Infinite Hopf Families of Algebras and Yang-Baxter Relations

17 Pages·0.21 MB·English

by  Niall MacKay

Infinite Hopf Families of Algebras and
Yang-Baxter Relations
1
0
0 Niall MacKay1∗ and Liu Zhao1,2,3†‡
2
1 Department of Mathematics, University of York, York YO10 5DD, UK

n 2 Institute of Modern Physics, Northwest University, Xian 710069, China
a
J 3 The Abdus-Salam International Centre for Theoretical Physics, 34100 Trieste, Italy

6
1

1
Abstract
v
8 A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of
1
an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given
0 explicitly on thelevelof L matrices. Therelation with theDrinfeld currentrealization is established in the
1
case of 4×4 R-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of
0
algebra “comorphisms” (which generalize the notion of algebra comodules for standard Hopf algebras), a
1
0 possible way of constructing infinitely many commuting operators out of the generalized RLL algebras is
/ given. Finally some examples of thegeneralized RLL algebras are briefly discussed.
h
p
-
h 1 Introduction
t
a
m
Yang-Baxter relations (RTT relations) have proved to be a central ingredient in 2D integrable models and
: quantum algebras over the last 20 years. In quantum algebras, the Yang-Baxter relations have been used in
v
i several different contexts, including
X
r • standard finite quantum groups (FRT formalism [8]), i.e. the RTT =TTR relation
a
R T T =T T R
12 1 2 2 1 12
in which R solves the spectral parameter-less Yang-Baxter equation
R R R =R R R
12 13 23 23 13 12
and T also does not depend on any spectral parameters;
• quantum affine algebras (RS formalism [24]), i.e. the RLL relations (here and below u± ≡u± ℏc unless
4
otherwise specified)
± ± ± ±
R (u−v)L (u)L (v) = L (v)L (u)R (u−v),
12 1 2 2 1 12
R12(u−−v+)L+1(u)L−2(v) = L−2(v)L+1(u)R12(u+−v−),
in which the first equation is defined for any spectral parameter u and v, while the second equation is
defined only for |u|<|v|, and R solves the spectral Yang-Baxter equation
R (u−v)R (v−w)R (u−w)=R (u−w)R (v−w)R (u−v);
12 13 23 23 13 12
∗e-mail: [email protected]
†e-mail: [email protected]
‡RoyalSocietyvisitingfellow
1
• the RLL=LLR∗ relations [11, 12]
± ± ± ± ∗
R (u−v)L (u)L (v) = L (v)L (u)R (u−v),
12 1 2 2 1 12
R12(u−−v+)L+1(u)L−2(v) = L−2(v)L+1(u)R1∗2(u+−v−),
wherethefirstequationisdefinedforanyspectralparameteruandv,whilethesecondequationisdefined
only for |u|<|v|, and R and R∗ both solve the spectral Yang-Baxter equation
R (u−v)R (v−w)R (u−w)=R (u−w)R (v−w)R (u−v),
12 13 23 23 13 12
and are related by some analytical relations (such as modular transformations when the entries of R and
R∗ are elliptic functions);
• dynamical RLL relations [7, 9, 10]
± ± ± ±
R (u−v,λ)L (u,λ)L (v,λ) = L (v,λ)L (u,λ)R (u−v,λ),
12 1 2 2 1 12
R12(u−−v+,λ)L+1(u,λ)L−2(v,λ) = L−2(v,λ)L+1(u,λ)R12(u+−v−,λ),
in which the first equation is defined for any spectral parameter u and v, while the second equation is
defined only for |u|<|v|, and R solves the dynamical spectral Yang-Baxter equation
R (u−v,λ−h(3))R (v−w,λ)R (u−w,λ−h(1))
12 13 23
=R (u−w,λ)R (v−w,λ−h(2))R (u−v,λ);
23 13 12
• the combination of RLL=LLR∗ and the dynamical RLL relations [14, 17],
± ± ± ± ∗
R (u−v,λ)L (u,λ)L (v,λ) = L (v,λ)L (u,λ)R (u−v,λ),
12 1 2 2 1 12
R12(u−−v+,λ)L+1(u,λ)L−2(v,λ) = L−2(v,λ)L+1(u,λ)R1∗2(u+−v−,λ),
where again the third relation is defined only for |u|<|v|.
Ofcourse,thislistdoesnotexhaustalltheusesofYang-Baxterrelationsindefiningquantumaffinealgebras
— firstofallthere areversionsofthe aboveformalismscorrespondingto deformationsof Lie super-algebras,in
whichallL-matricesappearingontheleft-handsidehavetobemultipliedbyanumericalsignaturematrixfrom
the right, while all L-matrices appearing on the right-hand side have to be multiplied by the same numerical
signaturematrixfromthe left[13]—andtherearesomeotherrarecasesinwhichsomeotherstrangedeformed
versions of RLL relations are used.
