Automatic Construction of Explicit R Matrices for the One-Parameter Families of Irreducible Typical Highest Weight (0|α) Representations of U_q[gl(m|n)]

65 Pages·0.6 MB·English

by  David De Wit

Automatic Construction of Explicit R Matrices
1
0 for the One-Parameter Families of Irreducible
0
2 Typical Highest Weight (0˙ |α˙ ) Representations
m n
n
a of U [gl(m|n)]
J q

4

] David De Wit
A
RIMS, Kyoto University, JAPAN
Q
. February 1, 2008
h
t
a
m
Abstract
[
WedetailtheautomaticconstructionofRmatricescorrespondingto(the

1 tensorproductsof)the(0˙m|α˙n)familiesofhighest-weightrepresentations
v of the quantum superalgebras Uq[gl(m|n)]. These representations are ir-
7 reducible, contain a free complex parameter α, and are2mn dimensional.
2
OurRmatricesareactually (sparse)rank4tensors,containingatotalof
0
24mn components, each of which is in general an algebraic expression in
1
the two complex variables q and α.
0
1 Although the constructions are straightforward, we describe them in
0 full here, to fill a perceived gap in the literature. As the algorithms are
/ generally impracticable for manualcalculation, wehaveimplemented the
th entire process in Mathematica;illustrating our results with Uq[gl(3|1)].
a
m
1 Introduction
:
v
i
X Broadly, R matrices are solutions to the various versions of the Yang–Baxter
equation, and as such, are of great interest in mathematical physics and knot
r
a theory (see, e.g. [18]), both in their algebraic (i.e. “universal”) forms, and
in their (matrix) representations (i.e. “quantum” forms), useful for explicit
computations. Here,wewillbespecificallyconcernedwithquantumRmatrices
associated with the quantum superalgebras U [gl(m|n)].
q
Althoughmuchis knownaboutthe originandpropertiesofquantumsuper-
algebraRmatrices(e.g. [19]providesuniversalRmatrices),explicitexamplesof
theirquantumRmatricesarerareintheliterature,duelargelytothecomputa-
tionaleffortinvolvedinobtainingthem. Thispaperdescribestheautomationof
analgorithmtogenerateasuiteofexplicitquantumRmatricesforU [gl(m|n)].
q
Asreadersofthis organmaynotbe familiarwiththesealgebraicstructures,we
provide a full description of their details.
1
Specifically, we construct trigonometric R matrices Rˇm,n(u) corresponding
to the α-parametric highest weight minimal representations labeled (0˙ |α˙ ) of
m n
the U [gl(m|n)]. These irreducible representations are 2mn dimensional, and
q
contain free complex parameters q and α; the real variable u is a ‘spectral’ pa-
rameter. QuantumRmatricesRˇm,n areimmediatelyobtainableasthe spectral
limits u→∞ of the Rˇm,n(u).
OurRmatricesareinfactgraded,astheyarebasedongradedvectorspaces,
hence they actually satisfy graded Yang–Baxter equations. However, it is a
simple matter to remove this grading and transform them into objects that
satisfy the usual Yang–Baxter equations.
TheconstructionshavebeenimplementedinMathematica,andresultsob-
tainedforn=1andm=1,2,3,4;weillustratethealgorithmsusingU [gl(3|1)].
q
Full listings of all our R matrices have been announced in [6].
As they are solutions to Yang–Baxter equations, our R matrices are of im-
mediate practicalinterest. Firstly, they are of physicalinterestin that they are
applicabletotheconstructionofexactlysolvablemodelsofinteractingfermions.
Corresponding to Rˇm,1(u), we may construct an integrable 2m state fermionic
modelonalattice. ModelsassociatedwithU [gl(2|1)]andU [gl(3|1)]havebeen
q q
discussedin[14]and[13],respectively. TheU [gl(4|1)]casehasanelegantinter-
q
pretationintermsofa2-legladdermodelforinteractingelectrons: adiscussion
of this is provided in [6].
Furthermore, corresponding to each Rˇm,n, we may obtain a polynomial
‘Links–Gould’ link invariant LGm,n [21], cf. the celebrated Jones polynomial.
