Skew representations of twisted Yangians

0.4 MB·English

by  A. I. Molev

Skew representations of twisted Yangians
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0 A. I. Molev
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a
J School of Mathematics and Statistics

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University of Sydney, NSW 2006, Australia
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[email protected]

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Abstract
[

2 Analogsoftheclassical Sylvestertheorem havebeenknownformatrices
v with entries in noncommutativealgebras includingthe quantizedalgebra of
3 functionsonGLN andtheYangianforglN.Weproveaversionofthistheo-
0 remforthetwistedYangiansY(gN)associatedwiththeorthogonalandsym-
3 plecticLiealgebrasgN =oN orspN.Thisgivesrisetorepresentationsofthe
8 twisted Yangian Y(gN−M) on the space of homomorphisms Homg (W,V),
0 M
whereW andV arefinite-dimensionalirreduciblemodulesovergM andgN,
4
respectively. In the symplectic case these representations turn out to be ir-
0
reducible and we identify them by calculating the corresponding Drinfeld
/
h polynomials. We also apply the quantum Sylvester theorem to realize the
t twisted Yangian as a projective limit of certain centralizers in universal en-
a
m velopingalgebras.
:
v
i
X
r
a
1
1 Introduction
Let g be a complex reductive Lie algebra and a⊂g a reductive subalgebra. Sup-
posethatV isafinite-dimensionalirreducibleg-moduleandconsideritsrestriction
to the subalgebra a. This restriction is isomorphic to a direct sum of irreducible
finite-dimensional a-modules W with certain multiplicities m ,
µ µ
V|a ∼=⊕mµWµ.
µ
IfeachW isprovidedwithabasisandthedecompositionismultiplicity-free(i.e.,
µ
m 6 1 for all µ) then it can be used to get a basis of V as the union of the
µ
basesofthe spacesW whichoccurinthe decomposition.Thisobservationplayed
µ
a key role inthe constructionof the Gelfand–Tsetlinbases for the representations
of the general linear and orthogonal Lie algebras. Although the restriction of an
irreducible finite-dimensional representationof the symplectic Lie algebra sp to
2n
the subalgebra sp is not multiplicity-free in general, this approach can be
2n−2
extended to the symplectic case with the use of the isomorphism
V ∼=⊕Uµ⊗Wµ, (1.1)
µ
where
U =Hom (W ,V), dimU =m .
µ a µ µ µ
The space U is an irreducible module over the algebra C(g,a) = U(g)a, the
µ
centralizerofaintheuniversalenvelopingalgebraU(g);seee.g.Dixmier[2,Section
9.1].Now,ifsomebasesofthespacesU andW aregiventhenthedecomposition
µ µ
(1.1) yields the natural tensor product basis of V. The general difficulty of this
approach is the complicated structure of the algebra C(g,a). For each pair of the
classical Lie algebras
(g,a) = (gl ,gl ), (o ,o ), (sp ,sp ),
N M N M N M
(with even N and M in the symplectic case), the centralizer C(g,a) and its rep-
resentations can be studied with the use of the quantum algebras called Yangians
and twisted Yangians. The Yangian Y(gl ) for the general linear Lie algebra gl
N N
is a deformation of the universal enveloping algebra U(gl ⊗C[x]) in the class of
N
Hopfalgebras;seee.g.Drinfeld[3].Thetwisted Yangian Y(g )fortheorthogonal
N
or symplectic Lie algebra (g = o or g = sp ) was introduced by Olshan-
N N N N
ski[19]. This is a subalgebraof Y(gl ) andit can alsobe presentedby generators
N
anddefiningrelations;seealso[14].Finite-dimensionalirreduciblerepresentations
of the algebras Y(gl ) and Y(g ) admit a complete parametrization; see Drin-
N N
feld [4] and Tarasov [20] for the Yangian case, and the author’s work [10] for
the twisted Yangiancase.The Olshanskicentralizer construction [18,19]provides
‘almost surjective’ algebra homomorphisms
Y(gl )→C(gl ,gl ), Y(g )→C(g ,g ) (1.2)
N−M N M N−M N M
2
which allow one to equip the corresponding C(g,a)-module U in (1.1) with the
µ
structure of a representation of the Yangian or twisted Yangian, respectively. In
particular, in the case N −M = 2 this module over the twisted Yangian Y(g )
2
admits a natural basis which leads to a construction of weight bases of Gelfand–
TsetlintypefortherepresentationsoftheorthogonalandsymplecticLiealgebras;
see [13] for a review of these results.
