A Recent Look at the Quantum Berezinian in the Yangian Y(gl_{m|n})

0.13 MB·English

by  Lucy Gow

A Recent Look at the Quantum Berezinian in the Yangian
6 Y (gl )
0 m|n
0
2

n Lucy Gow∗
a
J University of Sydney

0
2 February 1, 2008

]
A
Q BrundanandKleshchevrecentlyintroducedanewfamilyofpresentationsoftheYangianY(gl )
n
. associated to the general linear Lie algebra gl , and thus provided a fresh approach to its study. In
h n
this article, we would like to show how some of their ideas can be fruitfully extended to consider the
t
a YangianY(gl ) associatedto the Lie superalgebragl . In particular,we give a new proof of the
m m|n m|n
result by Nazarov that the quantum Berezinian is central.1
[

3
v 1 Definition of Y (gl )
m|n
1
4
0 TheYangianY(glm|n)isdefinedin[7]tobetheZ2-gradedassociativealgebraoverCwithgenerators
1 t(r) and certain relations described below. We define the formal power series
0 ij
5
0 t (u)=δ +t(1)u−1+t(2)u−2+...,
ij ij ij ij
/
h
and a matrix
t
a m+n
m T(u)= t (u)⊗E (−1)j(i+1), (1)
ij ij
: iX,j=1
v
i where Eij is the standard elementary matrix and i is the parity of the index i. In analogy with the
X
usual Yangian Y(gl ) (see for example [2], [5], [6]), the defining relations are then expressed by the
n
r matrix product
a
R(u−v)T (u)T (v)=T (v)T (u)R(u−v)
1 2 2 1
where
1
R(u−v)=1− P
(u−v) 12
and P is the permutation matrix:
12
m+n
P = E ⊗E (−1)j.
12 ij ji
iX,j=1
Then we have the following equivalent form of the defining relations:
(−1)ij+ik+jk
[t (u), t (v)] = (t (u)t (v)−t (v)t (u)).
ij kl kj il kj il
(u−v)
∗[email protected]
1Theauthor wouldliketoacknowledge theguidanceandsupportofherPhDsupervisorAlexander Molev.
1
Throughout this article we will observe the following notation for entries of the inverse matrix of
T(u):
T(u)−1 =: t′ (u) n .
ij i,j=1
(cid:0) (cid:1)
A straightforwardcalculation yields the following relation in Y(gl ):
m|n
(−1)ij+ik+jk m+n m+n
[tij(u),t′kl(v)]= (u−v) ·(δkj tis(u)t′sl(v)− δil t′ks(v)tsj(u)). (2)
Xs=1 Xs=1
2 Gauss Decomposition of T(u)
In[1],theDrinfeldpresentationisdescribedintermsofthequasideterminantsofGelfandandRetakh
([3], [4]). In this article we make use of the analogous set of generators of the Yangian Y(gl ).
m|n
First we recall the definition of the quasideterminants and some conventionalnotation.
Definition 2.1. Let X be a square matrix over a ring with identity such that its inverse matrix X−1
exists,andsuchthatitsjithentryisaninvertibleelementofthering. Thentheijth quasideterminant
of X is defined by the formula
|X| = (X−1) −1.
ij ji
(cid:0) (cid:1)
Itissometimesconvenienttoadoptthefollowingalternativenotationforthequasideterminants:
x ··· x ··· x
11 1j 1n
(cid:12) ··· ··· (cid:12)
(cid:12) (cid:12)
|X|ij =:(cid:12) xi1 ··· xij ··· xin (cid:12).
(cid:12) (cid:12)
(cid:12) ··· ··· (cid:12)
(cid:12) (cid:12)
(cid:12) x ··· x ··· x (cid:12)
(cid:12) n1 nj nn (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The matrixT(u)definedin(1)hasthe followingGaussdecompositioninterms ofquasideterminants
(by Theorem 4.96 in [3]; see §5 in [1]):
T(u)=F(u)D(u)E(u)
for unique matrices
d (u) ··· 0
1
.
 . 
d (u) .
D(u) = 2 ,
 ... ... 
 
 0 ··· dm+n(u) 
1 e12(u) ··· e1,m+n(u)
E(u) =  ... e2,m+n(u) ,
 ... ... 
 