These LGm,n are two-variable,integer-coefficientLaurent polynomials,and are
generallysubstantiallymorepowerfulthantheJonespolynomialindistinguish-
ingknots. (LG1,1degeneratestothewell-knownAlexander–Conwaypolynomial
in the single variable q2α (cf. [2]).) A fuller documentation of the suite LGm,n
hasbeenprovidedbymyselfincollaborationwithLouisKauffmanandJonLinks
in [4, 5, 7, 9]. Although the LGm,n are far from being complete invariants, as
they can distinguish neither mutants nor inversion[5, 9], it turns out that even
LG2,1 is infact morepowerfulthanthe well-knowntwo-variableHOMFLYand
Kauffman invariants, being able to distinguish (including chirality) all prime
knots of up to 10 crossings [5]. Their evaluation also involves automatic sym-
boliccomputation,butthecomputationalaspectsarecomparativelypedestrian.
Lastly, we mention explicitly that this paper contains no new theorems,
although it does contain two new technical lemmas, proven in Appendix A. It
is primarily intended to provide a proper foundation for the results presented
in [6, 7], although it also serves as a tutorial on an application of symbolic
computation. Whilst it specifically pertains to representations of U [gl(m|n)],
q
many of the algorithms have a much broader application.
The following subsections provide a synopsis of the paper.
2
1.1 Algebraic overview
Fixing m and n, we are initially interested in a 2mn dimensional vector space
V that is a module for the U [gl(m|n)] minimal typical highest weight repre-
q
sentation Λ = (0˙ |α˙ ). The algebra contains a free complex variable q, whilst
m n
the representation π acting on V contains a free complex variable α. Our
Λ
V is actually (Z ) graded; this ensures compatibility with the (Z ) grading of
2 2
U [gl(m|n)].
q
Using the properties of U [gl(m|n)], we apply a version of the Kac induced
q
module construction (KIMC) [16, 17] to establish a (weight) basis {|ii}2mn for
i=1
V. This involves postulating |1i as a highest weight vector, and recursively
acting on |1i with all possible distinct products of simple lowering generators
Ea+1 todefinetheotherbasisvectors,normalisingaswego. Thisconstruction
a
requires a ‘Poincar´e–Birkhoff–Witt (PBW) lemma’ for U [gl(m|n)] [8, 28], i.e.
q
aset ofcommutationssufficientto transformany productofalgebragenerators
into a normal form (see §4.2), together with a statement that the algebra is
spanned by the set of all such normal forms.
Where V has a graded weight basis {|ii}2mn, the tensor product module
i=1
V ⊗ V has a natural 22mn dimensional basis {|ii ⊗ |ji}2mn , which inherits
i,j=1
a weight system and a grading from V. For our particular representation, the
orthogonaldecompositionofV ⊗V isknown[11],andcontainsnomultiplicities,
viz:
V ⊗V = V ,
k
k
M
where the submodule V has highest weight λ , and these λ are known, and
k k k
all distinct. To build R matrices acting on V ⊗V, we require an alternative,
orthonormal weight basis B= B for V ⊗V, corresponding to this decom-
k k
position, viz B is a basis for V . Again using the KIMC, the basis vectors
k k
of each B are derived as lineaSr combinations of the form θ (|ii⊗|ji), where
k ij
the coefficients θ are algebraic expressions in q and α. This process initially
ij
yieldsabasisB thatis notnecessarilyorthonormal,sowealsoapplyaGram–
k
Schmidt process to orthonormalise B into B . The desired R matrix is then
k k
a weighted sum of projectors onto these V , where the weights are eigenvalues
k
of the appropriate second order Casimir invariants.