In this paper we exploit the relationship between the (twisted) Yangians
andtheclassicalLiealgebrasinthereversedirection:weusetheweightbasescon-
structedin[13]toinvestigatetherepresentationsofthetwistedYangiansY(g )
N−M
emerging from the homomorphisms (1.2).
Bytheresultsof[4]and[10]theisomorphismclassofeachfinite-dimensional
irreducible representationV of the (twisted) Yangian is determined by its highest
weight which is a tuple of formal series over C in a formal parameter. Moreover,
simultaneous multiplication of all components of the highest weight by a fixed
invertible formal series corresponds to a representation obtained from V by the
composition with a simple automorphism of the (twisted) Yangian. It is natu-
ral to combine these representations into a single similarity class. In the case of
the Yangian Y(gl ) these similarity classes correspond to finite-dimensional irre-
N
ducible representationsofthe Yangianfor the speciallinear Lie algebrasl . Both
N
in the case of the Yangian and the twisted Yangian the similarity classes are pa-
rameterized by families of the Drinfeld polynomials (P (u),...,P (u)) with some
1 r
additional data in the twisted case. Each P (u) is a monic polynomial in u, and r
i
is the rank of the corresponding Lie algebra.
Given partitions λ=(λ ,...,λ ) and µ=(µ ,...,µ ), let V(λ) and V(µ)
1 N 1 M
bethefinite-dimensionalirreduciblerepresentationofgl andgl withthehighest
N M
weightsλandµ,respectively.ThespaceHom (V(µ),V(λ))isthenanirreducible
gl
M
representationoftheYangianY(gl ).ItsDrinfeldpolynomialswerecalculated
N−M
by Nazarov and Tarasov [17]. The result is a simple combinatorial rule which
allows one to ‘read off’ each polynomial P (u) from the contents of the cells of
i
the skew diagram λ/µ. These skew representations of the Yangian (they were
called elementary in [17]), may be regarded as building blocks for the class of
tame representations. This class is characterized by the property that the action
of a natural commutative subalgebra of the Yangian in such a representation is
semisimple. By [17], each tame representation is isomorphic to a tensor product
of skew representations.
A different way to define the homomorphism (1.2) in the case of gl is pro-
N
vided by the quantum Sylvester theorem. Recall that the classical Sylvester theo-
rem is the following identity for a numerical N ×N matrix A=(a ):
ij
m−1
detB =detA· am+1···N ,
m+1···N
(cid:16) (cid:17)
where B =(b ) is the m×m matrix formed by the minors b =ai,m+1···N ofA.
ij ij j,m+1···N
The sequencesoftopandbottomindices indicate the rowandcolumnnumbers of
the minor, respectively. The most generalnoncommutative analog of this identity
3
wasgivenbyGelfandandRetakhinthecontextofthetheoryofquasideterminants
originated in their work [6]; see also [5] for a review of this theory. ‘Quantum’
versions of this identity apply to the matrices formed by the generatorsof certain
quantum algebras, and the determinants are replaced by appropriate quantum
determinants. In particular, such a version was given by Krob and Leclerc [8] for
thequantizedalgebraoffunctionsonGL .Theirapproachisalsoapplicabletothe
N
Yangian Y(gl ). A different proof for the Yangian case is given in [12] where the
N
corresponding quantum Sylvester theorem was used to give a modified version of
the Olshanskicentralizerconstruction.This provideda new definition of the skew
representations of the Yangian and the calculation of their Drinfeld polynomials.
