Yangian of the General Linear Lie Superalgebra
We prove several basic properties of the Yangian of the Lie superalgebra M|N.
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| citation_txt | Yangian of the General Linear Lie Superalgebra. Maxim Nazarov. SIGMA 16 (2020), 112, 24 pages |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 112, 24 pages
Yangian of the General Linear Lie Superalgebra
Maxim NAZAROV
Department of Mathematics, University of York, York YO10 5DD, UK
E-mail: maxim.nazarov@york.ac.uk
Received July 05, 2020, in final form November 01, 2020; Published online November 05, 2020
https://doi.org/10.3842/SIGMA.2020.112
Abstract. We prove several basic properties of the Yangian of the Lie superalgebra glM |N .
Key words: Berezinian; Hopf superalgebra; Yangian
2020 Mathematics Subject Classification: 16T20; 17B37; 81R50
1 Introduction
Let glM |N be the general linear Lie superalgebra over the complex field C. We will assume that
at least one of the non-negative integers M and N is not zero. The Yangian of glM |N has been
introduced in [8] by extending the definition of the Yangian of the general linear Lie algebra glM ,
see for instance [6]. We will denote this extension by Y(glM |N ). It is a deformation of the
universal enveloping algebra U(glM |N [u]) of the polynomial current Lie superalgebra glM |N [u]
in the class of Hopf superalgebras. The definition of Y(glM |N ) is reviewed in our Section 2.
In our Section 2 we will define two ascending filtrations on the associative algebra Y(glM |N ).
The graded algebra associated with the first filtration is supercommutative. We prove that its
elements corresponding to the defining generators (2.6) of Y(glM |N ) are free generators of this
supercommutative algebra. The graded algebra associated with the second ascending filtration is
isomorphic to U(glM |N [u]). We prove this by using the representation theory of Y(glM |N ). Our
proof follows [7] where the Yangian of the queer Lie superalgebra qM ⊂ glM |M was studied. The
freeness of the supercommutative graded algebra associated with the first filtration on Y(glM |N )
follows from this isomorphism. Another proof of the freeness property was given in [4].
Two different families of central elements of Y(glM |N ) have been defined in [8]. The definition
of the first family uses the Hopf superalgebra structure on Y(glM |N ). This definition is reviewed
in our Section 3. It was conjectured in [8] that the first family generates the centre of Y(glM |N ).
Shortly after the publication of [8] this conjecture was proved by the author. The method of
that proof was then used in [6] where the Yangian of glM was considered. This method was also
used in [4, 7]. We include the original proof of this conjecture for Y(glM |N ) in our Section 3.
The second definition extends the notion of a quantum determinant for the Yangian of glM ,
see again [6] and references therein. This definition is reviewed in our Section 4. The main result
of [8] was the relation between the two families of central elements of Y(glM |N ). However only
a summary of the proof of this relation was given in [8] while the details were left unpublished.
The main purpose of the present article is to publish the detailed original proof of this relation.
Since the Yangian Y(glM |N ) was introduced in [8] it has been studied by several other authors.
Here we do not aim to review the literature. Still let us mention the work [3] which contains
a direct proof of the centrality of the elements of Y(glM |N ) from our second family. Let us also
mention the work [9] which provides a generalization of Y(glM |N ) to arbitrary parity sequences.
This paper is a contribution to the Special Issue on Representation Theory and Integrable Systems in honor
of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
mailto:maxim.nazarov@york.ac.uk
https://doi.org/10.3842/SIGMA.2020.112
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
2 M. Nazarov
2 Definition of the Yangian
Throughout this article we will use the following general conventions. Let A and B be any two
associative Z2-graded algebras. Their tensor product A ⊗ B is also an associative Z2-graded
algebra such that for any homogeneous elements X,X ′ ∈ A and Y, Y ′ ∈ B
(X ⊗ Y )(X ′ ⊗ Y ′) = XX ′ ⊗ Y Y ′(−1)degX′ deg Y , (2.1)
deg(X ⊗ Y ) = degX + deg Y. (2.2)
For any Z2-graded modules U and V over A and B respectively, the vector space U ⊗ V is
a Z2-graded module over A⊗ B such that for any homogeneous elements x ∈ U and y ∈ V
(X ⊗ Y )(x⊗ y) = Xx⊗ Y y(−1)deg x deg Y , (2.3)
deg(x⊗ y) = deg x+ deg y. (2.4)
A homomorphism α : A→ B is a linear map such that α(XX ′) = α(X)α(X ′) for all X,X ′ ∈ A.
But an antihomomorphism β : A→ B is a linear map such that for all homogeneous X,X ′ ∈ A
β(XX ′) = β(X ′)β(X)(−1)degX degX′
. (2.5)
If A is unital, let ιh be its embedding into the tensor product A⊗n as the h-th tensor factor:
ιh(X) = 1⊗(h−1) ⊗X ⊗ 1⊗(n−h) for h = 1, . . . , n.
Here n can be any positive integer. We will also use various embeddings of the algebra A⊗m
into A⊗n for any m = 1, . . . , n. For any choice of pairwise distinct indices h1, . . . , hm ∈ {1, . . . , n}
and of an element X ∈ A⊗m of the form X = X(1) ⊗ · · · ⊗X(m) we will denote
Xh1...hm = ιh1
(
X(1)
)
· · · ιhm
(
X(m)
)
∈ A⊗n.
We will then extend the notation Xh1...hm to all elements X ∈ A⊗m by linearity.
Now let the indices i, j run through 1, . . . ,M +N . We will always write ı̄ = 0 if 1 6 i 6M
and ı̄ = 1 if M < i 6 M + N . Consider the Z2-graded vector space CM |N . Let ei ∈ CM |N be
an element of the standard basis. The Z2-grading on CM |N is defined so that deg ei = ı̄. Let
Eij ∈ EndCM |N be the standard matrix unit, so that Eijek = δjkei. The associative algebra
EndCM |N is Z2-graded so that degEij = ı̄+ ̄.
For any n we can identify the tensor product
(
EndCM |N
)⊗n
with the algebra End
((
CM |N
)⊗n)
acting on the vector space
(
CM |N
)⊗n
by repeatedly using the conventions (2.3) and (2.4).
Let us introduce the Yangian of the Lie superalgebra glM |N . This is the complex associative
unital Z2-graded algebra Y(glM |N ) with the countable set of generators
T
(r)
ij , where r = 1, 2, . . . and i, j = 1, . . . ,M +N. (2.6)
The Z2-grading on the algebra Y(glM |N ) is determined by setting deg T
(r)
ij = ı̄+ ̄ for r > 1. To
write down defining relations for these generators we will employ the series
Tij(u) = δij · 1 + T
(1)
ij u
−1 + T
(2)
ij u
−2 + · · · (2.7)
in a formal variable u with coefficients from Y(glM |N ). Then for all possible indices i, j, k, l
(u− v)[Tij(u), Tkl(v)](−1)ı̄k̄+ı̄l̄+k̄l̄ = Tkj(u)Til(v)− Tkj(v)Til(u), (2.8)
where v is another formal variable. The square brackets here stand for the supercommutator.
Notice that the series denoted by Tij(u) here and in [8] differ by the scalar factor (−1)(ı̄+1)̄.
Yangian of the General Linear Lie Superalgebra 3
We will also use the following matrix form of the defining relations (2.8). Take the element
P =
M+N∑
i,j=1
Eij ⊗ Eji(−1)̄ ∈
(
EndCM |N
)⊗2
. (2.9)
This element acts on the vector space
(
CM |N
)⊗2
so that ei⊗ej 7→ ej⊗ei(−1)ı̄̄. Here we identify
the algebra
(
EndCM |N
)⊗2
with the algebra End
((
CM |N
)⊗2)
by using (2.3).
For any n let Sn be the symmetric group acting on the set {1, . . . , n} by permutations. For
each m = 1, . . . , n− 1 denote by σm the element of Sn exchanging m and m+ 1. The group Sn
also acts on the vector space
(
CM |N
)⊗n
. This action is defined by the assignment σm 7→ Pm,m+1
for each m. Here we identify the algebra
(
EndCM |N
)⊗n
with End
((
CM |N
)⊗n)
via (2.3), (2.4).
The rational function R(u) = 1 − Pu−1 with values in the algebra
(
EndCM |N
)⊗2
is called
the Yang R-matrix. It satisfies the Yang–Baxter equation in the algebra
(
EndCM |N
)⊗3
(u, v, w)
R12(u− v)R13(u− w)R23(v − w) = R23(v − w)R13(u− w)R12(u− v). (2.10)
Since P 2 = 1, we also have the relation
R(−u)R(u) = 1− u−2. (2.11)
Now combine all the series (2.7) into the single element
T (u) =
M+N∑
i,j=1
Eij ⊗ Tij(u) ∈
(
EndCM |N
)
⊗Y(glM |N )
[[
u−1
]]
. (2.12)
For any n and any p = 1, . . . , n we will denote
Tp(u) = ιp ⊗ id(T (u)) ∈
(
EndCM |N
)⊗n ⊗Y(glM |N )
[[
u−1
]]
. (2.13)
By using this notation for n = 2 the relations (2.8) can be rewritten as
(R(u− v)⊗ 1)T1(u)T2(v) = T2(v)T1(u)(R(u− v)⊗ 1). (2.14)
Namely, after multiplying each side of (2.14) by u−v it becomes a relation of series in u, v with
coefficients in
(
EndCM |N
)⊗2 ⊗Y(glM |N ) equivalent to the collection of all the relations (2.8).