The algebraicstructureof U [gl(m|n)]is detailedin§2, andanintroduction
q
toitshighestweightrepresentationsisprovidedin§3. In§4,weprovideanormal
orderingandaPBWlemmaforU [gl(m|n)]. Theconstructionofourparticular
q
(0˙ |α˙ ) representations is detailed in §5. §6 describes the construction of the
m n
bases B , and §7 describes the construction of projectors and R matrices.
k
3
1.2 Implementation and results
Explicit computations within the representation theory of quantum superalge-
bras are tedious and error-prone when performed manually. The dimensions
of representations are generally large, and in our case, we have the presence of
the two variables q and α; these generally manifest themselves in complicated
rationalalgebraicexpressions,whose symmetries must be continually identified
and exploited to avoid the arising of intractable messes of algebra.
The construction of the basis {|ii}2mn involves many applications of the
i=1
PBW lemma to simplify long strings of algebra generators. This is compu-
tationally expensive; firstly as the simplification involves a minimally-efficient
sorting process, and second as it involves a geometric explosion in the number
of terms being sorted.
The construction of the weight space bases B is nontrivial, as each basis
k
vectorofeachB generallycontainsmanytermsoftheformθ (|ii⊗|ji),where
k ij
the coefficients θ are generally complicated rationalalgebraic expressions in q
ij
and α. (That said, we have avoided the more theoretically difficult situation of
computing weight space decompositions in cases where there are weight multi-
plicitiesintheunderlyingcarrierspaceV.) Althoughthe Rmatriceshave24mn
components, Nature is kind to us in that most of these components are zero,
and those that are not are generally simpler than the θ .
ij
To the best of our knowledge, computer implementation of the algebraic
structures and algorithms described herein has not previously been achieved.
We haveimplementedthe entireprocessasasuite of Mathematica functions;
the thousands of lines of code perform algebraic computations that a human
being could not ever realistically expect to perform correctly.
From §2.3 onwards, we use U [gl(3|1)] to illustrate our results. These are
q
summarised in Appendix B; where we list the explicit matrix elements for the
generators of the underlying 8 dimensional representation, orthonormal bases
B for the 4 submodules V ⊂V ⊗V, the components of the associated 4 pro-
k k
jectors P onto the V , and finally, the trigonometric and quantum R matrices,
k k
Rˇ3,1(u) and Rˇ3,1, respectively.
Whilsttherearenotheoreticallimitstomandn,acurrentpracticallimitfor
computation is mn 6 4. This is convenient, as an immediate application [6] of
the materialcritically requires Rˇ4,1(u). Although translationof the interpreted
Mathematica code into a compiled language would increase the speed of the
computations, storage requirements would still limit mn to perhaps 7 in the
general case.
Further discussion of implementational issues and results is provided in §8.
4
2 The quantum superalgebras U [gl(m|n)]
q
The algebraic structures labeled U [gl(m|n)] are quantum superalgebras,1 de-
q
scribed in many places, e.g. [1, 10, 24, 25, 26], and in the book [3, see §6.5].
For our purposes, m and n are positive integers, to be regarded as fixed,
and q is to be regarded as a nonzero complex variable. As U [gl(m|n)] may be
q
unfamiliar to the readers of this organ, in §2.1 we introduce its phylogeny, and
in §2.2, we provide a full description of its structure in terms of generatorsand
relations. Beyondthat, in§2.3wedescribeits rootsystem,andin§2.4weshow
how it may be regardedas a Hopf (super)algebra.
2.1 The phylogeny of U [gl(m|n)]
q
1. Wherenisapositiveinteger,recallthattheLiealgebra gl(n)isequivalent
totheusual(complex)vectorspaceofn×n(complex)matricesaugmented
by a ‘vector multiplication’ operation which is the usual matrix multipli-
cation. gl(n) is of course a unital algebra, and is of dimension n2 and
rank n−1. The n2 generators {ea }n of gl(n) satisfy a commutation
b a,b=1
relation:
[ea ,ec ]=δcea −δaec ,
b d b d d b
where[·,·]is the usualcommutator (bracket), definedforX,Y ∈gl(n)by:
[X,Y],XY −YX.