In this paper we produce a quantum Sylvester theorem for the twisted Yan-
gian Y(g ) with the use of the Sklyanin minors of the matrix of generators of
N
Y(g ). We firstobtainthe theoremfor the extended twisted Yangian X(g ) (Sec-
N N
tion 2), following the approachof [8]. The twisted Yangian Y(g ) is a quotient of
N
X(g ) which yields the corresponding result for Y(g ) (Section 3). In Section 4
N N
weapply the quantumSylvestertheoremtoconstructa newhomomorphism(1.2)
forthetwistedYangianandintroducethecorrespondingskewrepresentations.We
show that in the symplectic case each skew representation is irreducible and cal-
culate its highest weight and the Drinfeld polynomials. The Drinfeld polynomials
are found by the following simple combinatorial rule somewhat analogous to the
Yangian case [17] (see Section 4 below for a detailed formulation). Given a par-
tition ν = (ν ,...,ν ) we draw its diagram Γ(ν) as follows. First, place the row
1 n
with ν unit cells on the plane in such a way that the center of the leftmost cell
n
coincides with the origin. Then place the second row with ν −ν cells in such
n−1 n
a waythatthe southwestcornerofthis rowcoincideswith thenortheastcornerof
the first row. Continuing in this manner, we complete this procedure by placing
an infinite row of cells in such a way that its southwest corner coincides with the
northeastcorneroftherowwithν −ν cells.ThediagramΓ(ν)isobtainedasthe
1 2
union of the rows just placed and their images under the central symmetry with
respect to the southwest corner of the first row. The figure below represents the
diagram for the partition ν =(7,4), where the dot indicates the origin.
@ -
@
@q
@
@
(cid:27) @
To each cell of the diagram we attach its diagonal number, where by diagonals
we mean the lines passing northwest-southeast through the integer points of the
plane.Thelineonthefigureindicatesthe0-thdiagonalandthediagonalnumbers
are consecutive integers increasing from right to left. For any nonnegative integer
p denote by Γ(λ)(p) the diagram Γ(λ) lifted p units up.
4
Suppose now that V(λ) and V(µ) are the irreducible finite-dimensional rep-
resentations of sp and sp corresponding to partitions λ and µ having n and
2n 2m
m parts, respectively. Then the Drinfeld polynomials P (u),...,P (u) for the
1 n−m
skewrepresentationHom (V(µ),V(λ))ofthetwistedYangianY(sp )can
sp2m 2n−2m
be calculated by the following rule: all roots of the polynomial P (u) are simple
k
and they coincide with the diagonal numbers decreased by 1/2 of the cells of the
intersection Γ(µ)∩Γ(λ)(k−1) (see Theorem 4.9 and Example 4.10 below).
Finally, in Section 5 we give a realizationof the twisted Yangian Y(g ) as a
N
projective limit of centralizers in the universal enveloping algebras. This is a new
versionofthecentralizerconstruction(cf.[19,15])whichisbasedonthequantum
Sylvester theorem.
The recent work of Nazarov [16] is also devoted to the skew representations
of the twisted Yangians although from a different perspective. He uses the clas-
sical Weyl’s approach and gives a realization of the skew representations in the
tensor powers of the vector representation by applying certain generalized Young
symmetrizers.
2 Extended twisted Yangian
Westartbystatingandprovingsomeauxiliaryresultsabouttheextendedtwisted
Yangian X(g ); see [14] for more details.
N
2.1 Preliminaries
Weshallbeconsideringtheorthogonalandsymplecticcasessimultaneously,unless
otherwise stated. Given a positive integer N, we number the rows and columns
of N ×N matrices by the indices {−n,...,−1,0,1,...,n} if N = 2n+1, and
by {−n,...,−1,1,...,n} if N = 2n. Similarly, in the latter case the range of
summation indices −n 6 i,j 6 n will usually exclude 0. It will be convenient to
use the symbol θ which is defined by
ij
1 in the orthogonalcase,
θ =
ij
(sgni·sgnj in the symplectic case.
Throughout the paper, whenever the double sign ± or ∓ occurs, the upper sign
corresponds to the orthogonal case and the lower sign to the symplectic case. By
A7→At wewilldenotethe matrixtranspositionsuchthat(At) =θ A .Let
ij ij −j,−i
the E denote the standard basis vectors of the general linear Lie algebra gl .
ij N
Thesevectorsmaybealsoregardedaselementsoftheuniversalenvelopingalgebra
U(gl ). For this reason we want to distinguish the E from the standard matrix
N ij
units e which are considered as basis elements of the endomorphism algebra
ij
EndCN. Introduce the following elements of the Lie algebra gl :
N
F =E −θ E , −n6i,j 6n.
ij ij ij −j,−i
5
The Lie subalgebra g of gl spanned by the elements F is isomorphic to the
N N ij
orthogonalLie algebrao or the symplectic Lie algebrasp (in the latter case N
N N
is even).
