Yangian of the general linear Lie superalgebra

We prove several basic properties of the Yangian of the general linear Lie superalgebra.


Introduction
Let gl M |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 gl M |N has been introduced in [7] by extending the definition of the Yangian of the general linear Lie algebra gl M , see for instance [6]. We will denote this extension by Y(gl M |N ) . It is a deformation of the universal enveloping algebra U(gl M |N [u]) of the polynomial current Lie superalgebra gl M |N [u] in the class of Hopf superalgebras. The definition of Y(gl M |N ) is reviewed in our Section 1.
In our Section 1 we will define two ascending filtrations on the associative algebra Y(gl M |N ) . The graded algebra associated with the first filtration is supercommutative. We prove that its elements corresponding to the defining generators (1.6) of Y(gl M |N ) are free generators of this supercommutative algebra. The graded algebra associated with the second ascending filtration is isomorphic to U(gl M |N [u]) . We prove this by using the representation theory of Y(gl M |N ) . Our proof follows [8] where the Yangian of the queer Lie superalgebra q M ⊂ gl M |M was studied. The freeness of the supercommutative graded algebra associated with the first filtration on Y(gl M |N ) follows from this isomorphism. Another proof of the freeness property was given in [4].
Two different families of central elements of Y(gl M |N ) have been defined in [7]. The definition of the first family uses the Hopf superalgebra structure on Y(gl M |N ) . This definition is reviewed in our Section 2. It was conjectured in [7] that the first family generates the centre of Y(gl M |N ) . Shortly after the publication of [7] this conjecture was proved by the author. The method of that proof was then used in [6] where the Yangian of gl M was considered. This method was also used in [4,8]. We include the original proof of this conjecture for Y(gl M |N ) in our Section 2.
The second definition extends the notion of a quantum determinant for the Yangian of gl M , see again [6] and references therein. This definition is reviewed in our Section 3. The main result of [7] was the relation between the two families of central elements of Y(gl M |N ) . However only a summary of the proof of this relation was given in [7] while the details were left unpublished.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(gl M |N ) was introduced in [7] it has been studied by several other authors. Here we do not 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(gl M |N ) from our second family. Let us also mention the work [9] which provides a generalization of Y(gl M |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 http://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html arXiv:2011.02361v2 [math.QA] 19 Jun 2022 1 Definition of the Yangian Throughout this article we will use the following general conventions. Let A and B be any two associative Z 2 -graded algebras. Their tensor product A ⊗ B is also an associative Z 2 -graded algebra such that for any homogeneous elements X, X ∈ A and Y, Y ∈ B For any Z 2 -graded modules U and V over A and B respectively, the vector space U ⊗ V is a Z 2 -graded module over A ⊗ B such that for any homogeneous elements x ∈ U and y ∈ V deg (x ⊗ y) = deg x + deg y . (1.4) A homomorphism α : A → B is a linear map such that α(X X ) = 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(X X ) = β(X ) β(X) (−1) deg X deg X . (1.5) If A is unital, let ι h be its embedding into the tensor product A ⊗n as the h-th tensor factor: 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 h 1 , . . . , h m ∈ {1, . . . , n} and of an element X ∈ A ⊗m of the form X = X (1) ⊗ . . . ⊗ X (m) we will denote X h 1 ...hm = ι h 1 (X (1) ) . . . ι hm (X (m) ) ∈ A ⊗n .
We will then extend the notation X h 1 ...hm to all elements X ∈ A ⊗m by linearity. Now let the indices i , j run through 1, . . . , M + N . We will always writeī = 0 if 1 i M andī = 1 if M < i M + N . Consider the Z 2 -graded vector space C M |N . Let e i ∈ C M |N be an element of the standard basis. The Z 2 -grading on C M |N is defined so that deg e i =ī . Let E ij ∈ End C M |N be the standard matrix unit, so that E ij e k = δ jk e i . The associative algebra End C M |N is Z 2 -graded so that deg E ij =ī + .
For any n we can identify the tensor product (End C M |N ) ⊗n with the algebra End((C M |N ) ⊗n ) acting on the vector space (C M |N ) ⊗n by repeatedly using the conventions (1.3) and (1.4).
Let us introduce the Yangian of the Lie superalgebra gl M |N . This is the complex associative unital Z 2 -graded algebra Y(gl M |N ) with the countable set of generators where r = 1, 2, . . . and i , j = 1, . . . , M + N . (1.6) The Z 2 -grading on the algebra Y(gl M |N ) is determined by setting deg T (r) ij =ī + for r 1. To write down defining relations for these generators we will employ the series in a formal variable u with coefficients from Y(gl M |N ) . Then for all possible indices i , j , k , l where v is another formal variable. The square brackets here stand for the supercommutator. Notice that the series denoted by T ij (u) here and in [7] differ by the scalar factor (−1) (ī +1) .