Despite the numerous different forms of the Yang-Baxter relations in different quantum algebraic contexts,
there are essential factors that are common to most of these: there are alternative, essentially Yang-Baxter
relation-free definitions (or realizations) of the same quantum algebras (if the R-matrices are of the form
Rkl = a δkδl +b δlδk, anyway), and every known such algebra (i.e. one which admits any one of the above
ij ij i j ij i j
formalisms)fallsintotheclassofquasi-triangularquasi-Hopfalgebras–thatis,itisacertaintwistofastandard
triangular quasi-Hopf algebra [5, 17].
In practice, there are natural reasons to use many different realizations for the same quantum algebra.
For instance, to apply these algebras in integrable models in 2D physics, the best formalism to use is often the
RLL-likeform;tostudyinfinitedimensionalrepresentations,thebestformalismistheso-calledDrinfeldcurrent
realization; and to study structure and classification problems the best formalism is the Chevalley generating
relations. For these reasons the issue of the relations between different formalisms is an important subject,
and would be by no means an easy task were it not for the famous Ding-Frenkel theorem which establishes an
explicit relationship between the Yang-Baxter realization and the Drinfeld current realization.
Now our problem arises: on the one hand, in the study of two-parameter deformations of affine Lie (super)
algebras, we have accumulated a number of different two-parameterquantum affine (super-)algebras which are
2
members of the so-called infinite Hopf family of (super-)algebras [15, 16, 21, 27, 28] — a co-structure which
generalizes the standard Drinfeld coproduct [6] for standard quantum affine algebras — and we know nothing
aboutthecorrespondingRLLformalism;ontheotherhand,manyauthorshavestudiedtwo-parameterquantum
affine algebras [1, 2, 11, 12, 14, 17, 18, 19, 26], but from the point of view of Yang-Baxter realizations only,
and the co-structures studied are standard Drinfeld twists of Hopf algebras. Can these results be related by
use of an analogue of Ding-Frenkel theorem? That is, does there exist an RLL formalism for two-parameter
quantum affine algebraswiththe structure ofaninfinite Hopffamily of (super-)algebras? Howfar canwe goin
generalizingboththeRLLformalismandtheDing-Frenkeltheorem? Inthispaper,wewillpartlyaddressthese
problems by introducing a generalized RLL relation which contains all the above listed formalisms as special
cases. We shall consider some of the necessary conditions for such a generalized formalism to be consistent,
whichresultinsevererestrictionsonthepossiblechoicesoftheR-matrices. Then,weshallestablishtheinfinite
Hopffamily structure overthe generalizedRLLformalismandstudy the analogueofthe Ding-Frenkeltheorem
inthe specialcaseof4×4Rmatrices. We alsooutlineawayto constructinfinitely manycommutingoperators
out of our generalized RLL algebras and finally discuss briefly some possible examples of our construction.
2 The generalized formalism, associativity and co-structure
2.1 The generalized RLL relation
The generalized RLL formalism we shall study is
R(i)(u−v,λ)L±(i,j)(u,λ)L±(i,j)(v,λ) = L±(i,j)(v,λ)L±(i,j)(u,λ)R(j)(u−v,λ), (1)
12 1 2 2 1 12
R1(i2)(u−−v+,λ)L+1(i,j)(u,λ)L−2(i,j)(v,λ) = L−2(i,j)(v,λ)L+1(i,j)(u,λ)R1(j2)(u+−v−,λ), |u|<|v|, (2)
ℏc(i,j)
u± ≡ u± , (c(i,j) is central)
4
Unlike the usual RLL formalisms listed in the introduction, the R matrices are now labeled by an upper
index (i) and the L-matrices by two upper indices (i,j) where i,j ∈ Z. The parameters u,v are the usual
spectral parameters, and λ is a dynamical variable (which may or may not appear at all).
At this moment we assume no restrictions on the possible form of the R-matrices, and in particular we
have assigned no meaning to the extra upper indices of R and L and no relationship between the two R-
matrices carrying different upper indices at all. Such restrictions will emerge as we consider the associativity
and co-structure of the algebras defined by the above relation.
2.2 Associativity
Any work on quantum algebras should ensure that the algebra being studied is associative, and, in principle,
could be equipped with a co-structure, so that tensor products of representations may be defined. So we shall
first examine our generalized RLL relations to ensure that they really define associative algebras. To this end,
letusconsiderthecaseofproductof3L-matricesL(i,j)(u,λ)L(i,j)(v,λ)L(i,j)(w,λ). Ifthealgebraisassociative,
1 2 3
we should be able to apply this product in two different ways, i.e.