 0 1 
 
1 ··· 0
 f (u) ... ... 
F(u) = 21 ,
 ... ... 
 
 fm+n,1(u) fm+n,2(u) ··· 1 
2
where
t11(u) ··· t1,i−1(u) t1i(u)
di(u) = (cid:12)(cid:12) ... ... ... (cid:12)(cid:12),
(cid:12) (cid:12)
(cid:12)(cid:12) ti1(u) ··· ti,i−1(u) tii(u) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) t11(u) ··· t1,i−1(u)(cid:12)(cid:12) t1j(u)
eij(u) = di(u)−1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ti−1...,i(u) ·.·.·. ti−1,i...−1(u) ti−1,...j(u) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12),
(cid:12)(cid:12) ti1(u) ··· ti,i−1(u) tij(u) (cid:12)(cid:12)
(cid:12) (cid:12)
t11(u(cid:12)(cid:12)) ··· t1,i−1(u) t1i(u) (cid:12)(cid:12)
fji(u) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ti−1,...1(u) ·.·.·. ti−1,i...−1(u) ti−1...,i(u) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)di(u)−1.
(cid:12)(cid:12) tji(u) ··· tj,i−1(u) tji(u) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
It is easy to recovereachgenerat(cid:12)ing series t (u) by multiplying togeth(cid:12)er and taking commutators of
ij
d (u); 1 ≤ i ≤ m+n, and e (u) := e (u), f (u) = f (u); 1 ≤ i < m+n (see §5 of [1]). Thus
i i i,i+1 i i+1,i
the Yangian Y(gl ) is generated by the coefficients of the latter.
m|n
2.1 Some Useful Maps
HerewedefinesomeautomorphismsoftheYangianY(gl )andhomomorphismsbetweenYangians,
m|n
so that we may refer to them in the next section.
Let ω :Y(gl )→Y(gl ) be the automorphism defined by
m|n m|n m|n
ω :T(u)7→T(−u)−1.
Let τ :Y(gl )→Y(gl ) be the anti-automorphism defined by
m|n m|n
τ(t (u))=t (u)×(−1)i(j+1).
ij ji
Let ρ :Y(gl )→Y(gl ) be the isomorphism defined by
m|n m|n n|m
ρ (t (u))=t (−u).
m|n ij m+n+1−i,m+n+1−j
Let ϕ :Y(gl )֒→Y(gl ) be the inclusion which sends each generator t(r) ∈Y(gl )
m|n m|n m+k|n ij m|n
to the generator t(r) in Y(gl ).
k+i,k+j m+k|n
Finally, let ψ :Y(gl )→Y(gl ) be the injective homomorphism defined by
k m|n m+k|n
ψ =ω ◦ϕ ◦ω . (3)
k m+k|n m|n m|n
Thislasthomomorphismisusefulforstudyingquasideterminantssowediscussitinsomedetailwith
the following remarks.
Remark 2.1. We can calculate ψ (t (u)) explicitly for any 1≤i,j ≤m+n (see Lemma 4.2 of [1])
k ij
:
t (u) ··· t (u) t (u)
11 1k 1,k+j
(cid:12) .. .. .. .. (cid:12)
(cid:12) . . . . (cid:12)
ψk(tij(u))=(cid:12)(cid:12) t (u) ··· t (u) t (u) (cid:12)(cid:12).
(cid:12) k1 kk k,k+j (cid:12)
(cid:12) t (u) ··· t (u) t (u) (cid:12)
(cid:12) k+i,1 k+i,k k+i,k+j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
In particular, this means that for(cid:12)k ≥ 1, we have ψk(d1(u)) = dk+1(u), ψk((cid:12)e1(u)) = ek+1(u), and
ψ (f (u))=f (u).
k 1 k+1
Furthermore, by (3), we have for any k,l ≥ 1 that ψ ◦ψ = ψ , so we may generalise this
k l k+l
observation to give for instance ψ (d (u))=d (u).
k l k+l
3
Remark 2.2. Notice that the map ψ sends t′(r) ∈Y(gl ) to the element t′(r) in Y(gl ).
k ij m|n k+i,k+j m+k|n
Thus the subalgebra ψ (Y(gl )) is generated by the elements {t′(r) }n . Then, by (2), all
k m|n k+s,k+t s,t=1
elementsofthissubalgebracommutewiththoseofthesubalgebrageneratedbytheelements{t(r)}k .
ij i,j=1
By Remark 2.1, this implies in particular that for any i,j ≥1, the quasideterminants d (u) and
i
d (v) commute.
j
3 The Quantum Berezinian
The quantum Berezinian was defined by Nazarov [7] and plays a similar role in the study of the