We will also use the following matrix form of the defining relations (1.8). Take the element This element acts on the vector space (C M |N ) ⊗2 so that e i ⊗ e j → e j ⊗ e i (−1)ī . Here we identify the algebra (End C M |N ) ⊗2 with the algebra End((C M |N ) ⊗2 ) by using (1.3). For any n let S n be the symmetric group acting on the set {1, . . . , n} by permutations. For each m = 1, . . . , n − 1 denote by σ m the element of S n exchanging m and m + 1 . The group S n also acts on the vector space (C M |N ) ⊗n . This action is defined by the assignment σ m → P m,m+1 for each m . Here we identify the algebra (End The rational function R(u) = 1 − P u −1 with values in the algebra (End C M |N ) ⊗2 is called the Yang R-matrix . It satisfies the Yang-Baxter equation in the algebra (End (1.10) Since P 2 = 1 , we also have the relation Now combine all the series (1.7) into the single element For any n and any p = 1, . . . , n we will denote (1.13) By using this notation for n = 2 the relations (1.8) can be rewritten as (1.14) Namely, after multiplying each side of (1.14) by u − v it becomes a relation of series in u , v with coefficients in (End C M |N ) ⊗2 ⊗ Y(gl M |N ) equivalent to the collection of all the relations (1.8).
1. An antiautomorphism of Y(gl M |N ) can be defined by the assignment Proof . Due to the convention (1.1), by using the notation (1.13) for n = 2 we get By using the convention (1.5), the antihomomorphism property of (1.15) follows from the relation which is obtained from (1.14) by using (1.11). The antihomomorphism (1.15) is clearly involutive and therefore bijective.
For all indices i , j define the series T ij (u) by using the element inverse to (1.12) so that Proof . Similarly to (1.17), by using the notation (1.13) for n = 2 we get Comparing this with (1.16) the antihomomorphism property of (1.19) follows from the relation which is obtained by multiplying both sides of the defining relation (1.14) on the left and right by T 2 (v) −1 and then by T 1 (u) −1 . The bijectivity of (1.19) follows from Proposition 2.2 below.
Further, let τ be the antiautomorphism of End C M |N defined by the assignment Then by the definition (1.12) we have Proof . Observe that (τ ⊗ τ )(P ) = P and hence Therefore the antihomomorphism property of (1.21) follows from the relation which is obtained by applying τ ⊗ τ ⊗ id to both sides of(1.14), see the proofs of Propositions 1.1 and 1.2 above.
To prove the bijectivity of the antihomomorphism (1.21), observe that its square is given by In particular, the square is an automorphism of the Z 2 -graded algebra Y(gl M |N ) .
Put T (u) = τ ⊗ id ( T (u) −1 ) . Then by (1.18) Proof . The assignment (1.24) can also be obtained by first applying (1.21) to T ij (u) and then applying (1.19) to the result. Hence (1.24) defines an automorphism of the algebra Y(gl M |N ) as a composition of two antiautomorphisms.
By the definition (1.23) the homomorphism property of (1.24) is equivalent to the relation which can also be obtained by applying τ ⊗ τ ⊗ id to both sides of (1.20) and then using (1.22).
Hence ( where the tensor product is taken over the subalgebra . 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 (1.28) is an embedding of Hopf algebras as by the above definitions ij .
There are two natural ascending filtrations on the associative algebra Y(gl M |N ) . The first one is defined by assigning the degree r to the generator (1.6). Let gr Y(gl M |N ) be the corresponding graded algebra.
Let us denote by X ij in the degree r component of gr Y(gl M |N ) . Observe that the Z 2 -grading on the algebra Y(gl M |N ) descends to gr Y(gl M |N ) so that the degree of the image is againī + . It follows from the relations (1.8) that these images supercommute. We shall prove that gr Y(gl M |N ) is a free supercommutative algebra generated by these images. Now introduce another filtration on the associative algebra Y(gl M |N ) by assigning the degree r − 1 to the generator (1.6). Let gr Y(gl M |N ) be the corresponding graded algebra. Consider the latter algebra.