The extended twisted Yangian X(g ) corresponding to the Lie algebra g is
N N
the associativealgebrawithgeneratorss(1), s(2),... where −n6i,j 6n,subject
ij ij
to the defining relations written in terms of the generating series
s (u)=δ +s(1)u−1+s(2)u−2+···∈X(g )[[u−1]]
ij ij ij ij N
as follows
(u2−v2)[s (u),s (v)]=(u+v) s (u)s (v)−s (v)s (u)
ij kl kj il kj il
−(u−v)(cid:0)θk,−jsi,−k(u)s−j,l(v)−θi,−(cid:1)lsk,−i(v)s−l,j(u)
+θi,−j(cid:0)sk,−i(u)s−j,l(v)−sk,−i(v)s−j,l(u) , (cid:1)
whereuandvdenoteformalvariabl(cid:0)es.Thedefiningrelationscanalsob(cid:1)epresented
in a convenientmatrix form.Denote byS(u) the N×N matrix whoseij-thentry
iss (u).WemayregardS(u)asanelementofthealgebraX(g )[[u−1]]⊗EndCN
ij N
given by
S(u)= s (u)⊗e ,
ij ij
i,j
X
where the e denote the standard matrix units. For any positive integer m we
ij
shall be using the algebras of the form
X(g )[[u−1]]⊗EndCN ⊗···⊗EndCN, (2.1)
N
with m copies of EndCN. For any a∈{1,...,m} we denote by S (u) the matrix
a
S(u) which acts on the a-th copy of EndCN. That is, S (u) is an element of the
a
algebra (2.1) of the form
S (u)= s (u)⊗1⊗···⊗1⊗e ⊗1⊗···⊗1,
a ij ij
i,j
X
where the e belong to the a-th copy of EndCN and 1 is the identity matrix.
ij
Similarly, if
C = c e ⊗e ∈EndCN ⊗EndCN,
ijkl ij kl
i,j,k,l
X
then for distinct indices a,b ∈ {1,...,m} we introduce the element C of the
ab
algebra (2.1) by
C = c 1⊗1⊗···⊗1⊗e ⊗1⊗···⊗1⊗e ⊗1⊗···⊗1,
ab ijkl ij kl
i,j,k,l
X
6
where the e and e belong to the a-th and b-th copies of EndCN, respectively.
ij kl
Consider now the permutation operator
P = e ⊗e ∈EndCN ⊗EndCN.
ij ji
i,j
X
The rationalfunction R(u)=1−Pu−1 with values in the tensor product algebra
EndCN ⊗EndCN is called the Yang R-matrix. Introduce its transposed Rt(u)
by
Rt(u)=1−Qu−1, Q= θ e ⊗e . (2.2)
ij −j,−i ji
i,j
X
The defining relations for the extended twisted Yangian X(g ) are equivalent to
N
the quaternary relation
R(u−v)S (u)Rt(−u−v)S (v)=S (v)Rt(−u−v)S (u)R(u−v). (2.3)
1 2 2 1
Let u ,...,u be independent variables. For k > 2 consider the rational
1 k
function R(u ,...,u ) with values in (EndCN)⊗k defined by
1 k
R(u ,...,u )=(R )(R R )···(R ···R ),
1 k k−1,k k−2,k k−2,k−1 1k 12
where we abbreviate R =R (u −u ). Set
ij ij i j
S =S (u ), 16i6k and Rt =Rt =Rt(−u −u ), 16i<j 6k.
i i i ij ji ij i j
For an arbitrary permutation (p ,...,p ) of the numbers 1,...,k, we abbreviate
1 k
hS ,...,S i=S (Rt ···Rt )S (Rt ···Rt )···S .