Yangian Y(gl ) as the quantum determinant does in the case of the Yangian Y(gl ) (see [5]).
m|n n
Definition 3.1. The quantum Berezinian is the following power series with coefficients in the Yan-
gian Y(gl ):
m|n
b (u) := sgn(τ)t (u)t (u−1)···t (u−m+1)
m|n τ(1)1 τ(2)2 τ(m)m
τX∈Sm
× sgn(σ)t′ (u−m+1)···t′ (u−m+n)
m+1,m+σ(1) m+n,m+σ(n)
σX∈Sn
The first part of this expression for b (u) is quite special and so is given its own notation:
m|n
C (u):= sgn(τ)t (u)t (u−1)···t (u−m+1).
m τ(1)1 τ(2)2 τ(m)m
τX∈Sm
ItisclearthatC (u)isanelementofthesubalgebraofY(gl )generatedbytheset{t(r)} .
m m|n ij 1≤i,j≤m;r≥0
This subalgebra is isomorphic to the Yangian Y(glm) associated to the Lie algebra glm by the inclu-
sionY(glm)→Y(glm|n) whichsendeachgeneratorti(jr) in Y(glm)to the generatorof the same name
in Y(gl ). Moreover, C (u) is in fact the image under this map of the quantum determinant of
m|n m
the smaller Yangian Y(glm) (see [1], [5]). Then it is well known (see Theorem 2.32 in [6]) that we
have the alternative expression:
C (u)=d (u)d (u−1)···d (u−m+1).
m 1 2 m
We can extend this observation as follows:
Theorem 1. We have the following alternative expression for the quantum Berezinian:
b (u) = d (u)d (u−1)···d (u−m+1)
m|n 1 2 m
×d (u−m+1)−1···d (u−m+n)−1.
m+1 m+n
Proof. Notice that the second part of the expressionfor b (u) in Definition 3.1 is the image under
m|n
the isomorphism ρ ◦ω :Y(gl )→Y(gl ) of
n|m n|m n|m m|n
sgn(σ)t (u−m+1)···t (u−m+n−1)t (u+m−n) (4)
n,σ(n) 2,σ(2) 1,σ(1)
σX∈Sn
where in this expression (4) we are following the usual convention for denoting generators in the
Yangian Y(gl ). We recognise(by comparingwith(8.3)of[1]for example)thatthe expression(4)
n|m
isinfactC (u−m+n),theimageofthequantumdeterminantofY(gl )underthenaturalinclusion
n n
Y(gl ) ֒→ Y(gl ). So in order to verify the claim we must calculate the image of C (u−m+n)
n n|m n
under this map explicitly in terms of our quasideterminants d (v). Applying Proposition 1.6 of [4],
i
we find that the image of d (v) in Y(gl ) is (d (v))−1 in Y(gl ). This gives the desired
i n|m m+n+1−i m|n
result.
4
The following theorem is a result of Nazarov [7]. We give a new proof.
Theorem 2. The coefficients of the quantum Berezinian (3.1) are central in the algebra Y(gl ).
m|n
Proof. By Remark 2.2, we already know that the quantum Berezinian b (u) commutes with d (v)
m|n i
for 1 ≤ i ≤ m+n. So our problem reduces to showing that b (u) commutes with e (v) and with
m|n i
f (v) for each i between 1 and m+n−1. We proceed by breaking this problem into three cases.
i
Case 1: 1 ≤ i ≤ m−1. In this case, e (v) commutes with C (u) = d (u) ··· d (u−m+1)
i m 1 m
by Theorem 7.2 in [1]. On the other hand, e (v) is an element of the subalgebra generated by
i
{t(r)} and thus by Remark 2.2 commutes with d (u−m+s)−1 = t′ (u−m+s)
jk 1≤j,k≤m m+s m+s,m+s
for 1≤s≤n. Now we may use the anti-automorphismτ to show that f (v) also commutes with the
i
quantum Berezinian in this case because τ(e (v))=f (v) for 1≤i≤m−1.
i i
Case 2: m+1 ≤ i ≤ m+n−1. Applying Propositions 1.6 and 1.4 of [4] in turn to f (v), we
i
find an alternative expression:
t′ (v) t′ (v) ··· t′ (v) −1
i+1,i i+1,i+2 i+1,m+n t′ (v) ··· t′ (v)