L(i,j)(u,λ)L(i,j)(v,λ)L(i,j)(w,λ)=[L(i,j)(u,λ)L(i,j)(v,λ)]L(i,j)(w,λ)=L(i,j)(u,λ)[L(i,j)(v,λ)L(i,j)(w,λ)].
1 2 3 1 2 3 1 2 3
Now temporarily we omit the upper indices ± of L(i,j) and arguments of R(i) and L(i,j) (though the effects
of these arguments are not neglected). We now apply the generalized RLL relations first to reverse the order
3
of L(i,j)L(i,j)L(i,j) and then to return to the original order. We have
1 2 3
R(i)R(i)R(i)R(i)R(i)R(i)L(i,j)L(i,j)L(i,j) =L(i,j)L(i,j)L(i,j)R(j)R(j)R(j)R(j)R(j)R(j), (3)
32 31 21 23 13 12 1 2 3 1 2 3 32 31 21 23 13 12
where the order of lower indices of the L’s is changed in the following manne :
(123) → (213)→(231)→(321)
→ (312)→(132)→(123).
Onthe firstline wechangedL(i,j)L(i,j)L(i,j) intoL(i,j)L(i,j)L(i,j) usingleft-groupingofthe successiveproducts,
1 2 3 3 2 1
while onthe secondline we changedL(i,j)L(i,j)L(i,j) back into L(i,j)L(i,j)L(i,j) usingright-grouping,soassocia-
3 2 1 1 2 3
tivity has beenimplied inthe aboveoperation. Asufficientbutprobablynottoo restrictiveconditionto ensure
the correctness of the equation (3) is that
R(i)R(i)R(i)R(i)R(i)R(i) =R(j)R(j)R(j)R(j)R(j)R(j) =1.
32 31 21 23 13 12 32 31 21 23 13 12
Therefore, assuming unitarity of R-matrices, i.e.
R(i)(u,λ)R(i)(−u,λ) = 1, (4)
12 21
R(j)(u,λ)R(j)(−u,λ) = 1, (5)
12 21
we conclude that R(i) and R(j) must separately satisfy the (probably generalized) dynamical Yang-Baxter
equation
R(k)(u−v,λ−h(3))R(k)(v−w,λ)R(k)(u−w,λ−h(1))
12 13 23
= R(k)(u−w,λ)R(k)(v−w,λ−h(2))R(k)(u−v,λ), (k =i,j). (6)
23 13 12
LaterweshallseethatunitarityoftheR-matricesisalsorequirediftheDrinfeldcurrentrealizationisconsidered,
and will introduce a coproduct which is compatible with the infinite Hopf family in that realization.
Let us stress that so far we have not assumed any relationship between R(i) and R(j). The different upper
indices only indicate that the two R-matrices may be different. Thus associativity has not resulted in any
restrictions on the relation between R-matrices.
2.3 Co-structure
In contrast, however, the definition of a co-structure does require some relationship between R(i) and R(j), as
we discuss below.
Firstwe recallwhatwe meanby a co-structure. Algebraicallyaco-structureis a propertyofalgebraswhich
allows the definition of a tensor product between two algebras (usually two copies of the same algebra — in
that case the algebrais called co-closed). For instance, for algebrasdefined by the non-dynamicalRLL=LLR
relations,the co-structureis just the standardHopfalgebrastructure; inparticular,the coproductis simply an
operation which creates a generalized L matrix L(n) which obey the RL(n)L(n) = L(n)L(n)R relation with the
same R, where
L(n) = ∆(n)L
≡ (id⊗...⊗id⊗∆)◦(id⊗...⊗∆)◦...◦(id⊗∆)◦∆L
= L⊗˙L⊗˙...⊗˙L
is the n-th coproduct of L, in which L can be either L+ or L−, and L⊗˙L is defined via (L⊗˙L)a = La⊗Lc.
b c c b
The point is that applying R12 to the left hand side of L1L2 would result in P
R L(n)L(n) =L L R L(n−1)L(n−1) =...=L(n)L(n)R , (7)
12 1 2 2 1 12 1 2 2 1 12
4
so that the n-th coproduct ∆(n) is an algebra homomorphism.
Foralgebrasdefinedby relationslikeRLL=LLR∗,this isimpossible becauseafterthe firstuseofthe RLL
relation, a different R-matrix R∗ appears and the iteration stops. One possible way round is to introduce a
twist operationsuchthatthe R∗ whichappearsafter eachuse ofthe RLLrelationis twistedbackto R andthe
iteration can be continued. This amounts to the notion of the Drinfeld twist in quasi-Hopf algebras, and the
possibility of defining the twist operation imposes severe restrictions on the relations between R and R∗.