Proposition 2.1. An antiautomorphism of Y(glM |N ) can be defined by the assignment
Tij(u) 7→ Tij(−u). (2.15)
Proof. Due to the convention (2.1), by using the notation (2.13) for n = 2 we get
T1(u)T2(v) =
M+N∑
i,j,k,l=1
Eij ⊗ Ekl ⊗ Tij(u)Tkl(v)(−1)(ı̄+̄)(k̄+l̄), (2.16)
T2(−v)T1(−u) =
M+N∑
i,j,k,l=1
Eij ⊗ Ekl ⊗ Tkl(−v)Tij(−u). (2.17)
By using the convention (2.5), the antihomomorphism property of (2.15) follows from the relation
(R(u− v)⊗ 1)T2(−v)T1(−u) = T1(−u)T2(−v)(R(u− v)⊗ 1),
which is obtained from (2.14) by using (2.11). The antihomomorphism (2.15) is clearly involutive
and therefore bijective. �
4 M. Nazarov
For all indices i, j define the series T ′ij(u) by using the element inverse to (2.12) so that
T (u)−1 =
M+N∑
i,j=1
Eij ⊗ T ′ij(u). (2.18)
Proposition 2.2. An antiautomorphism of Y(glM |N ) can be defined by the assignment
Tij(u) 7→ T ′ij(u). (2.19)
Proof. Similarly to (2.17), by using the notation (2.13) for n = 2 we get
T2(v)−1T1(u)−1 =
M+N∑
i,j,k,l=1
Eij ⊗ Ekl ⊗ T ′kl(v)T ′ij(u).
Comparing this with (2.16) the antihomomorphism property of (2.19) follows from the relation
(R(u− v)⊗ 1)T2(v)−1T1(u)−1 = T1(u)−1T2(v)−1(R(u− v)⊗ 1), (2.20)
which is obtained by multiplying both sides of the defining relation (2.14) on the left and right
by T2(v)−1 and then by T1(u)−1. The bijectivity of (2.19) follows from Proposition 3.2 below. �
Further, let τ be the antiautomorphism of EndCM |N defined by the assignment
Eij 7→ Eji(−1)ı̄(̄+1).
Then by the definition (2.12) we have
τ ⊗ id(T (u)) =
M+N∑
i,j=1
Eji ⊗ Tij(u)(−1)ı̄(̄+1) =
M+N∑
i,j=1
Eij ⊗ Tji(u)(−1)̄(ı̄+1).
Proposition 2.3. An antiautomorphism of Y(glM |N ) can be defined by the assignment
Tij(u) 7→ Tji(u)(−1)̄(ı̄+1). (2.21)
Proof. Observe that (τ ⊗ τ)(P ) = P and hence
(τ ⊗ τ)(R(u− v)) = R(u− v). (2.22)
Therefore the antihomomorphism property of (2.21) follows from the relation which is obtained
by applying τ ⊗ τ ⊗ id to both sides of(2.14), see the proofs of Propositions 2.1 and 2.2 above.
To prove the bijectivity of the antihomomorphism (2.21), observe that its square is given by
Tij(u) 7→ Tij(u)(−1)ı̄+̄.
In particular, the square is an automorphism of the Z2-graded algebra Y(glM |N ). �
Put T ](u) = τ ⊗ id
(
T (u)−1
)
. Then by (2.18)
T ](u) =
M+N∑
i,j=1
Eji ⊗ T ′ij(u)(−1)ı̄(̄+1) =
M+N∑
i,j=1
Eij ⊗ T ′ji(u)(−1)̄(ı̄+1). (2.23)
Corollary 2.4. An automorphism of the algebra Y(glM |N ) can be defined by the assignment
Tij(u) 7→ T ′ji(u)(−1)̄(ı̄+1). (2.24)
Yangian of the General Linear Lie Superalgebra 5
Proof. The assignment (2.24) can also be obtained by first applying (2.21) to Tij(u) and then
applying (2.19) to the result. Hence (2.24) defines an automorphism of the algebra Y(glM |N ) as
a composition of two antiautomorphisms. �
By the definition (2.23) the homomorphism property of (2.24) is equivalent to the relation
(R(u− v)⊗ 1)T ]1(u)T ]2(v) = T ]2(v)T ]1(u)(R(u− v)⊗ 1), (2.25)
which can also be obtained by applying τ ⊗ τ ⊗ id to both sides of (2.20) and then using (2.22).
Proposition 2.5. The antiautomorphisms (2.15), (2.19), (2.21) of Y(glM |N ) pairwise commute.
Proof. The antiautomorphism (2.15) clearly commutes with either of (2.19), (2.21). To prove
the commutativity of the latter two, consider the tensor product of the antiautomorphisms τ−1
and (2.21) of EndCM |N and Y(glM |N ) respectively. This is is an antiautomorphism of the
algebra EndCM |N ⊗ Y(glM |N ), and the series (2.12) is invariant under this antiautomorphism.
Hence the series (2.18) is also invariant under it. The latter invariance implies that (2.21) maps
T ′ij(u) 7→ T ′ji(u)(−1)̄(ı̄+1).
Hence (2.24) can also be obtained by first applying (2.19) to Tij(u) and then applying (2.21) to
the result. Comparing this with the proof of Corollary 2.4 completes the argument. �
Consider the universal enveloping algebra U(glM |N ) of the Lie superalgebra glM |N . To avoid
confusion, the element of glM |N corresponding to Eij ∈ EndCM |N will be denoted by eij . By
definition, the bracket on glM |N is the supercommutator. Hence in U(glM |N )
[eij , ekl] = δjkeil − δliekj(−1)(ı̄+̄)(k̄+l̄). (2.26)
By using the defining relations (2.8) one can demonstrate that there is a homomorphism
Y(glM |N )→ U(glM |N ) : Tij(u) 7→ δij − ejiu−1(−1)̄. (2.27)
This homomorphism is surjective. The relations (2.8) also imply that there is a homomorphism
U(glM |N )→ Y(glM |N ) : eji 7→ −T (1)
ij (−1)̄. (2.28)
The composition of homomorphisms (2.28) and (2.27) is the identity map U(glM |N )→ U(glM |N ).
So (2.28) is an embedding of Z2-graded associative unital algebras. The homomorphism (2.27)
is identical on the subalgebra U(glM |N ). It is called the evaluation homomorphism for Y(glM |N ).
There is a natural Hopf algebra structure on Y(glM |N ). A coassociative comultiplication
homomorphism ∆: Y(glM |N )→ Y(glM |N )⊗Y(glM |N ) can be defined by the assignment
Tij(u) 7→
M+N∑
k=1
Tik(u)⊗ Tkj(u)(−1)(ı̄+k̄)(̄+k̄), (2.29)
where the tensor product is taken over the subalgebra C
[[
u−1
]]
in Y(glM |N )
[[
u−1
]]
. The counit
homomorphism ε : Y(glM |N ) → C is defined by the assignment Tij(u) 7→ δij . The antipodal
mapping S: Y(glM |N ) → Y(glM |N ) is the antiautomorphism (2.19). Justification of all these
definitions is similar to that in the case N = 0 considered for instance in [6, Section 1]. Here we
omit the details. Note that (2.28) is an embedding of Hopf algebras as by the above definitions
∆: T
(1)
ij 7→ T
(1)
ij ⊗ 1 + 1⊗ T (1)
ij , ε : T
(1)
ij 7→ 0, S: T
(1)
ij 7→ −T
(1)
ij .
6 M. Nazarov
There are two natural ascending filtrations on the associative algebra Y(glM |N ). The first one
is defined by assigning the degree r to the generator (2.6). Let gr Y(glM |N ) be the corresponding
graded algebra.
Let us denote by X
(r)
ij the image of T
(r)
ij in the degree r component of gr Y(glM |N ). Observe
that the Z2-grading on the algebra Y(glM |N ) descends to gr Y(glM |N ) so that the degree of the
image is again ı̄ + ̄. It follows from the relations (2.8) that these images supercommute. We
shall prove that gr Y(glM |N ) is a free supercommutative algebra generated by these images.
Now introduce another filtration on the associative algebra Y(glM |N ) by assigning the deg-
ree r−1 to the generator (2.6). Let gr′Y(glM |N ) be the corresponding graded algebra. Consider
the latter algebra.
Let us denote by Y
(r)
ij the image of T
(r)
ij in the degree r − 1 component of gr′Y(glM |N ). The
Z2-grading on the algebra Y(glM |N ) descends to gr′Y(glM |N ) so that the degree of the image
is ı̄+ ̄. The graded algebra gr′Y(glM |N ) inherits from Y(glM |N ) the Hopf algebra structure too.
Namely, by the above definitions in the graded Hopf algebra gr′Y(glM |N ) for r > 1
∆ : Y
(r)
ij 7→ Y
(r)
ij ⊗ 1 + 1⊗ Y (r)
ij , ε : Y
(r)
ij 7→ 0, S : Y
(r)
ij 7→ −Y
(r)
ij . (2.30)
Let us now consider the polynomial current Lie superalgebra glM |N [u]. The elements ejiu
r
with r = 0, 1, 2, . . . and i, j = 1, . . . ,M+N make a basis of glM |N [u]. The Z2-grading on glM |N [u]
is defined by deg ejiu
r = ı̄+ ̄. Take the universal enveloping algebra U(glM |N [u]).