fi(v) = −(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t′i+2...,i(v) t′i+2,i...+2(v) ·.·.·. ... (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) i+1,i...+1 i+1,m...+n (cid:12)(cid:12)(cid:12)(cid:12) .
(cid:12) (cid:12) (cid:12) t′ (v) ··· t′ (v) (cid:12)
(cid:12) t′ (v) t′ (v) ··· t′ (v) (cid:12) (cid:12) m+n,i+1 m+n,m+n (cid:12)
(cid:12) m+n,i m+n,i+2 m+n,m+n (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Thus, we find(cid:12)that for m+1≤i≤m+n−1, (cid:12)
ρ ◦ω (−f (v))=e (v),
n|m n|m m+n−i i
and similarly
ρ ◦ω (−e (v))=f (v).
n|m n|m m+n−i i
We apply this isomorphism to the results of Case 1 in the Yangian Y(gl ). This shows that e (v)
n|m i
and f (v) commute with the quantum Berezinian in the case where m+1≤i≤m+n−1.
i
Case 3: i=m. We begin by considering the Yangian Y(gl ). For this algebra we have that
1|1
the quantum Berezinian is b (u)=d (u)d (u)−1 and we would like to show that it commutes with
1|1 1 2
e (v) and f (v). So it will suffice to show
1 1
d (u)e (v)d (u)=d (u)e (v)d (u). (5)
1 1 2 2 1 1
We have
t (u) t (u) d (u) d (u)e (u)
11 12 = 1 1 1 (6)
(cid:18) t21(u) t22(u) (cid:19) (cid:18) f1(u)d1(u) f1(u)d1(u)e1(u)+d2(u) (cid:19)
t′ (v) t′ (v) d (v)−1+e (v)d (v)−1f (v) −e (v)d (v)−1
11 12 = 1 1 2 1 1 2 . (7)
(cid:18) t′21(v) t′22(v) (cid:19) (cid:18) −d2(v)−1f1(v) d2(v)−1 (cid:19)
An application of (2) gives
(u−v)[t (u),t′ (v)] = t (u)t′ (v)+t (u)t′ (v).
11 12 11 12 12 22
Substituting in the expressions from (6) and (7) then cancelling d (v), this gives
2
(u−v)[d (u),e (v)] = d (u)(e (v)−e (u)).
1 1 1 1 1
Similarly, by considering the commutator [t (u),t′ (v)], we derive the relation
12 22
(u−v)[d (u),e (v)] = d (u)(e (v)−e (u)).
2 1 2 1 1
We rewrite these relations to find
(u−v)e (v)d (u) = (u−v−1)d (u)e (v)+d (u)e (u),
1 1 1 1 1 1
(u−v)e (v)d (u) = (u−v−1)d (u)e (v)+d (u)e (u),
1 2 2 1 2 1
5
and by considering these expressions we see that (5) holds. Similarly, the quantum Berezinian com-
mutes with f (v).
1
NowwereturnourattentiontothegeneralYangianY(gl ). BysimilarappealstoRemark2.2
m|n
as in the first case, we see that e (v) and f (v) commute with d (u)···d (u − m + 2) and
m m 1 m−1
with d (u−m+2)−1··· d (u− m + n)−1. So we need only show they commute with
m+2 m+n
d (u−m+1)d (u−m+1)−1. Thisfollowsimmediatelywhenweapplythehomomorphismψ
m m+1 m−1
to the identity (5) in Y(gl ).
1|1
References
[1] JonathanBrundanandAlexanderKleshchev. Parabolicpresentationsofthe YangianY(gl ). To
n
appear in Commun.Math. Phys., 2004. Preprint: math.QA/0407011.
[2] VyjayanthiChariandAndrewPressley. A guide to quantum groups. CambridgeUniversityPress,
Cambridge, 1994.
[3] I. Gelfand, S. Gelfand, V. Retakh, and R.L. Wilson. Quasideterminants. to appear in Advances
in Mathematics, 2004. Preprint: arXiv:math.QA/0208146.
[4] I. M. Gelfand and V. S. Retakh. Determinants of matrices over noncommutative rings. Funct.
Anal. Appl., 25(2):91–102,1991.
[5] A. Molev, M. Nazarov, and G. Olshanskii. Yangians and classical Lie algebras. Russian Math.
Surveys, 51:205–282,1996.
[6] A. I. Molev. Yangians and their applications. In Handbook of algebra, Vol. 3, pages 907–959.
North-Holland, Amsterdam, 2003.
[7] M. L. Nazarov. Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys.,
21(2):123–131,1991.
6