Since we assumed no relationship between R(i) and R(j) for i 6= j so far, our situation is very much like
the case of RLL = LLR∗ relations. However, our aim here is not to resort to a Drinfeld twist but rather to
incorporate the structure of the infinite Hopf family in the generalized RLL formalism. Therefore, instead of
retaining the tensor product of many copies of the same algebra, we can think about making tensor products
between different algebras defined by our generalized RLL formalism — this is exactly what we did in the
Drinfeld current realization of infinite Hopf family of algebras.
InordertoincorporatetheinfiniteHopffamilystructuredirectlyatthelevelofRLLformalism,however,we
have to introduce a relationship between the two R-matrices R(i) and R(j) in our generalized RLL formalism.
Firstwechoosesome unitarysolutionofthe (dynamical)Yang-Baxterequation(6),denoted R(0)(u,λ). We
then introduce invertible operators ρ± which act on this R-matrix to give R(±1)(u,λ) = ρ±R(0)(u,λ). The
0 0
permitted operators ρ±aresuch that they preserve unitarity as well as the Yang-Baxterequation. OnR(±1)(u,λ)
0
we may again act with ρ± (with the same property as ρ±) to get R(±2)(u,λ) etc. Assume that we have a
±1 0
large (possibly infinite) set Υ of R-matrices whose elements are obtained by recursively applying the relation
R[±(n+1)](u,λ) ≡ (ρ±)R(±n)(u,λ). An additional requirement is that ρ− ◦ρ+ = ρ+ ◦ρ− = id , so that Υ
n n+1 n n−1 n n
forms a single ρ± orbit. We may then put these R-matrices into our generalized RLL relations.
n
These relations define a huge family of associative algebras{A ≡A(R(i),R(j)),i,j ∈Z} , each labeled by
i,j
two ordered integers corresponding to the upper indices of the two R-matrices in the above relations. Among
these algebras we are particularly interested in the ones labeled by two successive integers, i.e. A ≡ A .
n,n+1 n
For these algebraswe simplify the notation as follows: L±(n,n+1) →L±(n), c(n,n+1) →c . It is remarkablethat
n
for each pair of algebras An and An±1 there exists an algebra morphism
±
τn :An →An±1
defined via
τ+L±(n) =L±(n+1), τ−L±(n) =L±(n−1),
n n
τ±R(n)(u,λ)=ρ±R(n)(u,λ)=R(n±1)(u,λ),
n n
τ±R(n±1)(u,λ)=ρ± R(n±1)(u,λ)=R(n±2)(u,λ).
n n±1
The algebra morphisms τ± also obey the condition
n
τ− ◦τ+ =τ+ ◦τ− =id ,
n+1 n n−1 n n
because they are essentially the lift of the action of ρ± onto the algebras A . For any pair of integers n < m,
n n
the composition of τ±
n
Mor(A , A )∋τ(m,n) ≡τ+ ...τ+ τ+ : A →A ,
m n m−1 n+1 n n m
Mor(A , A )∋τ(n,m) ≡τ− ...τ− τ− : A →A (8)
n m n+1 m−1 m m n
arealgebramorphismsbetweenA andA ,andsinceτ(m,n)τ(n,m) =id , τ(n,m)τ(m,n) =id andτ(m,p)τ(p,n) =
n m m n
τ(m,n), they make the family of algebras {A , n∈Z} into a category.
n
Let us now define the following co-structure:
5
• Coproduct
ℏc ℏc
∆+L±(n)(u,λ) = L±(n)(u± n+1)⊗˙L±(n+1)(u∓ n),
n 4 4
∆−L±(n)(u,λ) = L±(n−1)(u± ℏcn)⊗˙L±(n)(u∓ ℏcn−1);
i 4 4
• Antipode
S+L±(n)(u,λ) = L±(n+1)(u,λ)−1,
n
S−L±(n)(u,λ) = L±(n−1)(u,λ)−1;
n
• Counit
ǫ L±(n)(u,λ)=id .
n n
By straightforward verification, we find that the above co-structure obeys the following axioms for infinite
Hopf family of algebras [15, 16, 27, 28, 21] :
• (ǫn⊗idn+1)◦∆+n =τn+, (idn−1⊗ǫn)◦∆−n =τn−
• mn+1◦(Sn+⊗idn+1)◦∆+n =ǫn+1◦τn+, mn−1◦(idn−1⊗Sn−)◦∆−n =ǫn−1◦τn−
• (∆−n ⊗idn+1)◦∆+n =(idn−1⊗∆+n)◦∆−n
in which m is the algebra multiplication for A . Moreover, it is easy to check that the images of ∆± satisfy
n n n
the defining relations of A with the center c +c , and hence ∆± is an algebra homomorphism. By
n,n+2 n n+1 n
using compositions of the coproduct, ∆(nm)+ ≡ (idn⊗idn+1⊗...⊗idn+m−2⊗∆+n+m−1)◦(idn⊗idn+1⊗...⊗
idn+m−3⊗∆+n+m−2)...◦(idn⊗∆+n+1)◦∆+n , we can show that all the algebras An,m are inter-related by the
co-structure. Indeed, denoting
L±(n,m)(u,λ)≡∆(m)+L±(n)(u,λ),
n
one can check that the following relations hold:
R(n)(u−v,λ)L±(n,m)(u,λ)L±(n,m)(v,λ) = L±(n,m)(v,λ)L±(n,m)(u,λ)R(m)(u−v,λ), (9)
12 1 2 2 1 12
R1(n2)(u−−v+,λ)L+1(n,m)(u,λ)L2−(n,m)(v,λ) = L2−(n,m)(v,λ)L+1(n,m)(u,λ)R1(m2)(u+−v−,λ), |u|<|v(1|,0)
u± ≡ u± ℏ(cn+cn+1+...+cm−1).