p1 pk p1 p1p2 p1pk p2 p2p3 p2pk pk
The identity
R(u ,...,u )hS ,...,S i=hS ,...,S iR(u ,...,u ) (2.4)
1 k 1 k k 1 1 k
can be deduced from the quaternary relation (2.3); see [14, Proposition 4.2]. Now
specialize the variables u by setting
i
u =u−i+1, i=1,...,k. (2.5)
i
It is well known that under this specialization, R(u ,...,u ) coincides with the
1 k
anti-symmetrization operator A on (CN)⊗k, where
k
A = sgnσ·P ,
k σ
σX∈Sk
and P denotes the image of σ ∈ S under the natural action of S on (CN)⊗k;
σ k k
see e.g. [14, Proposition 2.3]. Hence specializing the variables in (2.4) we get
A hS ,...,S i=hS ,...,S iA .
k 1 k k 1 k
7
This element of the tensor product X(g )[[u−1]]⊗(EndCN)⊗k can be written as
N
sa1···ak(u)⊗e ⊗···⊗e ,
b1···bk a1b1 akbk
summed overthe indiceXs a ,b ∈{−n,...,n}. We also set sa(u)=s (u). We call
i i b ab
the elements sa1···ak(u) of X(g )[[u−1]] the Sklyanin minors of the matrix S(u).
b1···bk N
Clearly, the Sklyanin minors are skew-symmetric with respect to permutations of
the upper indices and of the lower indices:
saσ(1)···aσ(k)(u)=sgnσ·sa1···ak(u) and sa1···ak (u)=sgnσ·sa1···ak(u)
b1···bk b1···bk bσ(1)···bσ(k) b1···bk
for any σ ∈S .
k
Proposition 2.1. We have the relations
(u2−v2)[s (u),sa1···ak(v)]
pq b1···bk
k
=(u+v) s (u)sa1···p···ak(v)−sa1 ··· ak (v)s (u)
aiq b1 ··· bk b1···q···bk pbi
Xi=1(cid:16) (cid:17)
k
−(u−v) θ s (u)sa1···−q···ak(v)−θ sa1 ··· ak (v)s (u)
ai,−q p,−ai b1 ··· bk p,−bi b1···−p···bk −bi,q
Xi=1(cid:16) (cid:17)
k
+θ s (u)sa1···−q···ak(v)−sa1 ··· ak (v)s (u)
p,−q ai,−p b1 ··· bk b1···−p···bk −q,bi
Xi=1(cid:16) (cid:17)
+ θ s (u)sa1···p···−q···ak(v)−θ sa1 ··· ak (v)s (u) ,
aj,−q ai,−aj b1 ··· bk p,−bi b1···−p···q···bk −bi,bj
Xi6=j(cid:16) (cid:17)
where in the Sklyanin minors the indices p and q replace a and b , respectively, in
i i
the first sum; the indices −q and −p replace a and b , respectively, in the second
i i
and third sums; in the fourth sum p and −q replace a and a , respectively, and
i j
−p and q replace b and b , respectively.
i j
Proof. By (2.4), we have the relation
R(u,v,v−1,...,v−k+1)hS ,...,S i
0 k
=hS ,...,S iR(u,v,v−1,...,v−m+1), (2.6)
k 0
where we have used an extra copy of the algebra EndCN labelled by 0 and the
parameters are specialized as follows
u =u, and u =v−i+1 for i=1,...,k.
0 i
Then one easily verifies (see e.g. [12]) that the product of R-matrices in (2.6)
simplifies to
1
R(u,v,v−1,...,v−k+1)=A 1− (P +···+P ) .
k 01 0k
u−v
(cid:16) (cid:17)
8
ApplyingthetranspositionoverthezerothcopyofEndCN andreplacinguby−u
we also deduce that
1
A Rt ···Rt =A 1+ (Q +···+Q ) . (2.7)
k 01 0k k u+v 01 0k
(cid:16) (cid:17)
Hence (2.6) takes the form
1 1
1− (P +···+P ) S (u) 1+ (Q +···+Q ) A hS ,...,S i
01 0k 0 01 0k k 1 k
u−v u+v
(cid:16) 1 (cid:17) (cid:16) 1 (cid:17)
=hS ,...,S iA 1+ (Q +···+Q ) S (u) 1− (P +···+P ) .
k 1 k 01 0k 0 01 0k
u+v u−v
(cid:16) (cid:17) (cid:16) (cid:17)
It remains to apply both sides to the vector e ⊗e ⊗···⊗e and compare the
q b1 bk
coefficients at the vector e ⊗e ⊗···⊗e , where the e denote the canonical
p a1 ak i
basis vectors of CN.