Proposition 2.6. One can define a surjective homomorphism U(glM |N [u]) → gr′Y(glM |N ) of
Z2-graded associative algebras by mapping for r > 0
ejiu
r 7→ −Y (r+1)
ij (−1)̄. (2.31)
Proof. Due to (2.26) the supercommutation relations with r, s > 0
[ejiu
r, elku
s] = δilejku
r+s − δkjeliur+s(−1)(ı̄+̄)(k̄+l̄) (2.32)
define the Lie superalgebra glM |N [u]. Multiplying the relation (2.32) by (−1)ı̄k̄+ı̄l̄+k̄l̄+̄+l̄ and
replacing the basis elements of glM |N [u] there by their images under the mapping (2.31) we get
[Y
(r+1)
ij , Y
(s+1)
kl
]
(−1)ı̄k̄+ı̄l̄+k̄l̄ = δkjY
(r+s+1)
il − δilY
(r+s+1)
kj .
The latter relation in gr′Y(glM |N ) follows from (2.8), see for instance [6, Section 1]. This proves
the homomorphism property of the assignment (2.31). This homomorphism is clearly surjective
and preserves the Z2-grading. �
It immediately follows from (2.30) that (2.31) is a homomorphism of Hopf algebras. Here we
use the standard Hopf algebra structure on U(glM |N [u]) as the universal enveloping algebra of
a Lie superalgebra. We shall demonstrate that the homomorphism (2.31) is also injective.
By comparing (2.10) with (2.14) we obtain that for any z ∈ C the assignment
EndCM |N ⊗Y(glM |N )
[[
u−1
]]
→
(
EndCM |N
)⊗2[[
u−1
]]
: T (u) 7→ R(u− z) (2.33)
defines a representation Y(glM |N )→ EndCM |N . More explicitly, under (2.33) for any r > 0
T
(r+1)
ij 7→ −Ejizr(−1)̄. (2.34)
Note that this representation of Y(glM |N ) can be also obtained from the standard representation
U(glM |N )→ EndCM |N by pulling back through the evaluation homomorphism (2.27) and then
back through the automorphism of Y(glM |N ) defined by mapping Tij(u) 7→ Tij(u− z).
Yangian of the General Linear Lie Superalgebra 7
The comultiplication on Y(glM |N ) now allows us to define for n = 1, 2, . . . a representation
Y(glM |N ) → (EndCM |N )⊗n depending on z1, . . . , zn ∈ C. This is the tensor product of the
representations (2.33) where z = z1, . . . , zn. Due to (2.12) and (2.29) under this representation
T (u) 7→ R12(u− z1) · · ·R1,n+1(u− zn). (2.35)
Proposition 2.7. Let the complex parameters z1, . . . , zn and the positive integer n vary. Then
the kernels of the representations (2.35) of Y(glM |N ) have the zero intersection.
Proof. Take any finite linear combination of the products T
(r1+1)
i1j1
· · ·T (rm+1)
imjm
∈ Y(glM |N ) with
Ar1...rmi1j1...imjm
∈ C being the respective coefficients. In this linear combination the number m > 0
and indices r1, . . . , rm > 0 may vary. Consider the image of this linear combination under the
representation Y(glM |N )→
(
EndCM |N
)⊗n
defined by (2.35). This image depends on z1, . . . , zn
polynomially. Let A be the sum of those terms of this polynomial which have the maximal total
degree in z1, . . . , zn. Let d be this degree.
Consider the second of our two ascending filtrations on the associative algebra Y(glM |N ), the
corresponding graded algebra being gr′Y(glM |N ). Equip the tensor product Y(glM |N )⊗n with
the ascending filtration where the degree is the sum of the degrees on the tensor factors. Then
by the definition (2.29) under the comultiplication Y(glM |N )→ Y(glM |N )⊗n for r > 0
T
(r+1)
ij 7→
n∑
h=1
1⊗(h−1) ⊗ T (r+1)
ij ⊗ 1⊗(n−h) + lower degree terms.
Therefore the sum A ∈
(
EndCM |N
)⊗n
coincides with the image of the sum∑
r1+···+rm=d
Ar1...rmi1j1...imjm
ej1i1u
r1 · · · ejmimurm(−1)m+̄1+···+̄m ∈ U(glM |N [u])
under the tensor product of the evaluation representations
U(glM |N [u])→ EndCM |N : ejiu
r 7→ Ejiz
r (2.36)
at the points z = z1, . . . , zn. Here we used the explicit description (2.34) of the representation
Y(glM |N )→ EndCM |N corresponding to z ∈ C. Due to Proposition 2.6 it now suffices to show
that when the complex parameters z1, . . . , zn and the positive integer n vary, the kernels of the
tensor products of the evaluation representations of the algebra U(glM |N [u]) at z = z1, . . . , zn
have the zero intersection. This will also imply that the homomorphism (2.31) is injective.
Choose any Z2-homogeneous basis f1, . . . , f(M+N)2 of glM |N such that the first vector f1 is
M+N∑
i=1
eii. (2.37)
The corresponding elements of EndCM |N will be denoted by F1, . . . , F(M+N)2 . Hence F1 = 1.
The elements fpu
r with r = 0, 1, 2, . . . and p = 1, . . . , (M +N)2 constitute a basis of glM |N [u].
Choose any total ordering of this basis which ends with the infinite sequence . . . , f1u
2, f1u, f1.
Take any finite linear combination L of the products
fp1u
r1 · · · fpmurm ∈ U(glM |N [u]) (2.38)
with Lr1...rmp1...pm ∈ C being the coefficients. We assume that the factors in the products are arranged
according to our ordering of the basis of glM |N [u]. Due to the supercommutation relations in
U(glM |N [u]) we assume it without any loss of generality. The basis elements of Z2-degree 0 may
occur in any product (2.38) with a multiplicity but those of Z2-degree 1 may occur at most once.
8 M. Nazarov
Let us denote by ρz the evaluation representation (2.36). More generally, denote by ρz1...zn
the tensor product ρz1 ⊗ · · · ⊗ ρzn pulled back through n-fold comultiplication on U(glM |N [u]).
Hence ρz1...zn is a homomorphism of associative algebras
U(glM |N [u])→
(
EndCM |N
)⊗n
: fpu
r 7→ Fpz
r
1 ⊗ 1⊗(n−1) + · · ·+ 1⊗(n−1) ⊗ Fpzrn.
Suppose that ρz1...zn(L) = 0 for every n and all z1, . . . , zn ∈ C. We need to prove that L = 0.
For each product (2.38) there is a number a such that p1, . . . , pa > 1 but pa+1, . . . , pm = 1.
This is due to our ordering of the basis of glM |N [u]. The numbers a for different products (2.38)
may differ, and we do not exclude the case a = 0. Let h be the maximum of the numbers a.
Suppose n > h. Let ωh be the supersymmetrisation map of the tensor product glM |N [u]⊗h
normalised so that ω2
h = h!ωh. Let W be the subspace of
(
EndCM |N
)⊗n
spanned by the vectors
Fq1 ⊗ · · · ⊗ Fqn where at least one of the indices q1, . . . , qh is 1. If h = 0 then this subspace is
assumed to be zero. Applying the homomorphism ρz1...zn to a product (2.38) with a = h gives
(ρz1 ⊗ · · · ⊗ ρzh)
(
ωh
(
fp1u
r1 ⊗ · · · ⊗ fphu
rh
))
⊗ 1⊗(n−h)
m∏
b=h+1
(
zrb1 + · · ·+ zrbn
)
(2.39)
modulo the subspace W . Applying ρz1...zn to a product (2.38) with a < h gives an element of W .
But a linear combination of the expressions (2.39) belongs to W only if this combination is zero.
For each product (2.38) with a = h there is a number c > h such that rh+1 > · · · > rc > 0
but rc+1, . . . , rm = 0. This is due to our ordering of the basis of glM |N [u]. Then (2.39) equals
(ρz1 ⊗ . . .⊗ ρzh)
(
ωh
(
fp1u
r1 ⊗ . . .⊗ fphu
rh
))
⊗ 1⊗(n−h)
c∏
b=h+1
(
zrb1 + · · ·+ zrbn
)
(2.40)
multiplied by nm−c. Let g be the maximum of the numbers c for all products (2.38) with a = h.
Suppose n > g. Consider the pairs of sequences of indices p1, . . . , pm and r1, . . . , rm showing
in L in the products (2.38) with a = h. For every such a pair there is c ∈ {h, . . . , g}. Then
we have ph+1, . . . , pm = 1 and rc+1, . . . , rm = 0. Take all different pairs of sequences p1, . . . , ph
and r1, . . . , rc arising in this way. The expressions (2.40) corresponding to the latter pairs are
linearly independent as polynomials in z1, . . . , zn with values in
(
EndCM |N
)⊗n
. This is again
due to our ordering of the basis of glM |N [u]. Here we also employ the observation that in (2.40)
the image of ρz1 ⊗ · · · ⊗ ρzh does not depend on the parameters zh+1, . . . , zn while the product
over b = h + 1, . . . , c in (2.40) does depend on these parameters whenever c > h. Therefore if
ρz1...zn(L) = 0 for a certain n > g and for all z1, . . . , zn ∈ C then for each of the latter pairs
∞∑
m=c
Lr1...rc0...0
p1...ph1...1n
m−c = 0.
There are exactly m lower indices and also m upper indices in the coefficient Lr1...rc0...0
p1...ph1...1 above.
By letting the number n vary we now prove that all these coefficients vanish. �
In the course of the proof of Proposition 2.7 we established that the homomorphism (2.31) is
injective. Together with Proposition 2.6 and the observation made just after its proof this yields
Theorem 2.8. The Hopf algebras U(glM |N [u]) and gr′Y(glM |N ) are isomorphic via (2.31).
Let us now invoke the Poincaré–Birkhoff–Witt theorem for the universal enveloping alge-
bras of Lie superalgebras [5, Theorem 5.15]. By applying this theorem to the Lie superalge-
bra glM |N [u] we now obtain its analogue for the Yangian Y(glM |N ).