4
TheserelationsareexactlythegeneratingrelationsofA withthecenterc(n,m) givenbyc(n,m) =c +c +
n,m n n+1
...+cm−1. Equations (9) and (10) are the analogues of (7) in the case of infinite Hopf family of algebras.
Remark. The co-structure of an infinite Hopf family of algebras may look quite unnatural at first sight to
readers who are familiar with standard Hopf algebra co-structures, because it makes use of tensor products
between different algebras. However, the idea of making tensor product between different algebras is actually
notsosurprising—ithasalreadybeenstudiedin severaldifferentcontexts[3,22,23,25]whicharenotrelated
to our purpose.
3 Gauss Decomposition
In this section, we assume that the R-matrices have the form
[R(n)]kl(u−v,λ)=a(n)(u−v,λ)δkδl +b(n)(u−v,η,λ)δlδk. (11)
ij ij i j ij i j
6
This is because it is known that for R-matrices ofmore complex form, there is no analogueof the Ding-Frenkel
theorem (such “bad” cases include the 8-vertex R-matrix [11, 12]).
The proof of the Ding-Frenkel theorem for R-matrices of arbitrary size is a tedious and complicated piece
of work, and should in general be accomplished by induction over the size of R-matrices [4]. In this paper,
however, we shall not make the induction over the size of R-matrices, but rather will illustrate our generalized
RLL formalism only in the simplest case of 4×4 R-matrices.
Beforeweproceed,letus remarkthatthe constructionofthe lastsectionworksfineifweunderstandallthe
productsbetweenRandLandthoseamongLsasgradedmatrixproductsforsupermatrices,andsimultaneously
understand all the tensor products as graded tensor products. In this way we would have the RLL formalism
for an infinite Hopf family of super-algebras. In practice, however, it is usually better to change the graded
matrix product into the usual matrix products. For members of the infinite Hopf family of super-algebras,this
amounts to altering the RLL relations into the form
R(i)(u−v,λ)L±(i,j)(u,λ)̟L±(i,j)(v,λ)̟ = ̟L±(i,j)(v,λ)̟L±(i,j)(u,λ)R(j)(u−v,λ), (12)
12 1 2 2 1 12
R1(i2)(u−−v+,λ)L+1(i,j)(u,λ)̟L−2(i,j)(v,λ)̟ = ̟L−2(i,j)(v,λ)̟L+1(i,j)(u,λ)R1(j2)(u+−v−,λ), |u|<|v|(13)
ℏc(i,j)
u± ≡ u± , (c(i,j) is central)
4
where̟ isadiagonalnumericalmatrixgivenby̟kl =(−1)[i][j]δkδl whichreflectsthe gradingoftheR andL-
ij i j
matrices. Wemayalsowritetheusual(non-super)RLLrelationsintheaboveform,providedweunderstand̟
astheidentitymatrix. ThissectionwillbebasedonRLLrelationsoftheaboveform,putting thedeformations
corresponding to the usual and super root systems on an equal footing.
The most general 4×4 R-matrix of the form (11) is written
a(i)(u,λ) 0 0 0
R(i)(u,λ)= 0 b(i)(u,λ) t(i)(u,λ) 0  (14)
0 s(i)(u,λ) c(i)(u,λ) 0
 
 0 0 0 d(i)(u,λ) 
 
The corresponding numerical signature matrix ̟ is given by
1
 1 
̟ = , ǫ=±1.