Let us denote by Y The Z 2 -grading on the algebra Y(gl M |N ) descends to gr Y(gl M |N ) so that the degree of the image isī + . The graded algebra gr Y(gl M |N ) inherits from Y(gl M |N ) the Hopf algebra structure too. Namely, by the above definitions in the graded Hopf algebra gr Y(gl M |N ) for r 1 ( The latter relation in gr Y(gl M |N ) follows from (1.8), see for instance [6,Section 1]. This proves the homomorphism property of the assignment (1.31). This homomorphism is clearly surjective and preserves the Z 2 -grading.
It immediately follows from (1.30) that (1.31) is a homomorphism of Hopf algebras. Here we use the standard Hopf algebra structure on U(gl M |N [u]) as the universal enveloping algebra of a Lie superalgebra. We shall demonstrate that the homomorphism (1.31) is also injective.
By comparing (1.10) with (1.14) we obtain that for any z ∈ C the assignment More explicitly, under (1.33) for any r 0 The comultiplication on Y(gl M |N ) now allows us to define for n = 1, 2, . . . a representation Y(gl M |N ) → (End C M |N ) ⊗n depending on z 1 , . . . , z n ∈ C . This is the tensor product of the representations (1.33) where z = z 1 , . . . , z n . Due to (1.12) and (1.29) under this representation ( Proof . Take any finite linear combination of the products T Therefore the sum A ∈ (End C M |N ) ⊗n coincides with the image of the sum under the tensor product of the evaluation representations at the points z = z 1 , . . . , z n . Here we used the explicit description (1.34) of the representation Y(gl M |N ) → End C M |N corresponding to z ∈ C . Due to Proposition 1.6 it now suffices to show that when the complex parameters z 1 , . . . , z n and the positive integer n vary, the kernels of the tensor products of the evaluation representations of the algebra U(gl M |N [u]) at z = z 1 , . . . , z n have the zero intersection. This will also imply that the homomorphism (1.31) is injective.
The corresponding elements of End C M |N will be denoted by Choose any total ordering of this basis which ends with the infinite sequence . . . , f 1 u 2 , f 1 u , f 1 . Take any finite linear combination L of the products with L r 1 ...rm p 1 ...pm ∈ C being the coefficients. We assume that the factors in the products are arranged according to our ordering of the basis of gl M |N [u] . Due to the supercommutation relations in U(gl M |N [u]) we assume it without any loss of generality. The basis elements of Z 2 -degree 0 may occur in any product (1.38) with a multiplicity but those of Z 2 -degree 1 may occur at most once.
Let us denote by ρ z the evaluation representation (1.36). More generally, denote by ρ z 1 ...zn the tensor product ρ z 1 ⊗ . . . ⊗ ρ zn pulled back through n-fold comultiplication on U(gl M |N [u]) . Hence ρ z 1 ...zn is a homomorphism of associative algebras Suppose that ρ z 1 ...zn (L) = 0 for every n and all z 1 , . . . , z n ∈ C . We need to prove that L = 0 . For each product (1.38) there is a number a such that p 1 , . . . , p a > 1 but p a+1 , . . . , p m = 1 . This is due to our ordering of the basis of gl M |N [u] . The numbers a for different products (1.38) may differ, and we do not exclude the case a = 0 . Let h be the maximum of the numbers a .
Suppose n h . Let ω h be the supersymmetrisation map of the tensor product gl M |N [u] ⊗h normalised so that ω 2 h = h! ω h . Let W be the subspace of (End C M |N ) ⊗n spanned by the vectors F q 1 ⊗ . . . ⊗ F qn where at least one of the indices q 1 , . . . , q h is 1 . If h = 0 then this subspace is assumed to be zero. Applying the homomorphism ρ z 1 ...zn to a product (1.38) with a = h gives  Here we also employ the observation that in (1.40) the image of ρ z 1 ⊗ . . . ⊗ ρ z h does not depend on the parameters z h+1 , . . . , z n while the product over b = h + 1, . . . , c in (1.40) does depend on these parameters whenever c > h . Therefore if ρ z 1 ...zn (L) = 0 for a certain n g and for all z 1 , . . . , z n ∈ C then for each of the latter pairs

Centre of the Yangian
Here we will give a description of the centre of the algebra Y(gl M |N ) . An element of Y(gl M |N ) is called central if it supercommutes with each element of Y(gl M |N ) . However, we will see that the central elements of Y(gl M |N ) have the Z 2 -degree 0 . Hence they commute with each element of Y(gl M |N ) in the usual sense. Our description comes from computing the square of the antipodal mapping S of Y(gl M |N ) . Another description of the centre of Y(gl M |N ) will be given in the next section. In that section we will also establish a correspondence between the two descriptions.