2. Letting both m and n be positive integers, the Lie superalgebra gl(m|n)
may be obtained from gl(m+n) by retaining the generators {ea }m+n,
b a,b=1
butmodifyingthe definitionofthecommutatorbracketandcommutation
relations to include some ‘parity factors’ of ±1. Specifically, we have the
commutation relation:
[ea ,ec ]=δcea −(−)[eab][ecd]δaec , (1)
b d b d d b
where [·,·] is now the graded commutator (bracket), defined for homoge-
neous (see below) X,Y ∈gl(m|n) by:
[X,Y] , XY −(−)[X][Y]YX, (2)
and extended by linearity. In both (1) and (2), [X]∈ {0,1} refers to the
grading ofthehomogeneouselementX. Forthisreason,Liesuperalgebras
are sometimes called “gradedLie algebras”. From the gl(m+n) case, we
see that gl(m|n) is of dimension (m+n)2 and rank m+n−1.
1These structures are sometimes called “quantum supergroups”, but they are actually
(associative,noncommutative) algebras.
5
3. U[gl(m|n)] is then the usual universal enveloping algebra obtained from
gl(m|n)byregardingthegl(m|n)generatorsasletters inwordscontained
inU[gl(m|n)],wherethe(graded)commutatorbracketactsasarelationto
reducethealgebrasomewhat. U[gl(m|n)]isinfinitedimensional,although
of finite rank, viz, again m+n−1.
4. Thequantumsuperalgebra U [gl(m|n)]isthenaso-called‘q-deformation’2
q
ofU[gl(m|n)],whichmaintainsitsviabilityasaHopf(super)algebrastruc-
ture (see below) [12]. Roughly speaking, the deformation amounts to
‘exponentiation by q’; indeed U[gl(m|n)] may be recovered as the limit
q → 1 of U [gl(m|n)]. U [gl(m|n)] is of course also infinite dimensional,
q q
and again of rank m+n−1.
2.2 Generators and relations for U [gl(m|n)]
q
Following Zhang [28, pp1237-1238],we providea full descriptionof U [gl(m|n)]
q
intermsofgeneratorsandrelations. ForvariousinvertibleX,wewillrepeatedly
use the notation X ,X−1.
2.2.1 U [gl(m|n)] generators
q
Where I , {1,...,m +n} is the set of the gl(m|n) indices, we define a Z
2
grading [·]:I →Z :
2
0 if a6m even indices
[a],
1 else odd indices.
(cid:26)
Throughout, we shall use dummy indices a,b ∈ I where meaningful. A set of
(m+n)2 generators for U [gl(m|n)] is then:
q
K , m+n Cartan
a
Eb , a<b 1(m+n)(m+n−1) lowering . (3)
 a 2 
Eab, a<b 21(m+n)(m+n−1) raising 
Let usnow introduce the notation, for any a∈I: 
q ,q(−)[a],
a
For any power N, replacing q with qN immediately shows that (q )N =(qN) ,
a a
so we may write qN with impunity, specifically, we will write q ≡ q−1. Next,
a a a
an equivalent notation for Ka is qaEaa; where the exponential is defined in the
usual manner as an infinite sum, thus powers KN are meaningful; specifically,
a
we will often be working with N ∈ 1Z. Thus, under the mapping q 7→q, K is
2 a
mapped to K , where we intend K ≡K−1. As expected, for arbitrary powers
a a a
M,N, we have:
KMKN =KM+N where K0 ≡Id,
a a a a
where Id is the U [gl(m|n)] identity element. Apart from N ∈ N, powers (i.e.
q
products) of the non-Cartan generators (Ea )N for a6=b, are not meaningful.
b
2Wemightsay‘quantumdeformation’here,6buttherelationtoquantummechanicsismore
ofanalogythanofrigor.