Corollary 2.9. The supercommutative algebra gr Y(glM |N ) is freely generated by X
(r)
ij of the
Z2-degree ı̄+ ̄ where r = 1, 2, . . . and i, j = 1, . . . ,M +N .
Yangian of the General Linear Lie Superalgebra 9
3 Centre of the Yangian
Here we will give a description of the centre of the algebra Y(glM |N ). An element of Y(glM |N )
is called central if it supercommutes with each element of Y(glM |N ). However, we will see
that the central elements of Y(glM |N ) have the Z2-degree 0. Hence they commute with each
element of Y(glM |N ) in the usual sense. Our description comes from computing the square of
the antipodal mapping S of Y(glM |N ). Another description of the centre of Y(glM |N ) will be
given in the next section. In that section we will also establish a correspondence between the
two descriptions.
Proposition 3.1. There is a formal power series Z(u) in u−1 with coefficients in the centre
of Y(glM |N ) and with leading term 1 such that for all indices i and j
M+N∑
k=1
Tkj(u+M −N)T ′ik(u) = δijZ(u), (3.1)
M+N∑
k=1
T ′kj(u)Tik(u+M −N) = δijZ(u). (3.2)
Proof. By using the definition (2.9) introduce the element of the algebra
(
EndCM |N
)⊗2
Q = (id⊗ τ)(P ) =
M+N∑
i,j=1
Eij ⊗ Eij(−1)ı̄̄. (3.3)
The image of the action of Q on
(
CM |N
)⊗2
is one dimensional and is spanned by the vector
M+N∑
i=1
ei ⊗ ei.
Here we regard Q as an element of the algebra End
((
CM |N
)⊗2)
by identifying this algebra with(
EndCM |N
)⊗2
via (2.3). We also have Q2 = (M −N)Q. By using the latter relation we get
((id⊗ τ)(R(u)))−1 =
(
1−Qu−1
)−1
= 1 +Q(u−M +N)−1. (3.4)
The rational function of the variable u given by the equalities (3.4) will be denoted by R](u).
Now multiply both sides of the relation (2.14) by T−1
2 (v) on the left and right, and then
apply τ relative to the second tensor factor EndCM |N in
(
EndCM |N
)⊗2⊗Y(glM |N ). Multiplying
the resulting relation on the left and right by R](u− v)⊗ 1 yields(
R](u− v)⊗ 1
)
T1(u)T ]2(v) = T ]2(v)T1(u)(R](u− v)⊗ 1). (3.5)
Multiplying the latter relation by u− v −M +N and then setting u = v +M −N we get
(Q⊗ 1)T1(v +M −N)T ]2(v) = T ]2(v)T1(v +M −N)(Q⊗ 1), (3.6)
see (3.4). As the image of Q in
(
CM |N
)⊗2
is one dimensional, either side of the relation (3.6) must
be equal to Q⊗Z(v) where Z(v) is a certain power series in v−1 with coefficients from Y(glM |N ).
The equality of the left-hand side and of the right-hand side of (3.6) to Q⊗ Z(v) is equivalent
respectively to (3.1) and (3.2). We just need to replace the variable v in (3.6) by u.
It is immediate from (2.7) and (3.1) that every coefficient of the series Z(u) has Z2-degree 0
and that its leading term is 1. Let us prove that all these coefficients are central in Y(glM |N ).
10 M. Nazarov
To this end we will work with the elements (2.13) where n = 3. By using (2.14) and (3.5),(
R]13(u− v)R12(u− v −M +N)⊗ 1
)
T1(u)T2(v +M −N)T ]3(v)
=
(
R]13(u− v)⊗ 1
)
T2(v +M −N)T1(u)T ]3(v)(R12(u− v −M +N)⊗ 1)
= T2(v +M −N)T ]3(v)T1(u)
(
R]13(u− v)R12(u− v −M +N)⊗ 1
)
. (3.7)
On the other hand, due to (2.11) we have the identity in the algebra
(
EndCM |N
)⊗3
(u)
R13(−u)P23R12(u) =
(
1− u−2
)
P23.
By applying to it the antiautomorphism τ relative to the third tensor factor of
(
EndCM |N
)⊗3
and then using the definition (3.4) we get
Q23R
]
13(u+M −N)R12(u) =
(
1− u−2
)
Q23. (3.8)
Therefore multiplying the first and third lines of the display (3.7) by Q23 ⊗ 1 on the left yields(
1− (u− v −M +N)−2
)
T1(u)(Q23 ⊗ Z(v))
= (Q23 ⊗ Z(v))T1(u)
(
1− (u− v −M +N)−2
)
.
We just need to replace the variable u in (3.8) by u− v−M +N and use the relation (3.6). The
last relation implies that any generator T
(r)
ij commutes with every coefficient of Z(v). �
The square S2 of antipodal mapping is an automorphism of the associative algebra Y(glM |N ).
Proposition 3.2. The automorphism S2 of Y(glM |N ) is given by the assignment
Tij(u) 7→ Z(u)−1Tij(u+M −N).
Proof. By the definition (2.18) for any indices i, j we have the identity
M+N∑
k=1
Tik(u)T ′kj(u)(−1)(ı̄+k̄)(̄+k̄) = δij . (3.9)
Here we use the convention (2.1). By using the definition of S this identity can be written as
M+N∑
k=1
Tik(u) S(Tkj(u))(−1)(ı̄+k̄)(̄+k̄) = δij .
Let us apply the antiautomorphism S to both sides of the latter identity. We get
M+N∑
k=1
S2(Tkj(u)) S(Tik(u)) = δij or
M+N∑
k=1
S2(Tkj(u))T ′ik(u) = δij .
By comparing the last displayed identity with (3.1) we see that for any indices k, j
Z(u) S2(Tkj(u)) = Tkj(u+M −N). �
Corollary 3.3. For the formal power series Z(u) in u−1 we have
∆: Z(u) 7→ Z(u)⊗ Z(u), ε : Z(u) 7→ 1, S: Z(u) 7→ Z(u)−1. (3.10)
Yangian of the General Linear Lie Superalgebra 11
Proof. The square of the antipodal mapping is a coalgebra homomorphism. Hence the images
of Tij(u) relative to the two compositions ∆ S2 and
(
S2⊗S2
)
∆ are the same. By Proposition 3.2
these images are respectively equal to
∆
(
Z(u)−1Tij(u+M −N)
)
= ∆
(
Z(u)−1
)M+N∑
k=1
Tik(u+M −N)⊗ Tkj(u+M −N)(−1)(ı̄+k̄)(̄+k̄),
and
M+N∑
k=1
S2(Tik(u))⊗ S2(Tkj(u))(−1)(ı̄+k̄)(̄+k̄)
=
(
Z(u)−1 ⊗ Z(u)−1
)M+N∑
k=1
Tik(u+M −N)⊗ Tkj(u+M −N)(−1)(ı̄+k̄)(̄+k̄).
By equating the two images of Tij(u) we obtain that
∆: Z(u)−1 7→ Z(u)−1 ⊗ Z(u)−1.
Since the comultiplication ∆ is an algebra homomorphism, we get the first statement in (3.10).
The second statement in (3.10) immediately follows from (3.1) and from the definition of the
counit homomorphism ε. The third statement follows from the first and the second because the
multiplication µ : Y(glM |N ) ⊗ Y(glM |N ) → Y(glM |N ) and the unit mapping δ : C → Y(glM |N )
satisfy the identity µ(S⊗id)∆ = δε. Indeed, by applying to the coefficients of the series Z(u)
the homomorphisms at the two sides of this identity we get the equality S(Z(u))Z(u) = 1. �
In Section 4 we will also use the following observation. For any indices i, j by (2.18) we have
M+N∑
k=1
T ′ik(u)Tkj(u)(−1)(ı̄+k̄)(̄+k̄) = δij
similarly to (3.9). The collection of last displayed identities can be written as a single relation
(Q⊗ 1)T ]2(u)T1(u) = Q⊗ 1 (3.11)
of series in u with coefficients in the algebra
(
EndCM |N
)⊗2 ⊗Y(glM |N ).
Proposition 3.4. The series Z(u) is invariant under the antiautomorphism (2.21) of Y(glM |N ).
Proof. The series Z(u) is equal to the sum at the left-hand side of the relation (3.1) with i = j.
By applying the antiautomorphism (2.21) to that sum and using Proposition 2.5 we get
M+N∑
k=1
T ′ki(u)Tik(u+M −N)(−1)k̄(ı̄+1)+ı̄(k̄+1)+(k̄+ı̄)(ı̄+k̄),
which is equal to the sum the left-hand side of the relation (3.2) with i = j. �
Corollary 3.5. The automorphism (2.24) of Y(glM |N ) maps Z(u) 7→ Z(u)−1.
Proof. The automorphism (2.24) is the composition of the antiautomorphisms (2.19) and (2.21)
of Y(glM |N ). Hence the required statement follows from Corollary 3.3 and Proposition 3.4. �
12 M. Nazarov
Due to the definitions (2.18) and (3.1) the coefficient of the series Z(u) at u−1 is zero. Thus
Z(u) = 1 + Z(2)u−2 + Z(3)u−3 + · · · (3.12)
for certain central elements Z(2), Z(3), . . . ∈ Y(glM |N ). The main result of the present section is
Theorem 3.6. The elements Z(2), Z(3), . . . are free generators of the centre of Y(glM |N ).
We shall prove Theorem 3.6 at the end of this section. We will use the following proposition.