Proof . By using the definition (1.9) introduce the element of the algebra (End 3) The image of the action of Q on (C M |N ) ⊗2 is one dimensional and is spanned by the vector Here we regard Q as an element of the algebra End((C M |N ) ⊗2 ) by identifying this algebra with (End C M |N ) ⊗2 via (1.3). We also have Q 2 = (M − N ) Q . By using the latter relation we get The rational function of the variable u given by the equalities (2.4) will be denoted by R (u) . Now multiply both sides of the relation (1.14) by T −1 2 (v) on the left and right, and then apply τ relative to the second tensor factor End C M |N in (End C M |N ) ⊗2 ⊗ Y(gl M |N ) . Multiplying the resulting relation on the left and right by R (u − v) ⊗ 1 yields Multiplying the latter relation by u − v − M + N and then setting u = v + M − N we get see (2.4). Since the image of Q in (C M |N ) ⊗2 is one dimensional, either side of the relation (2.6) must be equal to Q ⊗ Z(v) where Z(v) is a certain power series in v −1 with coefficients from Y(gl M |N ) . The equality of the left hand side and of the right hand side of (2.6) to Q ⊗ Z(v) is equivalent respectively to (2.1) and (2.2). We just need to replace the variable v in (2.6) by u . It is immediate from (1.7) and (2.1) that every coefficient of the series Z(u) has Z 2 -degree 0 and that its leading term is 1 . Let us prove that all these coefficients are central in Y(gl M |N ) .
To this end we will work with the elements (1.13) where n = 3 . By using (1.14) and (2.5), On the other hand, due to (1.11) we have the identity in the algebra (End C M |N ) ⊗3 (u) By applying to it the antiautomorphism τ relative to the third tensor factor of (End C M |N ) ⊗3 and then using the definition (2.4) we get Therefore multiplying the first and third lines of the display (2.7) by Q 23 ⊗ 1 on the left yields We just need to replace the variable u in (2.8) by u − v − M + N and use the relation (2.6). The last relation implies that any generator T (r) ij commutes with every coefficient of Z(v) .
The square S 2 of antipodal mapping is an automorphism of the associative algebra Y(gl M |N ) .
Proposition 2.2. The automorphism S 2 of Y(gl M |N ) is given by the assignment Proof . By the definition (1.18) for any indices i , j we have the identity Here we use the convention (1.1). By using the definition of S this identity can be written as Let us apply the antiautomorphism S to both sides of the latter identity. We get By comparing the last displayed identity with (2.1) we see that for any indices k , j Z(u) S 2 (T kj (u)) = T kj (u + M − N ) .
Proof . The square of the antipodal mapping is a coalgebra homomorphism. Hence the images of T ij (u) relative to the two compositions ∆ S 2 and (S 2 ⊗ S 2 ) ∆ are the same. By Proposition 2.2 these images are respectively equal to By equating the two images of T ij (u) we obtain that Since the comultiplication ∆ is an algebra homomorphism, we get the first statement in (2.10). The second statement in (2.10) immediately follows from (2.1) and from the definition of the counit homomorphism ε . The third statement follows from the first and the second because the multiplication µ : Y(gl M |N ) ⊗ Y(gl M |N ) → Y(gl M |N ) and the unit mapping δ : C → Y(gl M |N ) satisfy the identity µ (S ⊗id) ∆ = δ ε . 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 3 we will also use the following observation. We shall prove Theorem 2.6 at the end of this section. We will use the following proposition.
Proposition 2.7. For any r 2 the element Z (r) ∈ Y(gl M |N ) has degree r − 2 relative to the second filtration on Y(gl M |N ) . Its image in the graded algebra gr Y(gl M |N ) is equal to (2.14) For any i and j denote byṪ ij (u) the formal derivative of the series T ij (u) so thaṫ By tending in the relation (2.14) the parameter u to v + M − N we then get Let us now observe that where O ij (v) is a certain formal power series in v −1 with coefficients in Y(gl M |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 2 is zero. By taking the sum of the relations (2.16) over the indices i = 1, . . . , M + N and then using the definition (2.1) along with this observation we get By using the definition (1.18) the latter relation can be rewritten as (2.17) For any indices i and j the leading term of the series T ij (v) is δ ij while for every r 1 the coefficient of this series at v −r has degree r − 1 relative to the second filtration on Y(gl M |N ) . Furthermore for any given r 1 the coefficients at v −r of the seriesṪ ij (v) andṪ ij (v + M − N ) have the same image in the graded algebra gr Y(gl M |N ) , see (2.15). Therefore by taking the coefficients at v −r with any r 2 in the relation (2.17) the image of Z (r) in gr Y(gl M |N ) equals In our proof of Theorem 2.6 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] . C r pq f r where C r pq ∈ C are the structure constants. Let d be the maximal degree such that A depends on at least one of the basis elements where the coefficients A  f 1 u d+1 , . . . , f K u d+1 must be zero. Thus where we used the notation Similarly to (2.19) here we used the notation Let us now fix any non-negative integers m 1 , . . . , m K and observe that the elements also form a basis of a . Since the centre of the Lie superalgebra a is trivial, the system of linear equations on w 1 , . . . , w K ∈ C has only zero solution. This system can be written as K q=1 w q (m q + 1) C r pq (−1) h q = 0 where p , r = 1, . . . , K .