Onthe generatorswe define anaturalZ gradingintermsofthe gradingon
2
the indices:
[KN],0, [Ea ],[a]+[b] (mod2), (4)
a b
wheretheformermaybeseenasaspecialcaseofthelatterbysettinga=band
making the identification Ka ≡q(−)[a]Eaa. We use the terms “even” and “odd”
forgeneratorsinthesamemanneraswedoforindices. ElementsofU [gl(m|n)]
q
are said to be homogeneous if they are linear combinations of generators of
the same grading. The product XY of homogeneous X,Y ∈ U [gl(m|n)] has
q
grading:
[XY],[X]+[Y] (mod2). (5)
Thus, for example, inspection of (4) and (5) shows that we may cheerfully
substitute [Ea ] for [Ea Eb ]. Further, we also have the following useful results
c b c
for a<b<c<d:
[Ea ][Eb ]=[Eb ] and [Ea ][Eb ]=[Ea ][Ec ]=0.
d c c b c b d
2.2.2 U [gl(m|n)] simple generators
q
The full set of generators (3) includes some redundancy; some can be regarded
as simple in that the rest may be expressedin terms of them. We shall call the
following subset of 3(m+n)−2 generators the U [gl(m|n)] simple generators:
q
K , m+n Cartan
a
Ea+1 , a<m+n m+n−1 simple lowering .
a
 
Ea , a<m+n m+n−1 simple raising
 a+1 
The fact that there are m +n−1 simple lowering generators indicates that
U [gl(m|n)] has rank m + n − 1. Note that there are only two odd simple
q
generators: Em+1 (lowering) and Em (raising).
m m+1
2.2.3 U [gl(m|n)] nonsimple generators
q
In the gl(m|n) case, the remaining nonsimple (non-Cartan) generators satisfy
the same commutation relations as the simple generators. The situation is
verydifferentforU [gl(m|n)];the nonsimple generatorsdo not satisfy the same
q
commutationrelationsasdothesimplegenerators. Instead,theyarerecursively
definedintermsofsumsofproductsofthesimplegenerators(see[27,p1971,(3)]
and [28, p1238,(2)]). Strictly speaking, they are not explicitly required for the
definitionofthealgebra;theirusecanhelpsimplifyotherwiselargeexpressions.
Towhit,asetofU [gl(m|n)]nonsimplegeneratorsmaybedefinedrecursively
q
for a<b by:
(a) Eb , Eb Ec −q Ec Eb nonsimple lowering
a c a c a c (6)
(b) Ea , Ea Ec −q Ec Ea nonsimple raising,
b c b c b c (cid:27)
where a<c<b; viz c is an arbitrary index, we do not intend a sum here.
7
In §5, we will have use for an alternative set of nonsimple generators,again
defined recursively for a<b by:
(a) Eb , Eb Ec −q Ec Eb alternative nonsimple lowering
a c a c a c (7)
(b) Ea , Ea Ec −q Ec Ea alternative nonsimple raising,
b c b c b c (cid:27)
where we intend Ea , Ea when Ea is simple, viz, for any |a−b|= 1. Note
b b b
that we use a boldface Ea where the original source [27] uses an overline Ea ;
b b
the use of the boldface notation saves the overline for indicating inverses.
These definitions may be written more concisely with some more notation.
Writing Sa ,sign(a−b), we may replace (6) and (7), for all a6=b, by:
b
(a) Ea = Ea Ec −qSbaEc Ea
b c b c b c (8)
(b) Eab = EacEcb−qScbaEcbEac. )
The two different sets of generators are in fact Hermitian conjugates. For
all meaningful indices a,b, we have:
(Ea )† =Eb , (Ea )† =Eb , (KN)† =KN,
b a b a a a
and these definitions ensure that (X†)† = X for all U [gl(m|n)] generators X.
q
Notethattheseareordinary,notZ gradedHermitianconjugates,meaningthat
2
we have (XY)† =Y†X†; expressly not (XY)† =(−)[X][Y]Y†X†.
Lastly, we mention a result of Zhang [27, Lemma 3], which gives us a more
efficient formula than (8b) for expanding the alternative nonsimple generators:
Ea =Ea +Sa ∆ Ec Ea , (9)
b b b c b c
c
X
for any indices a 6= b, where the sum is over all c strictly between a and b. (If
|a−b|=1, then the sum is ignored, and the result is trivial.)