1
 
 ǫ 
 
In such cases the L-matrices can be represented as operator-valued 2×2 matrices and we adopt the following
Gauss decompositions (we omit the upper indices (i,j) here and below):
L±(u,λ) = (cid:18) 1f±(u,λ) 01 (cid:19)(cid:18) k01±(u,λ) k0±(u,λ) (cid:19)(cid:18) 10 e1±(u,λ) (cid:19)
2
k±(u,λ) k±(u,λ)e±(u,λ)
= (cid:18) f1±(u,λ)k±(u,λ) k1±(u,λ)+f±(u,λ)k±(u,λ)e±(u,λ) (cid:19),
1 2 1
L±(u,λ)−1 = (cid:18) 10 −1e±(u,λ) (cid:19)(cid:18) k01±(u,λ)−1 0k±(u,λ)−1 (cid:19)(cid:18) 1−f±(u,λ) 01 (cid:19)
2
k±(u,λ)−1+e±(u,λ)k±(u,λ)−1f±(u,λ) −e±(u,λ)k±(u,λ)−1
= (cid:18) −1k±(u,λ)−1f±(u,λ) 2 k±(u,λ)−12 (cid:19).
2 2
Forsimplicityofnotation,wealsodenotek±(u),e±(u)andf±(u)withoutexplicitlyreferringtothedependence
i
on λ. This dependence, however,is always implicitly assumed.
7
To get the desired analogue of the Ding-Frenkel theorem, it is convenient to rewrite the RLL relations in
several equivalent ways. These include
L±(v,λ)−1̟R(i)(u−v,λ)L±(u,λ)̟ = ̟L±(u,λ)R(j)(u−v,λ)̟L±(v,λ)−1,
2 12 1 1 12 2
L−2(v,λ)−1̟R1(i2)(u−−w+,λ)L+1(u,λ)̟ = ̟L+1(u,λ)R1(j2)(u+−v−,λ)̟L−2(v,λ)−1; (15)
L±(u,λ)−1̟L±(v,λ)−1̟R(i)(u−v,λ) = R(j)(u−v,λ)̟L±(v,λ)−1̟L±(u,λ)−1,
1 2 12 12 2 1
L+1(u,λ)−1̟L−2(v,λ)−1̟R1(i2)(u−−w+,λ) = R1(j2)(u+−v−,λ)̟L−2(v,λ)−1̟L+1(u,λ)−1; (16)
L±(u,λ)−1R(i)(v−u,λ)̟L±(v,λ)̟ = ̟L±(v,λ)̟R(j)(v−u,λ)L±(u,λ)−1,
1 21 2 2 21 1
L+1(u,λ)−1R2(i1)(v+−u−,λ)̟L−2(v,λ)̟ = ̟L−2(v,λ)̟R2(j1)(v−−u+,λ)L+1(u,λ)−1, (17)
where the unitarity of the R-matrices is implied by the equivalence of these different equations.
Expanding the aboveequations into matrix components, we get, after some algebra,the followingrelations:
±
1. Relations among k (u):
i
a(i)(u−v,λ)k±(u)k±(v) = k±(v)k±(u)a(j)(u−v,λ), (18)
1 1 1 1
a(i)(u−−v+,λ)k1+(u)k1−(v) = k1−(v)k1+(u)a(j)(u+−v−,λ), (19)
k±(v)−1b(i)(u−v,λ)k±(u) = k±(u)b(j)(u−v,λ)k±(v)−1, (20)
2 1 1 2
k2−(v)−1b(i)(u−−v+,λ)k1+(u) = k1+(u)b(j)(u+−v−,λ)k2−(v)−1, (21)
k±(u)−1k±(v)−1d(i)(u−v,λ) = d(j)(u−v,λ)k±(v)−1k±(u)−1, (22)
2 2 2 2
k2+(u)−1k2−(v)−1d(i)(u−−v+,λ) = d(j)(u+−v−,λ)k2−(v)−1k2+(u)−1, (23)
k±(u)−1b(i)(v−u,λ)k±(v) = k±(v)b(j)(v−u,λ)k±(u)−1, (24)