Proposition 3.7. For any r > 2 the element Z(r) ∈ Y(glM |N ) has degree r − 2 relative to the
second filtration on Y(glM |N ). Its image in the graded algebra gr′Y(glM |N ) is equal to
(1− r)
M+N∑
i=1
Y
(r−1)
ii (−1)ı̄. (3.13)
Proof. Let us expand the relation (3.5) of series with coefficients in
(
EndCM |N
)⊗2⊗Y(glM |N )
by using the basis of End
((
CM |N
)⊗2)
constituted by the elements Eij ⊗Ekl. By taking (−1)ı̄+̄
times the terms in the expansion corresponding to the basis element Eij ⊗ Eij with any given
indices i and j we obtain the relation of series with coefficients in the algebra Y(glM |N )
Tij(u)T ′ji(v) +
M+N∑
k=1
Tkj(u)T ′jk(v)(u− v −M +N)−1(−1)ı̄
= T ′ji(v)Tij(u)(−1)ı̄+̄ +
M+N∑
k=1
T ′ki(v)Tik(u)(u− v −M +N)−1(−1)ı̄. (3.14)
For any i and j denote by Ṫij(u) the formal derivative of the series Tij(u) so that
Ṫij(u) = −T (1)
ij u
−2 − 2T
(2)
ij u
−3 − · · · . (3.15)
By tending in the relation (3.14) the parameter u to v +M −N we then get
Tij(v +M −N)T ′ji(v) +
M+N∑
k=1
Ṫkj(v +M −N)T ′jk(v)(−1)ı̄
= T ′ji(v)Tij(v +M −N)(−1)ı̄+̄ +
M+N∑
k=1
T ′ki(v)Ṫik(v +M −N)(−1)ı̄. (3.16)
Let us now observe that
Tij(v +M −N) = Tij(v) + (M −N)Ṫij(v) +Oij(v),
where Oij(v) is a certain formal power series in v−1 with coefficients in Y(glM |N ) such that the
coefficient at v−r with r > 3 has degree r − 3 relative to the second fitration. The coefficient
of this series at v−r with any r 6 2 is zero. By taking the sum of the relations (3.16) over the
indices i = 1, . . . ,M +N and then using the definition (3.1) along with this observation we get
Z(v) + (M −N)
M+N∑
k=1
Ṫkj(v +M −N)T ′jk(v) =
M+N∑
i,k=1
T ′ki(v)Ṫik(v +M −N)(−1)ı̄
+
M+N∑
i=1
T ′ji(v)(Tij(v) + (M −N)Ṫij(v) +Oij(v))(−1)ı̄+̄.
Yangian of the General Linear Lie Superalgebra 13
By using the definition (2.18) the latter relation can be rewritten as
Z(v) + (M −N)
M+N∑
k=1
Ṫkj(v +M −N)T ′jk(v)
=
M+N∑
i,k=1
T ′ki(v)Ṫik(v +M −N)(−1)ı̄
+ 1 +
M+N∑
i=1
T ′ji(v)((M −N)Ṫij(v) +Oij(v))(−1)ı̄+̄. (3.17)
For any indices i and j the leading term of the series T ′ij(v) is δij while for every r > 1 the
coefficient of this series at v−r has degree r − 1 relative to the second filtration on Y(glM |N ).
Furthermore for any given r > 1 the coefficients at v−r of the series Ṫij(v) and Ṫij(v +M −N)
have the same image in the graded algebra gr′Y(glM |N ), see (3.15). Therefore by taking the
coefficients at v−r with any r > 2 in the relation (3.17) the image of Z(r) in gr′Y(glM |N ) equals
−(M −N)
M+N∑
k=1
(1− r)Y (r−1)
kj δjk +
M+N∑
i,k=1
δki(1− r)Y
(r−1)
ik (−1)ı̄
+
M+N∑
i=1
δji(M −N)(1− r)Y (r−1)
ij (−1)ı̄+̄ = (1− r)
M+N∑
i=1
Y
(r−1)
ii (−1)ı̄. �
In our proof of Theorem 3.6 we will also use the following general proposition. Let a be any
finite-dimensional Lie superalgebra over C. Take the polynomial current Lie superalgebra a[u].
Proposition 3.8. Suppose the centre of the Lie superalgebra a is trivial. Then the centre of the
universal enveloping algebra U(a[u]) is also trivial, that is equal to C.
Proof. Consider adjoint action of the Lie superalgebra a[u] on its supersymmetric algebra. By
Poincaré–Birkhoff–Witt theorem for U(a[u]) it suffices to show that the space of invariants of this
action is trivial, see [5, Theorem 5.15].
Let A be an element of the supersymmetric algebra of a[u] invariant under the adjoint action.
Let K = dim a. Choose any Z2-homogeneous basis f1, . . . , fK of a and write the Lie brackets as
[fp, fq] =
K∑
r=1
Crpqfr,
where Crpq ∈ C are the structure constants. Let d be the maximal degree such that A depends
on at least one of the basis elements f1u
d, . . . , fKu
d of a[u]. Write
A =
∑
m1,...,mK
Am1...mK
(
f1u
d
)m1 · · ·
(
fKu
d
)mK , (3.18)
where the coefficients Am1...mK are certain elements of the supersymmetric algebra which do not
depend on f1u
d, . . . , fKu
d. For each p = 1, . . . ,K let p̄ be the Z2-degree of the basis element fp
of a. We allow m1, . . . ,mK in (3.18) to range over 0, 1, 2, . . . but assume that Am1...mK = 0 if
mp > 1 for at least one index p with p̄ = 1.
For each p = 1, . . . ,K we have the equation ad(fpu)(A) = 0 in the supersymmetric algebra
of a[u]. In particular, the component of the left-hand side of this equation that involves any of
the basis elements f1u
d+1, . . . , fKu
d+1 must be zero. Thus∑
m1,...,mK
Am1...mK
K∑
q,r=1
mqC
r
pq
(
f1u
d
)m1 · · ·
(
fqu
d
)mq−1 · · ·
(
fKu
d
)mKfru
d+1(−1)hq = 0,
14 M. Nazarov
where we used the notation
hq =
q−1∑
s=1
p̄s̄ms +
K∑
s=q+1
r̄s̄ms. (3.19)
If follows that for any non-negative integers m′1, . . . ,m
′
K we have the equations
M∑
q=1
Am′
1...m
′
q+1...m′
K
(m′q + 1)Crpq(−1)h
′
q = 0, where p, r = 1, . . . ,K. (3.20)
Similarly to (3.19) here we used the notation
h′q =
q−1∑
s=1
p̄s̄m′s +
K∑
s=q+1
r̄s̄m′s.
Let us now fix any non-negative integers m′1, . . . ,m
′
K and observe that the elements
f ′q = (m′q + 1)fq(−1)h
′
q with q = 1, . . . ,K
also form a basis of a. Since the centre of the Lie superalgebra a is trivial, the systemfp, K∑
q=1
wqf
′
q
= 0 with p = 1, . . . ,K
of linear equations on w1, . . . , wK ∈ C has only zero solution. This system can be written as
K∑
q=1
wq(m
′
q + 1)Crpq(−1)h
′
q = 0, where p, r = 1, . . . ,K.
Hence by comparing the latter system with (3.20) we obtain that Am′
1...m
′
q+1...m′
K
= 0 for each
index q = 1, . . . ,K. It follows that A ∈ C and that d = 0 in particular. �
We can now prove Theorem 3.6. Due to Proposition 3.1 the elements
Z(2), Z(3), . . . ∈ Y(glM |N )
are central. Therefore it suffices to prove that the images of these elements in the graded algebra
gr′Y(glM |N ) are free generators of its centre, see Proposition 3.7. By Theorem 2.8 the graded
algebra is isomorphic to the universal enveloping algebra U(glM |N [u]) via (2.31). Under this
isomorphism the element (3.13) of gr′Y(glM |N ) with any r > 2 corresponds to the element
(r − 1)
M+N∑
i=1
eiiu
r−2 ∈ U(glM |N [u]).
Let us demonstrate that all the latter elements are free generators of the centre of U(glM |N [u]).
They are algebraically independent by the Poincaré–Birkhoff–Witt theorem for U(glM |N [u]),
see [5, Theorem 5.15]. To show that they generate the centre of U(glM |N [u]) consider the
quotient of U(glM |N [u]) by the ideal they generate. This quotient is isomorphic to the universal
enveloping algebra U(a[u]) where the Lie superalgebra a is the quotient of glM |N by the span of
the element (2.37). The centre of the Lie superalgebra a is trivial. Hence the centre of U(a[u])
is also trivial, see Proposition 3.8. This argument completes our proof of Theorem 3.6.
Yangian of the General Linear Lie Superalgebra 15
4 Quantum Berezinian
The element (2.12) can be also regarded as an (M + N) × (M + N) matrix whose ij entry
is Tij(u). The quantum Berezinian of that matrix is the series B(u) defined as the product∑
σ∈SM
(−1)σTσ(1)1(u+M −N − 1)Tσ(2)2(u+M −N − 2) · · ·Tσ(M)M (u−N)
×
∑
σ∈SN
(−1)σT ′M+1,M+σ(1)(u−N)T ′M+2,M+σ(2)(u−N + 1) · · ·T ′M+N,M+σ(N)(u− 1)
of two alternated sums over the symmetric groups SM and SN . Here (−1)σ denotes the sign
of the permutation σ. The purpose of the present section is to prove the following theorem.
Theorem 4.1. We have the equality B(u+ 1) = Z(u)B(u) of formal power series in u−1.
By the definition (2.7) the leading term of the formal power series Tij(u) in u−1 is δij . Due
to the definition (2.18) the leading term of the series T ′ij(u) is also δij . It follows that the
leading term of B(u) is 1. Using this observation, the coefficients of the series B(u) are uniquely
determined by those of the series Z(u) via the equality in Theorem 4.1, see the expansion (3.12).