Note that in (9), we have introduced the following handy notation:
∆ , q−q, ∆ ,q −q =(−)[a](q−q)=(−)[a]∆
a a a (10)
∆ , (∆)−1 ∆a ,(∆a)−1. )
2.2.4 The graded commutator
The graded commutator [·,·] : U [gl(m|n)]×U [gl(m|n)] → U [gl(m|n)], is de-
q q q
fined for homogeneous X,Y ∈U [gl(m|n)] by (2), viz:
q
[X,Y],XY −(−)[X][Y]YX,
andextendedbylinearity. Forcompleteness,wementionthatforassociative su-
peralgebras,ofwhichU [gl(m|n)]iscertainlyanexample,wehavethefollowing
q
useful graded commutator identities:
(a) [XY,Z] = X[Y,Z]+(−)[Y][Z][X,Z]Y
(11)
(b) [X,YZ] = [X,Y]Z+(−)[X][Y]Y[X,Z]. )
8
2.2.5 U [gl(m|n)] relations
q
With this notation, we have the following U [gl(m|n)] relations:
q
1. The Cartan generators all commute; for any powers M,N:
KMKN =KNKM. (12)
a b b a
2. The Cartan generators commute with the simple raising and lowering
generators in the following manner:
K Eb K =q(δba−δba±1)Eb . (13)
a b±1 a a b±1
From (13), we have the following useful interchange:
K Eb =q(δba−δba±1)Eb K . (14)
a b±1 a b±1 a
In Lemma 2 (proved in Appendix A.1), we show that (14) may be much
strengthened to:
KNEb =qN(δba−δca)Eb KN, (15)
a c a c a
for any meaningful indices b,c (viz b<c, b>c, and even b=c), and any
power N.
3. The non-Cartan simple generators satisfy the following commutation re-
lations (this is the really interesting part!):
K K −K K
[Ea ,Eb+1 ]=δa a a+1 a a+1. (16)
a+1 b b q −q
a a
Alternatively, again employing the notation of (10), we may write this:
[Ea ,Eb+1 ]=δa(−)[a] (−)[a]Ea −(−)[a+1]Ea+1 , (17)
a+1 b b a a+1
q
h i
where we have introduced the q-bracket, defined for various invertible
X ∈U [gl(m|n)], including scalars (well, scalar multiples of Id):
q
qX −qX
[X] , , observe that lim[X] =X. (18)
q q−q q→1 q
Note that in (17), Zhang [27] replaces (−)[a]Ea −(−)[a+1]Ea+1 with
a a+1
the more convenient expression h .
a
9
In fact, (16) generalises to a more useful result (proven in [8]):
K K −K K
[Ea ,Eb ]= a b a b =∆ (K K −K K ), (19)
b a a a b a b
q −q
a a
viz:
[Ea ,Eb ]=(−)[a] (−)[a]Ea −(−)[b]Eb .
b a a b
q
h i
We also have, for |a−b|>1, the commutations:
Ea+1 Eb+1 =Eb+1 Ea+1 and Ea Eb =Eb Ea . (20)
a b b a a+1 b+1 b+1 a+1
4. The squares of the odd simple generators are zero:
(Em )2 =(Em+1 )2 =0.
m+1 m
In fact, we may show that this implies that the squares of nonsimple odd
generators are also zero:
(Ea )2 =0, [a]6=[b]. (21)
b
5. Lastly, we have the U [gl(m|n)] Serre relations; their inclusion ensures
q
that the algebra is reduced enough to be simple. For a6=m:
(Ea+1a)2Ea±1+a1±1−∇Ea+1aEa±1+a1±1Ea+1a+Ea±1+a1±1(Ea+1a)2 =0
(Eaa+1)2Ea±a1±1+1−∇Eaa+1Ea±a1±1+1Eaa+1+Ea±a1±1+1(Eaa+1)2 =0.
where to save space, we have introduced the notation: ∇, q+q. These
may be more succinctly expressed using nonsimple generators. Noting
that for a6=m, we have q =q , the above become, again for a6=m:
a a+1
(a) Ea+1 Ea+2 = q Ea+2 Ea+1
a a a a a
(b) Ea Ea = q Ea Ea
a+1 a+2 a a+2 a+1
 (22)
((dc)) EEaa+−11a−1EEaa+1a == qqaEEaa+1aEEaa−+11a−1. 
a+1 a+1 a a+1 a+1

10