2 1 1 2
k2+(u)−1b(i)(v+−u−,λ)k1−(v) = k1−(v)b(j)(v−−u+,λ)k2+(u)−1, (25)
2. relations between k±(u) and e±(v), f±(v):
i
k±(u)a(j)(u−v,λ)e±(v)−e±(v)k±(u)b(j)(u−v,λ)−k±(u)e±(u)s(j)(u−v,λ)=0,
1 1 1
b(i)(u−v,λ)k±(u)f±(v)+t(i)(u−v,λ)f±(u)k±(u)−f±(v)a(i)(u−v,λ)k±(u)=0;
1 1 1
k1+(u)a(j)(u+−v−,λ)e−(v)−e−(v)k1+(u)b(j)(u+−v−,λ)−k1+(u)e+(u)s(j)(u+−v−,λ)=0,
b(i)(u−−v+,λ)k1+(u)f−(v)+t(i)(u−−v+,λ)f+(u)k1+(u)−f−(v)a(i)(u−−v+,λ)k1+(u)=0;
k±(v)−1d(i)(u−v,λ)f±(u)−f±(u)k±(v)−1b(i)(u−v,λ)−ǫk±(v)−1f±(v)s(i)(u−v,λ)=0,
2 2 2
k2−(v)−1d(i)(u−−v+,λ)f+(u)−f+(u)k2−(v)−1b(i)(u−−v+,λ)−ǫk2−(v)−1f−(v)s(i)(u−−v+,λ)=0;
b(j)(u−v,λ)k±(v)−1e±(u)−e±(u)d(j)(u−v,λ)k±(v)−1+ǫt(j)(u−v,λ)e±(v)k±(v)−1 =0,
2 2 2
b(j)(u+−v−,λ)k2−(v)−1e+(u)−e+(u)d(j)(u+−v−,λ)k2−(v)−1+ǫt(j)(u+−v−,λ)e−(v)k2−(v)−1 =0;
b(i)(v+−u−,λ)k1−(v)f+(u)−f+(u)a(i)(v+−u−,λ)k1−(v)+t(i)(v+−u−,λ)f−(v)k1−(v)=0,
e+(u)k1−(v)b(j)(v−−u+,λ)−k1−(v)a(j)(v−−u+,λ)e+(u)+k1−(v)e−(v)s(j)(v−−u+,λ)=0;
e−(v)d(j)(v−−u+,λ)k2+(u)−1−ǫt(j)(v−−u+,λ)e+(u)k2+(u)−1−b(j)(v−−u+,λ)k2+(u)−1e−(v)=0,
f−(v)k2+(u)−1b(i)(v+−u−,λ)−k2+(u)−1d(i)(v+−u−,λ)f−(v)+ǫk2+(u)−1f+(u)s(i)(v+−u−,λ)=0;
3. Relations containing two e±’s or two f±’s:
a(i)(u−v,λ)k±(u)e±(u)k±(v)e±(v)−ǫk±(v)e±(v)k±(u)e±(u)d(j)(u−v,λ)=0,
1 1 1 1
a(i)(u−−v+,λ)k1+(u)e+(u)k1−(v)e−(v)−ǫk1−(v)e−(v)k1+(u)e+(u)d(j)(u+−v−,λ)=0,
d(i)(u−v,λ)f±(u)k±(u)f±(v)k±(v)−ǫf±(v)k±(v)f±(u)k±(u)a(j)(u−v,λ)=0,
1 1 1 1
d(i)(u−−v+,λ)f+(u)k1+(u)f−(v)k1−(v)−ǫf−(v)k1−(v)f+(u)k1+(u)a(j)(u+−v−,λ)=0;
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4. Mixed relations between e± and f±’s:
−ǫe±(u)f±(v)+f±(v)e±(u) = k±(u)−1b(i)(u−v,λ)−1t(i)(u−v,λ)k±(u)
1 2
−k±(v)b(j)(u−v,λ)−1t(j)(u−v,λ)k±(v)−1, (26)
2 1
−ǫe+(u)f−(v)+f−(v)e+(u) = k1+(u)−1b(i)(u−−v+,λ)−1t(i)(u−−v+,λ)k2+(u)
−k2−(v)b(j)(u+−v−,λ)−1t(j)(u+−v−,λ)k1−(v)−1, (27)
−ǫe−(u)f+(v)+f+(v)e−(u) = k1−(u)−1b(i)(u+−v−,λ)−1t(i)(u+−v−,λ)k2−(u)
−k2+(v)b(j)(u−−v+,λ)−1t(j)(u−−v+,λ)k1+(v)−1, (28)
where the equations in (26) are defined for both |u| > |v| and |u| < |v|, equation (27) is defined only for
|u|>|v| and equation (28) is defined only for |u|<|v|.