Theorems 3.6 and 4.1 now imply that the coefficients of the series B(u) at u−1, u−2, . . . are
free generators of the centre of Y(glM |N ). Corollary 3.3 and Theorem 4.1 now imply that
∆: B(u) 7→ B(u)⊗B(u), ε : B(u) 7→ 1, S: B(u) 7→ B(u)−1.
Note that the second assignment here also follows directly from the definition of B(u) because
ε : Tij(u) 7→ δij and ε : T ′ij(u) 7→ δij .
For N = 0 the Hopf algebra Y(glM |N ) is the Yangian Y(glM ) of the Lie algebra glM , see [6].
In this case the second sum in the above definition of B(u) is assumed to be 1, and B(u) equals∑
τ∈SM
(−1)σTσ(1)1(u+M − 1)Tσ(2)2(u+M − 2) · · ·Tσ(M)M (u). (4.1)
This is the quantum determinant of the M×M matrix whose ij entry is the series Tij(u). It has
been well known that the coefficients of the series (4.1) at u−1, u−2, . . . are free generators of
the centre of Y(glM ), see [6, Section 2] and references therein. A detailed proof of Theorem 4.1
in this particular case was given in [6, Section 5] by following [8].
For any M , N denote by X(u) the (M +N)× (M +N) matrix whose ij entry is the series
Xij(u) = δij · 1 +X
(1)
ij u
−1 +X
(2)
ij u
−2 + · · ·
with the coefficients of Z2-degree ı̄ + ̄, see Section 2. This matrix is invertible. Let X ′ij(u) be
the ij entry of the inverse matrix. Note that under the correspondence Y(glM |N )→ gr Y(glM |N )
the series B(u) gets mapped to the product∑
σ∈SM
(−1)σXσ(1)1(u)Xσ(2)2(u) · · ·Xσ(M)M (u)
×
∑
σ∈SN
(−1)σX ′M+1,M+σ(1)(u)X ′M+2,M+σ(2)(u) · · ·X ′M+N,M+σ(N)(u)
of two determinants. This is just the Berezinian or the superdeterminant of the matrix X(u) as
defined in [1, Section I.3.1]. These two observations explain our choice of terminology for B(u).
16 M. Nazarov
Let us now consider the other particular case when M = 0. In this case B(u) equals the sum∑
σ∈SN
(−1)σT ′1σ(1)(u−N)T ′2σ(2)(u−N + 1) · · ·T ′Nσ(N)(u− 1).
This sum can also be obtained by applying the automorphism (2.24) of Y(gl0|N ) to the series∑
σ∈SN
(−1)σTσ(1)1(u−N)Tσ(2)2(u−N − 1) · · ·Tσ(N)N (u− 1).
Let us denote by C(u) the latter series. Then due to Corollary 3.5 the statement of Theorem 4.1
for M = 0 becomes equivalent to the relation
Z(u)C(u+ 1) = C(u). (4.2)
Now observe that the Yangian Y(gl0|N ) is isomorphic to Y(glN |0) = Y(glN ). A Hopf algebra
isomorphism Y(gl0|N )→ Y(glN ) can be defined by the assignment Tij(u) 7→ Tij(−u), see (2.8).
Under this isomorphism Z(u) 7→ Z(−u), see (3.1). Denote by D(u) the quantum determinant
for the Yangian Y(glN ). This is the series (4.1) with M replaced by N . Then C(u) 7→ D(1− u)
under the isomorphism. Therefore by applying the isomorphism to the relation (4.2) we get
Z(−u)D(−u) = D(1− u).
The latter relation holds by Theorem 4.1 for Y(glN ). Hence Theorem 4.1 also holds for M = 0.
From now on until the end of this section we will be assuming that M,N > 0. The next
proposition goes back to [2, Theorem 2.4], see also [4, Section 1]. In particular, this proposition
implies that the two alternated sums in the definition of B(u) commute with each other.
Proposition 4.2. If i, j 6M < k, l then the coefficients of the series Tij(u) commute with the
coefficients of the series T ′kl(v).
Proof. Let I ∈ EndCM |N and J ∈ EndCM |N be the projections of the Z2-graded vector
space CM |N onto its even and odd subspaces respectively, so that Iei = δ0ı̄ei and Jei = δ1ı̄ei for
every index i = 1, . . . ,M + N . These two subspaces of CM |N are denoted by CM |0 and C0|N .
By the definition (3.3) we have the relations in
(
EndCM |N
)⊗2
(I ⊗ J)Q = Q(I ⊗ J) = 0. (4.3)
Hence by multiplying (3.5) on the left and on the right by I ⊗ J ⊗ 1 we get the relation
(I ⊗ J ⊗ 1)T1(u)T ]2(v)(I ⊗ J ⊗ 1) = (I ⊗ J ⊗ 1)T ]2(v)T1(u)(I ⊗ J ⊗ 1) (4.4)
of series in u, v with coefficients in the algebra
(
EndCM |N
)⊗2⊗Y(glM |N ), see (3.4). The latter
relation is equivalent to the collection of all commutation relations stated in Proposition 4.2. �
We will keep using the projectors I and J introduced in the proof of Proposition 4.2 above.
Note that they also satisfy the relations in the algebra
(
EndCM |N
)⊗2
(J ⊗ I)Q = Q(J ⊗ I) = 0,
Q(I ⊗ I) = Q(I ⊗ 1) = Q(1⊗ I) and Q(J ⊗ J) = Q(J ⊗ 1) = Q(1⊗ J).
These relations together with (4.3) imply that
Q = Q(I ⊗ I + I ⊗ J + J ⊗ I + J ⊗ J) = Q(I ⊗ I + J ⊗ J) = Q(I ⊗ 1 + 1⊗ J). (4.5)
The next two technical propositions will be employed in our proof of Theorem 4.1 for M,N > 0.
Yangian of the General Linear Lie Superalgebra 17
Proposition 4.3. We have the equality in the algebra
(
EndCM |N
)⊗(M+N+2)
Q1,M+N+2
(
1− 1
M
QM+1,M+N+2
)(
1 +
1
N
Q1,M+2
)
× I1 · · · IM (IM+1 + JM+2)JM+3 · · · JM+N+2
= I2 · · · IM+1JM+2 · · · JM+N+1Q1,M+N+2
×
(
− 1
M
P1,M+1JM+N+2 +
1
N
PM+2,M+N+2I1
)
.
Proof. The second relation in (4.3) implies that
Q1,M+N+2I1JM+N+2 = 0,
QM+1,M+N+2IM+1JM+N+2 = 0,
Q1,M+2I1JM+2 = 0.
Therefore by opening the parentheses at the left-hand side of required equality we get the sum
− 1
M
Q1,M+N+2QM+1,M+N+2I1 · · · IMJM+2JM+3 · · · JM+N+2
+
1
N
Q1,M+N+2Q1,M+2I1 · · · IMIM+1JM+3 · · · JM+N+2. (4.6)
We have the relation P23P13 = P13P12 in
(
EndCM |N
)⊗3
. By applying to this relation the
antiautomorphism τ of the third tensor factor EndCM |N we get the relation Q13Q23 = Q13P12.
It follows that in the algebra
(
EndCM |N
)⊗(M+N+2)
Q1,M+N+2QM+1,M+N+2 = Q1,M+N+2P1,M+1. (4.7)
Further, since (τ ⊗ τ)(P ) = P we have the equality
Q =
(
τ−1 ⊗ id
)
(P )
by the definition (3.3). Hence applying to the relation P12P13 = P13P23 the antiautomor-
phism τ−1 of the first tensor factor of
(
EndCM |N
)⊗3
yields the relation Q13Q12 = Q13P23. It
follows that in the algebra
(
EndCM |N
)⊗(M+N+2)
Q1,M+N+2Q1,M+2 = Q1,M+N+2PM+2,M+N+2. (4.8)
Therefore the sum displayed in the two lines of (4.6) equals
− 1
M
Q1,M+N+2P1,M+1I1 · · · IMJM+2JM+3 · · · JM+N+2
+
1
N
Q1,M+N+2PM+2,M+N+2I1 · · · IMIM+1JM+3 · · · JM+N+2.
This can be written as the right-hand side of the equality in Proposition 4.3 by using the relations
P1,M+1I1 = IM+1P1,M+1,
PM+2,M+N+2JM+N+2 = JM+2PM+2,M+N+2. �
For any positive integer n denote respectively by G(n) and H(n) the images of the elements∑
σ∈Sn
(−1)σσ and
∑
σ∈Sn
σ
18 M. Nazarov
of the group ring CSn in the algebra
(
EndCM |N
)⊗n
. We use the representation σm 7→ Pm,m+1
for m = 1, . . . , n− 1 introduced in Section 2. Note the relations in the algebra
(
EndCM |N
)⊗n
G(n) = (1− P1,n − · · · − Pn−1,n)
(
G(n−1) ⊗ 1
)
, (4.9)
H(n) = (1 + P1,n + · · ·+ Pn−1,n)
(
H(n−1) ⊗ 1
)
. (4.10)
Here we assume that G(0) = H(0) = 1. To avoid cumbrous notation, below we will simply write G
for G(M) and H for H(N). These two images act by antisymmetrization on the subspaces(
CM |0
)⊗M ⊂ (CM |N)⊗M and
(
C0|N)⊗N ⊂ (CM |N)⊗N . (4.11)
Proposition 4.4. We have the equality in the algebra
(
EndCM |N
)⊗(M+N+2)
I2 · · · IM+1JM+2 · · · JM+N+1G2···M+1HM+2...M+N+1Q1,M+N+2
×
(
1− 1
M
QM+1,M+N+2
)(
1 +
1
N
Q1,M+2
)
G1...MHM+3...M+N+2
= (M − 1)!(N − 1)!P1,M+1PM+2,M+N+2
× I1 · · · IMJM+3 · · · JM+N+2G1...MHM+3...M+N+2QM+1,M+2.