Hence by comparing the latter system with (2.20) we obtain that A m 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 2.6. Due to Proposition 2.

Quantum Berezinian
The element (1.12) can be also regarded as an (M + N ) × (M + N ) matrix whose ij entry is the series T ij (u) . The quantum Berezinian of that matrix is the series B(u) defined as the product of two alternated sums over the symmetric groups S M and S N . Here (−1) σ denotes the sign of the permutation σ . The main purpose of the present section is to prove the following theorem. Note that the second assignment here also follows directly from the definition of B(u) because For N = 0 the Hopf algebra Y(gl M |N ) is the Yangian Y(gl M ) of the Lie algebra gl M , see [6]. In this case the second sum in the above definition of B(u) is assumed to be 1 , and B(u) equals This is the quantum determinant of the M × M matrix whose ij entry is the series T ij (u) . It has been well known that the coefficients of the series (3.1) at u −1 , u −2 , . . . are free generators of the centre of Y(gl M ) , see [6, Section 2] and references therein. A detailed proof of Theorem 3.1 in this particular case was given in [6, Section 5] by following [7]. For any M , N denote by X(u) the (M + N ) × (M + N ) matrix whose ij entry is the series Let us now consider the other particular case when M = 0 . In this case B(u) equals the sum This sum can also be obtained by applying the automorphism (1.24) of Y(gl 0|N ) to the series Let us denote by C(u) the latter series. Then due to Corollary 2.5 the statement of Theorem 3.1 for M = 0 becomes equivalent to the relation Z(u) C(u + 1) = C(u) .  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. Hence by multiplying (2.5) on the left and on the right by I ⊗ J ⊗ 1 we get the relation of series in u , v with coefficients in the algebra (End C M |N ) ⊗2 ⊗ Y(gl M |N ) , see (2.4). The latter relation is equivalent to the collection of all commutation relations stated in Proposition 3.2.
We will keep using the projectors I and J introduced in the proof of Proposition 3.2 above. Note that they also satisfy the relations in the algebra (End   Further, since (τ ⊗ τ )(P ) = P we have the equality by the definition (2.3). Hence applying to the relation P 12 P 13 = P 13 P 23 the antiautomorphism τ −1 of the first tensor factor of (End C M |N ) ⊗3 yields the relation Q 13 Q 12 = Q 13 P 23 . It follows that in the algebra (End C M |N ) ⊗(M +N +2) Therefore the sum displayed in the two lines of (3.6) equals 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 Therefore by multiplying (3.13) by the first summand at the right hand side of (3.14) we get which is equal to the product at the right hand side of the equality stated in Proposition 3.4. (G (n) ⊗ 1) T 1 (u) . . . T n (u − n + 1) = T n (u − n + 1) . . . T 1 (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 i < j n then let σ ij ∈ S n be the transposition of the numbers i and j . There is a well known identity in the symmetric group ring CS n σ∈Sn 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 (End C M |N ) ⊗n The first equality stated in Proposition 3.5 follows from this relation by repeatedly using (1.14).
The second equality in Proposition 3.5 follows from this relation by repeatedly using (1.25).
We will now prove Theorem 3.1. Using the relations (1.14),(1.25) and (2.5) we get an equality of series in u with coefficients in the algebra (End Now recall that the supertrace on End C M |N is the linear function defined by the assignment str : E ij → δ ij (−1)ī .
For any homogeneous elements X, X ∈ End C M |N we have the equality str(X X ) = str(X X ) (−1) deg X deg X . as applying str to all tensor factors of (End C M |N ) ⊗(M +N +2) except the first and the last ones. We will relate elements of the source vector space in (3.23) 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.

Maxim Nazarov
Let us now right multiply S(u) by (3.25) and also by the element We could now show by direct calculation that applying the map (3.23) to the product in the last two lines of (3.27) yields (−1) N (M − 1)! (N − 1)! Q ⊗ Z(u) B(u) . (3.28)