The appearance of these can be drastically simplified by defining
E(u) = e+(u−)−e−(u+),
F(u) = f+(u+)−f−(u−),
giving
±
• Relations between the k ’s and the E and F:
i
k1+(u)a(j)(u−v−,λ)E(v) = E(v)k1+(u)b(j)(u−v−,λ), (29)
b(i)(u−v ,λ)k+(u)F(v) = F(v)a(i)(u−v ,λ)k+(u), (30)
+ 1 + 1
E(u)k1−(v)b(j)(v−−u,λ) = k1−(v)a(j)(v−−u,λ)E(u), (31)
b(i)(v −u,λ)k−(v)F(u) = F(u)a(i)(v −u,λ)k−(v), (32)
+ 1 + 1
b(j)(v−−u,λ)k2+(u)−1E(v) = E(v)d(j)(v−−u,λ)k2+(u)−1, (33)
k+(u)−1d(i)(v −u,λ)F(v) = F(v)k+(u)−1b(i)(v −u,λ). (34)
2 + 2 +
b(j)(u−v−,λ)k2−(v)−1E(u) = E(u)d(j)(u−v−,λ)k2−(v)−1, (35)
k−(v)−1d(i)(u−v ,λ)F(u) = F(u)k−(v)−1b(i)(u−v ,λ). (36)
2 + 2 +
• Relations of the kind EE and FF:
a(j)(v−u,λ)E(u)b(j)(v−u,λ)−1E(v)−ǫE(v)b(j)(u−v,λ)−1E(u)d(j)(u−v,λ) = 0, (37)
d(i)(u−v,λ)F(u)b(i)(u−v,λ)−1F(v)−ǫF(v)b(i)(v−u,λ)−1F(u)a(i)(v−u,λ) = 0; (38)
• Exchange relation between E and F :
[E(u),F(v)]ǫ = k2−(v−)[Φ+(j)(u+−v−,λ)−Φ−(j)(u+−v−,λ)]k1−(v−)−1
−k2+(v+)[Φ+(j)(u−−v+,λ)−Φ−(j)(u−−v+,λ)]k1+(v+)−1, (39)
where [E(u),F(v)] =E(u)F(v)−ǫF(v)E(u) and Φ±(i)(u) are defined via
ǫ
Φ±(i)(u−v,λ)≡b(i)(u−v,λ)−1t(i)(u−v,λ) for |u|>|v| .
(cid:26) |u|<|v|
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Notice that equation (39) can only be understood as an analytical continuation, because Φ±(i)(u) are never
simultaneously well-defined for the same u. In the sense of analytical continuations, Φ−(i)(u) can be regarded
as the same as Φ+(i)(u) for almost any value of u except u = 0, where there is a singularity. Therefore, it is
convenient to express the difference between Φ+(i)(u) and Φ−(i)(u) in terms of Dirac delta function:
Φ+(i)(u)−Φ−(i)(u)=N(i)(~,λ)δ(u),
where N(i)(~,λ) is a normalization coefficient which may depend on the dynamical variable λ.
Using this notation, the exchange relation between E(u) and F(v) can be rewritten as
[E(u),F(v)]ǫ =δ(u+−v−)k2−(v−)N(j)(~,λ)k1−(v−)−1−δ(u−−v+)k2+(v+)N(j)(~,λ)k1+(v+)−1. (40)
Summarizing the above results, we conclude that the generating relations for the algebra defined by our
generalized RLL relations (12-13) with the 4 × 4 R-matrices (14) can be written in terms of the Drinfeld
±
currents k (u) and E(u), F(u) via equations (18-25), (29-36), (37-38) and (40), provided the R-matrices (14)
1,2
satisfy the (dynamical) Yang-Baxter equation (6) and the unitarity conditions (4-5).
Let us stress that in the above calculations we made no use of explicit formulae for the R-matrix entries;
the only condition used is unitarity. We also assumed no explicit commutation relations between entries of
the (dynamical) R-matrices and the Drinfeld currents, so our result should hold for all unitary dynamical R-
matrices. Such a result is very useful when we consider concrete examples of the generalized RLL algebras —
to get the Drinfeld current realization, for example, the only thing we need to do is to substitute the concrete
R-matrix entries into the equations (18-25), (29-36), (37-38) and (40).
Now let us make some more comments about the essential role of the unitarity conditions (4-5). In section
two, these conditions are assumed only because they are sufficient to make the generalized RLL relations
associative. In this section, however, we see that to have the Drinfeld current realization, we need these
conditions to hold. Actually there are other reasons to impose the unitarity conditions on the R-matrices. For
instance, let us consider the cases when the R-matrices are non-dynamical and ǫ = +1. In such cases, the
equations (37) and (38) can be written as
b(j)(v−u)d(j)(u−v)
E(u)E(v) = E(v)E(u), (41)
a(j)(v−u)b(j)(u−v)
a(i)(v−u)b(i)(u−v)
F(u)F(v) = F(v)F(u). (42)
b(i)(v−u)d(i)(u−v)
It is proven in [27] that the co-structure of the Drinfeld current realization of the infinite Hopf family of
algebras is characterized solely by the structure functions appearing in the commutation relations
E (u)E (v) = Ψ (u−v|q)E (v)E (u),
i j ij j i
F (u)F (v) = Ψ (u−v|q˜)−1F (v)F (u),
i j ij j i
and the only condition that these structure functions have to obey is
Ψ (u|q)=Ψ (−u|q)−1.
ij ji
Now looking at the equations (41) and (42) we find that the condition imposed above on Ψ (u|q) is just
ij
b(j)(−u)d(j)(u) a(j)(u)b(j)(−u)
= ,
a(j)(−u)b(j)(u) b(j)(u)d(j)(−u)
which holds trivially if the unitarity condition for R(j) is satisfied. Thus the unitarity condition is not merely
among the sufficient conditions for the RLL relations to be associative, but also necessary for the Drinfeld
current realization.
10