is defined bygenerators �i, ejf j, i=1, . . . ,N, j=1, . . . ,N−1, and the exchange relations of the Uq�gl�m �n��algebra are given below,

q�iq−�i = q−�iq�i = 1,

q�iej = q�− 1��j��ij−�− 1��j+1��ij+1ejq�i,

q�i f j = q−�− 1��j��ij+�− 1��j+1��ij+1f jq�i,

eif j − �− 1��i�+�i+1��j�+�j+1��f jei = �ijq�i−�i+1 − q−�i+�i+1

q − q−1 ,

xixj = �− 1��i�+�i+1��j�+�j+1��xjxi, xi � ei, f i� , �A1�

and q Chevallay–Serre relations,

xi2xi 1 − �q + q−1�xixi 1xi + xi 1xi

2 = 0, xi � ei, f i�, i � m. �A2�

Now set hi=�i, then the Uq�sl�m �n�� algebra is defined by generators ei, f i, and hi. Let aij be theelements of the related Cartan N�N matrix, which, for instance, for the distinguished Dynkindiagram is



a =�2 − 1

− 1 2 − 1

�

− 1 0 1

�

1 − 2 1

1 − 2

� , �A3�

the zero diagonal element occurs in the m position. Define also

�h�q =qh − q−h

q − q−1 �A4�

then the Uq�sl�m �n�� superalgebra for the distinguished Dynkin diagram reads as

�ei, f i� = �hi�q, i � m, em, fm� = − �hm�q, �ei, f i� = − �hi�q, i � m,

�hj,hk� = 0, �hi,ej� = aijej, �hi, f j� = − aijf j , �A5�

plus the Chevalley–Serre relations �A2�. The above algebra is equipped with a nontrivial coprod-uct,

��i� = �i � I + I � �i,

��x� = q−hi/2 � x + x � qhi/2, x � ei, f i� . �A6�

There is also an isomorphism between the FRT representation of the algebra and theChevalley–Serre basis. Recall that

L�� = L+ − e−2�L−,

L�� = L+ − O�e−2��, � →

L�� = L− − O�e2��, � → − . �A7�

Recall for the reflection algebra L��=L−1�−�� and also

L = �i,j

eij � lij , L = �

i,jeij � li j

. �A8�

Then we have the following identifications:27,34

lii+ = �− 1��i�q�i, lii+1

+ = �− 1��i+1��q − q−1�q��i+�i+1�/2f i, li+1i+ = 0,

lii− = �− 1��i�q−�i, li+1i

− = − �− 1��i��q − q−1�eiq−��i+�i+1�/2, lii+1

− = 0,

lii+ = �− 1��i�q�i, li+1i

+ = �− 1��i+1��q − q−1�q−aii/2q��i+�i+1�/2ei, lii+1+ = 0,



lii− = �− 1��i�q−�i, li+1i

− = − �− 1��i��q − q−1�qaii/2f iq−��i+�i+1�/2, li+1i

− = 0. �A9�

Here, lij+ , li j

− , i� j and lij− , li j

+ , i� j are also nonzero and are expressed as combinations of theUq�gl�m �n�� generators, but are omitted here for brevity �see, e.g., expressions in Refs. 35 and 7for the Uq�gl�n�� case�.

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