Corollary 2.2. Suppose that for some indices i,j,l,m∈{1,...,k} we have a =
i
−b and b =−a . Then
l j m
[s (u),sa1···ak(v)]=0.
aibj b1···bk
Proof. By the skew-symmetry property, the Sklyanin minor is zero if it has two
repeated upper or lower indices. Hence we may assume that i = m if and only
if j = l. Suppose first that i 6= m. Then using the skew-symmetry of Sklyanin
minors, we derive from Proposition 2.1 that
(u−v−1)(u+v+1)[s (u),sa1···ak(v)]=θ [s (u),sa1···ak(v)].
aibj b1···bk ai,−bj −bj,−ai b1···bk
The same relation holds with i and j replaced by m and l, respectively, which
provesthe claim in the case under consideration.If a =−b then Proposition2.1
i j
immediately gives
(u−v−1)(u+v+1)[s (u),sa1···ak(v)]=0,
aibj b1···bk
completing the proof.
The series
sdetS(u)=s−n···n(u)∈X(g )[[u−1]]
−n···n N
is called the Sklyanin determinant of the matrix S(u). Corollary 2.2 implies that
allthe coefficientsofthis seriesbelong tothe centerofthe algebraX(g );seealso
N
[14, Theorem 4.8] for a slightly different proof.
The matrix S(u) is invertible and we shall denote by S−1(u) the inverse
matrix. The mapping
̟ :S(u)7→S−1 −u−N/2 (2.8)
N
(cid:0) (cid:1)
9
definesaninvolutiveautomorphismofthealgebraX(g );see[14,Proposition6.5].
N
The Sklyanin comatrix S(u) is defined by the relation
S(u)S(u−N +1)=sdetS(u). (2.9)
b
Due to (2.8), the mapping
b
S(u)7→S(−u+N/2−1) (2.10)
defines a homomorphism of X(g ) into itself.
N b
We shall also use the auxiliary minors sˇa1···ak (u)∈X(g )[[u−1]] defined
b1···bk−1,c N
by
A hS ,...,S iRt ···Rt
k 1 k−1 1k k−1,k
= sˇa1···ak (u)⊗e ⊗···⊗e ⊗e , (2.11)
b1···bk−1,c a1b1 ak−1bk−1 akc
X
summed over a ,b ,c∈{−n,...,n}. Since
i i
A hS ,...,S iRt ···Rt S =A hS ,...,S i,
k 1 k−1 1k k−1,k k k 1 k
we immediately obtain the relation
n
sa1···ak(u)= sˇa1···ak (u)s (u−k+1). (2.12)
b1···bk b1···bk−1,c cbk
c=−n
X
We obviously have
sˇaσ(1)···aσ(k)(u)=sgnσ·sˇa1···ak (u)
b1···bk−1,c b1···bk−1,c
for any σ ∈S . Also,
k
sˇa1···ak (u)=sgnσ·sˇa1···ak (u)
bσ(1)···bσ(k−1),c b1···bk−1,c
foranyσ ∈S ;see[9].Furthermore,itisstraightforwardtoobtainthefollowing
k−1
property of the auxiliary minors from their definition (cf. [9, Proposition 4.4]): if
c∈/ {a ,...,a } and c∈/ {−b ,...,−b } then
1 k−1 1 k−1
sˇa1···ak (u)=0 (2.13)
b1···bk−1,c
if c6=a , while
k
sˇa1···ak−1,c(u)=sa1···ak−1(u). (2.14)
b1···bk−1,c b1···bk−1
Set(a ,...,a )=(−n,...,n).Thenthematrixelementss (u)oftheSklyanin
1 N aiaj
comatrix S(u) are given by
b
s (u)=(−1)N−isˇa1···aN (u), (2.15)
b aiaj a1···ai···aN,aj
where the hat on the right hand side indicates the index to be omitted; see [9,
b b
Section 6].
10