Proof. By again using the relation (4.7) we get the equalities
IM+1JM+2Q1,M+N+2QM+1,M+N+2Q1,M+2 = IM+1JM+2Q1,M+N+2P1,M+1Q1,M+2
= IM+1JM+2Q1,M+N+2QM+1,M+2P1,M+1 = IM+1JM+2QM+1,M+2Q1,M+N+2P1,M+1.
The product at the right-hand side of these equalities vanishes by (4.3). Hence by opening the
parentheses at the left-hand side of the equality in Proposition 4.4 and using (4.7), (4.8) we get
I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1Q1,M+N+2
×
(
1− 1
M
QM+1,M+N+2 +
1
N
Q1,M+2
)
G1...MHM+3...M+N+2
= I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1Q1,M+N+2
×
(
1− 1
M
P1,M+1 +
1
N
PM+2,M+N+2
)
G1...MHM+3...M+N+2
= I2 · · · IM+1JM+2 · · · JM+N+1Q1,M+N+2(G2...M+1HM+2...M+N+1
− 1
M
P1,M+1G1...MHM+2...M+N+1 +
1
N
PM+2,M+N+2G2...M+1HM+3...M+N+2)
×G1...MHM+3...M+N+2. (4.12)
Observe that G2
1...M = M !G1...M while due to (4.9) the product G2...M+1G1...M equals
(1− P2,M+1 − · · · − PM,M+1)G
(M−1)
2...M G1...M
= (M − 1)!(1− P2,M+1 − · · · − PM,M+1)G1...M .
Similarly H2
M+3...M+N+2 = N !HM+3...M+N+2 while HM+2...M+N+1HM+3...M+N+3 equals
(1 + PM+2,M+3 + · · ·+ PM+2,M+N+1)H
(N−1)
M+3...M+N+1HM+3...M+N+2
= (N − 1)!(1 + PM+2,M+3 + · · ·+ PM+2,M+N+1)HM+3...M+N+2.
Here we employed (4.10). Hence the right-hand side of the equalities (4.12) can be rewritten as
(M − 1)!(N − 1)!I2 · · · IM+1JM+2 · · · JM+N+1Q1,M+N+2 (4.13)
Yangian of the General Linear Lie Superalgebra 19
multiplied on the right by
((1− P2,M+1 − · · · − PM,M+1)(1 + PM+2,M+3 + · · ·+ PM+2,M+N+1)
− P1,M+1(1 + PM+2,M+3 + · · ·+ PM+2,M+N+1)
+ PM+2,M+N+2(1− P2,M+1 − · · · − PM,M+1))G1...MHM+3...M+N+2
= ((1− P1,M+1 − · · · − PM,M+1)(1 + PM+2,M+3 + · · ·+ PM+2,M+N+2)
+ P1,M+1PM+2,M+N+2)G1...MHM+3...M+N+2
= G
(M+1)
1...M+1H
(N+1)
M+2...M+N+2 + P1,M+1PM+2,M+N+2G1...MHM+3...M+N+2. (4.14)
Due to (4.5) we have the equality
Q1,M+N+2 = Q1,M+N+2(I1 + JM+N+2).
Therefore by multiplying (4.13) by the first summand at the right-hand side of (4.14) we get
(M − 1)!(N − 1)!Q1,M+N+2(I1 · · · IM+1JM+2 · · · JM+N+1
+ I2 · · · IM+1JM+2 · · · JM+N+2)G
(M+1)
1...M+1H
(N+1)
M+2...M+N+2.
The latter product vanishes because zero is the only antisymmetric tensor in the subspaces(
CM |0
)⊗(M+1) ⊂
(
CM |N
)⊗(M+1)
and
(
C0|N)⊗(N+1) ⊂
(
CM |N
)⊗(N+1)
.
Multiplying (4.13) by the second summand at the right-hand side of the equalities (4.14) we get
(M − 1)!(N − 1)!I2 · · · IM+1JM+2 · · · JM+N+1Q1,M+N+2
× P1,M+1PM+2,M+N+2G1...MHM+3...M+N+2,
which is equal to the product at the right-hand side of the equality stated in Proposition 4.4. �
Proposition 4.5. For any positive integer n we have equalities of series in u with coefficients
in the algebra
(
EndCM |N
)⊗n ⊗Y(glM |N )(
G(n) ⊗ 1
)
T1(u) · · ·Tn(u− n+ 1) = Tn(u− n+ 1) · · ·T1(u)
(
G(n) ⊗ 1
)
,(
H(n) ⊗ 1
)
T ]1(u) · · ·T ]n(u+ n− 1) = T ]n(u+ n− 1) · · ·T ]1(u)
(
H(n) ⊗ 1
)
.
Proof. If 1 6 i < j 6 n then let σij ∈ Sn be the transposition of the numbers i and j. There
is a well known identity in the symmetric group ring CSn
∑
σ∈Sn
(−1)σσ =
n∏
j=2
(
j−1∏
i=1
(
1− σij
j − i
))
, (4.15)
where the factors at the right-hand side are arranged from left to right as i and j are increasing,
see for instance [6, Section 2.3]. The identity implies the relation in the algebra
(
EndCM |N
)⊗n
G(n) =
n∏
j=2
(
j−1∏
i=1
Rij(j − i)
)
. (4.16)
The first equality stated in Proposition 4.5 follows from this relation by repeatedly using (2.14).
Further, (4.15) is equivalent to another well known identity in CSn
∑
σ∈Sn
σ =
n∏
j=2
(
j−1∏
i=1
(
1 +
σij
j − i
))
,
20 M. Nazarov
which implies the relation in the algebra
(
EndCM |N
)⊗n
H(n) =
n∏
j=2
(
j−1∏
i=1
Rij(i− j)
)
. (4.17)
The second equality in Proposition 4.5 follows from this relation by repeatedly using (2.25). �
We will now prove Theorem 4.1. Using the relations (2.14), (2.25) and (3.5) we get an equality
of series in u with coefficients in the algebra
(
EndCM |N
)⊗(M+N+2) ⊗Y(glM |N )(
R]1,M+2(M)R1M (M − 1) · · ·R12(1)⊗ 1
)
×
(
R]M+1,M+N+2(−N)RM+3,M+N+2(1−N) · · ·RM+N+1,M+N+2(−1)⊗ 1
)
× T1(u+M −N)T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)T ]M+N+2(u)
= T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
R]1,M+2(M)R1M (M − 1) · · ·R12(1)⊗ 1
)
×
(
R]M+1,M+N+2(−N)RM+3,M+N+2(1−N) · · ·RM+N+1,M+N+2(−1)⊗ 1
)
. (4.18)
Let us multiply both sides of the equality (4.18) respectively on the left and on the right by
I2 · · · IM+1JM+2 · · · JM+N+1G
(M−1)
2...M H
(N−1)
M+3...M+N+1Q1,M+N+2 ⊗ 1
and by P1,M+1PM+2,M+N+2 ⊗ 1. Due to (4.16) and (4.17) we have in
(
EndCM |N
)⊗(M+N+2)
G
(M−1)
2...M R1M (M − 1) · · ·R12(1) = G1...M , (4.19)
H
(N−1)
M+3...M+N+1RM+3,M+N+2(1−N) · · ·RM+N+1,M+N+2(−1) = HM+3...M+N+2. (4.20)
Hence after the multiplication the left-hand side of (4.18) becomes(
I2 · · · IM+1JM+2 · · · JM+N+1Q1,M+N+2R
]
1,M+2(M)R]M+1,M+N+2(−N)⊗ 1
)
× (G1...M ⊗ 1)T1(u+M −N)T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× (HM+3...M+N+2 ⊗ 1)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
× T ]M+N+2(u)(P1,M+1PM+2,M+N+2 ⊗ 1). (4.21)
After the same multiplication the right-hand side of (4.18) becomes(
I2 . . . IM+1JM+2 . . . JM+N+1G
(M−1)
2...M H
(N−1)
M+3...M+N+1Q1,M+N+2 ⊗ 1
)
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
R]1,M+2(M)R1M (M − 1) · · ·R12(1)⊗ 1
)
×
(
R]M+1,M+N+2(−N)RM+3,M+N+2(1−N) · · ·RM+N+1,M+N+2(−1)⊗ 1
)
× (P1,M+1PM+2,M+N+2 ⊗ 1). (4.22)
Yangian of the General Linear Lie Superalgebra 21
Now recall that the supertrace on EndCM |N is the linear function defined by the assignment
str : Eij 7→ δij(−1)ı̄.
For any homogeneous elements X,X ′ ∈ EndCM |N we have the equality
str(XX ′) = str(X ′X)(−1)degX degX′
.
Let us define a linear map(
EndCM |N
)⊗(M+N+2) ⊗Y(glM |N )→
(
EndCM |N
)⊗2 ⊗Y(glM |N ) (4.23)
as applying str to all tensor factors of
(
EndCM |N
)⊗(M+N+2)
except the first and the last ones.
We will relate elements of the source vector space in (4.23) by the symbol ∼ if their images by
this map are the same. We extend the relation ∼ to series in u with coefficients in the source.
By Proposition 4.5 the product displayed in the four lines of (4.21) is divisible on the right
by G2...M+1HM+2...M+N+1 ⊗ 1. Therefore (4.21) is related by ∼ to the product
(I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1Q1,M+N+2 ⊗ 1)
×
(
R]1,M+2(M)R]M+1,M+N+2(−N)G1...MHM+3...M+N+2 ⊗ 1
)
× T1(u+M −N)T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)T ]M+N+2(u)
× (P1,M+1PM+2,M+N+2 ⊗ 1)
1
M !N !
.
Due to the definition (3.4) and to Proposition 4.4 the latter product equals
(P1,M+1PM+2,M+N+2 ⊗ 1)
× (I1 · · · IMJM+3 · · · JM+N+2G1...MHM+3...M+N+2QM+1,M+2 ⊗ 1)
× T1(u+M −N)T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)T ]M+N+2(u)
× (P1,M+1PM+2,M+N+2 ⊗ 1)
1
MN
= (P1,M+1PM+2,M+N+2 ⊗ 1)
× (I1 · · · IMJM+3 · · · JM+N+2G1...MHM+3...M+N+2QM+1,M+2 ⊗ 1)
× T1(u+M −N)T2(u+M −N − 1) · · ·TM (u−N + 1)
× T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)T ]M+N+2(u)(P1,M+1PM+2,M+N+2 ⊗ 1)
1
MN
.
Here we have used (3.11). Since G and H antisymmetrize the subspaces (4.11), by applying our
map (4.23) to the right-hand side of this equality and using the definition of B(u+ 1) we get
(−1)N (M − 1)!(N − 1)!Q⊗B(u+ 1). (4.24)
Let us now consider the product (4.22) which is equal to (4.21) due to (4.18). By again using
Proposition 4.5 the product (4.22) can be rewritten as(
I2 · · · IM+1JM+2 · · · JM+N+1G
(M−1)
2...M H
(N−1)
M+3...M+N+1Q1,M+N+2 ⊗ 1
)
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
22 M. Nazarov
×
(
R]1,M+2(M)G
(M−1)
2...M R1M (M − 1) · · ·R12(1)⊗ 1
)
×
(
R]M+1,M+N+2(−N)H
(N−1)
M+3...M+N+1RM+3,M+N+2(1−N) · · ·
×RM+N+1,M+N+2(−1)⊗ 1
)
(P1,M+1PM+2,M+N+2 ⊗ 1)
1
(M − 1)!(N − 1)!
.
By again using (4.19) and (4.20) the latter product equals(
I2 · · · IM+1JM+2 · · · JM+N+1G
(M−1)
2...M H
(N−1)
M+3...M+N+1Q1,M+N+2 ⊗ 1
)
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
R]1,M+2(M)R]M+1,M+N+2(−N)G1...MHM+3...M+N+2 ⊗ 1
)
× (P1,M+1PM+2,M+N+2 ⊗ 1)
1
(M − 1)!(N − 1)!
=
(
I2 · · · IM+1JM+2 · · · JM+N+1G
(M−1)
2...M H
(N−1)
M+3...M+N+1Q1,M+N+2 ⊗ 1
)
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
R]1,M+2(M)R]M+1,M+N+2(−N)⊗ 1
)
× (P1,M+1PM+2,M+N+2G2...M+1HM+2...M+N+1 ⊗ 1)
1
(M − 1)!(N − 1)!
∼ (I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1Q1,M+N+2 ⊗ 1)
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)T1(u+M −N)
× T ]M+N+2(u)TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
R]1,M+2(M)R]M+1,M+N+2(−N)P1,M+1PM+2,M+N+2 ⊗ 1
)
= (I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1 ⊗ Z(u))
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
×
(
Q1,M+N+2R
]
1,M+2(M)R]M+1,M+N+2(−N)P1,M+1PM+2,M+N+2 ⊗ 1
)
.
To obtain the last equality we used the definition of the series Z(u) and the centrality in Y(glM |N )
of the coefficients of this series, see the proof of Proposition 3.1.
Let us denote by S(u) the product in the latter four displayed lines. It is related by ∼ to the
product (4.22) which is equal to (4.21). We have already proved that the image of (4.21) under
our map (4.23) is equal to (4.24). Hence the image of S(u) is also equal to (4.24). In particular,
the image of S(u) under (4.23) does not change if we multiply this image on the right by
(I ⊗ 1 + 1⊗ J)⊗ 1 ∈
(
EndCM |N
)⊗2 ⊗Y(glM |N ),
see (4.5). Equivalently, the product S(u) is related by ∼ to itself multiplied on the right by
(I1 + JM+N+2)⊗ 1 ∈
(
EndCM |N
)⊗(M+N+2) ⊗Y(glM |N ). (4.25)
Let us now right multiply S(u) by (4.25) and also by the element
I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1 ∈
(
EndCM |N
)⊗(M+N+2) ⊗Y(glM |N ). (4.26)
Yangian of the General Linear Lie Superalgebra 23
The result is still related by ∼ to S(u) because (4.26) is a projector dividing S(u) on the left.
However, by multiplying the last line of S(u) by both (4.25) and (4.26) we obtain the product(
Q1,M+N+2R
]
1,M+2(M)R]M+1,M+N+2(−N)P1,M+1PM+2,M+N+2 ⊗ 1
)
× ((I1 + JM+N+2)I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1)
=
(
Q1,M+N+2R
]
1,M+2(M)R]M+1,M+N+2(−N)⊗ 1
)
× (I1 · · · IM (IM+1 + JM+2)JM+3 · · · JM+N+2P1,M+1PM+2,M+N+2 ⊗ 1).
Due to the definition (3.4) and to Proposition 4.3 the latter product equals
(I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1)
×
(
Q1,M+N+2
(
− 1
M
P1,M+1JM+N+2 +
1
N
PM+2,M+N+2I1
)
P1,M+1PM+2,M+N+2 ⊗ 1
)
= (I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1)
×
(
Q1,M+N+2
(
− 1
M
PM+2,M+N+2JM+2 +
1
N
P1,M+1IM+1
)
⊗ 1
)
.
Therefore S(u) is related by ∼ to
(I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1 ⊗ Z(u))
× T2(u+M −N − 1) · · ·TM (u−N + 1)T ]M+2(u−N)
× TM+1(u−N)T ]M+3(u−N + 1) · · ·T ]M+N+1(u− 1)
× (I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1)
×
(
Q1,M+N+2
(
− 1
M
PM+2,M+N+2JM+2 +
1
N
P1,M+1IM+1
)
⊗ 1
)
= (I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1 ⊗ Z(u)B(u))
× (I2 · · · IM+1JM+2 · · · JM+N+1 ⊗ 1)
×
(
Q1,M+N+2
(
− 1
M
PM+2,M+N+2JM+2 +
1
N
P1,M+1IM+1
)
⊗ 1
)
∼ (I2 · · · IM+1JM+2 · · · JM+N+1G2...M+1HM+2...M+N+1 ⊗ Z(u)B(u))
×
(
Q1,M+N+2
(
− 1
M
PM+2,M+N+2 +
1
N
P1,M+1
)
⊗ 1
)
. (4.27)
To obtain the equality in (4.27) we also used the relation (4.4) which in this instance implies that
(IM+1JM+2 ⊗ 1)T ]M+2(u−N)TM+1(u−N)(IM+1JM+2 ⊗ 1)
= (IM+1JM+2 ⊗ 1)TM+1(u−N)T ]M+2(u−N)(IM+1JM+2 ⊗ 1).
We could now show by direct calculation that applying the map (4.23) to the product in the
last two lines of (4.27) yields
(−1)N (M − 1)!(N − 1)!Q⊗ Z(u)B(u). (4.28)
Theorem 4.1 would then follow because the equality of (4.21) and (4.22) implies the equality
of (4.24) and (4.28). However, we will complete the proof of Theorem 4.1 by an indirect ar-
gument. We have already proved that the image of the product in the last two lines of (4.27)
equals (4.24). Since the image of the action of Q on
(
CM |N
)⊗2
is one dimensional, the latter
equality implies that Z(u)B(u) equals B(u+ 1) up to a scalar factor. This scalar factor is 1 be-
cause the leading terms of both series B(u)Z(u) and B(u+1) are 1. Theorem 4.1 is now proved.
24 M. Nazarov
References
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https://doi.org/10.1007/978-94-017-1963-6
https://doi.org/10.1007/978-94-017-1963-6
https://doi.org/10.1215/S0012-7094-87-05423-8
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https://doi.org/10.2307/1970615
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1 Introduction
2 Definition of the Yangian
3 Centre of the Yangian
4 Quantum Berezinian
References
|
| id | nasplib_isofts_kiev_ua-123456789-211008 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T03:14:51Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nazarov, Maxim 2025-12-22T09:26:27Z 2020 Yangian of the General Linear Lie Superalgebra. Maxim Nazarov. SIGMA 16 (2020), 112, 24 pages 1815-0659 2020 Mathematics Subject Classification: 16T20; 17B37; 81R50 arXiv:2011.02361 https://nasplib.isofts.kiev.ua/handle/123456789/211008 https://doi.org/10.3842/SIGMA.2020.112 We prove several basic properties of the Yangian of the Lie superalgebra M|N. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Yangian of the General Linear Lie Superalgebra Article published earlier |
| spellingShingle | Yangian of the General Linear Lie Superalgebra Nazarov, Maxim |
| title | Yangian of the General Linear Lie Superalgebra |
| title_full | Yangian of the General Linear Lie Superalgebra |
| title_fullStr | Yangian of the General Linear Lie Superalgebra |
| title_full_unstemmed | Yangian of the General Linear Lie Superalgebra |
| title_short | Yangian of the General Linear Lie Superalgebra |
| title_sort | yangian of the general linear lie superalgebra |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211008 |
| work_keys_str_mv | AT nazarovmaxim yangianofthegenerallinearliesuperalgebra |