Representations of the Yangians associated with Lie superalgebras ${\frak{osp}}(1|2n)$

We classify the finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\frak{osp}}_{1|2n}$ in terms of the Drinfeld polynomials. The arguments rely on the description of the representations in the particular case $n=1$ obtained in our previous work.


Introduction
The finite-dimensional irreducible representations of the Yangian Y(g) associated with a simple Lie algebra g were classified by Drinfeld [5]. The arguments rely on the work of Tarasov [9] on the particular case of Y(sl 2 ), where the classification was carried over in the language of monodromy matrices within the quantum inverse scattering method; see [7,Sec. 3.3] for a detailed adapted exposition of these results. This description of the representations of the Yangian Y(sl 2 ), along with some other low rank cases, should also play an essential role in the classification of the finite-dimensional irreducible representations of the Yangians associated with simple Lie superalgebras. One of these cases was considered in our previous work [8], where the representations of the Yangian Y(osp 1|2 ) were described.
These two basic cases turn out to be sufficient to complete the classification in the case of the Yangians associated with the orthosymplectic Lie superalgebras osp 1|2n . We prove in this paper that, similar to the classification results of [5], the finite-dimensional irreducible representations of the Yangian Y(osp 1|2n ) are in one-to-one correspondence with the n-tuples of monic polynomials (P 1 (u), . . . , P n (u)), and so we call them the Drinfeld polynomials.
To describe the results in more detail, recall that the Yangian Y(osp M |2n ), as introduced by Arnaudon et al. [1], can be considered as a quotient of the extended Yangian X(osp M |2n ) defined via an RT T relation. A standard argument shows that every finite-dimensional irreducible representation of X(osp M |2n ) is a highest weight representation. It is isomorphic to the irreducible quotient L(λ(u)) of the Verma module M(λ(u)) associated with an (n + 1)-tuple λ(u) = (λ 1 (u), . . . , λ n+1 (u)) of formal series λ i (u) ∈ 1 + u −1 C[[u −1 ]]. The tuple is called the highest weight of the representation. The key step in the classification is to find the conditions on the highest weight for the representation L(λ(u)) to be finite-dimensional. The required necessary conditions are derived by induction from those for the associated actions of the Yangians Y(gl 2 ) and X(osp 1|2 ) on the respective cyclic spans of the highest vector of L(λ(u)). The sufficiency of these conditions is verified by constructing the fundamental representations of the Yangian X(osp M |2n ); cf. [3], [4]. The following is our main result.
Main Theorem. Every finite-dimensional irreducible representation of the algebra X(osp 1|2n ) is isomorphic to L(λ(u)) for a certain highest weight λ(u). The representation L(λ(u)) is finitedimensional if and only if for some monic polynomials P i (u) in u. The finite-dimensional irreducible representations of the Yangian Y(osp 1|2n ) are in a one-to-one correspondence with the n-tuples of monic polynomials (P 1 (u), . . . , P n (u)).
The endomorphism algebra End C 1|2n gets a Z 2 -gradation with the parity of the matrix unit e ij found byī + mod 2.
We will consider even square matrices with entries in Z 2 -graded algebras, their (i, j) entries will have the parityī + mod 2. The algebra of even matrices over a superalgebra A will be identified with the tensor product algebra End C 1|2n ⊗ A, so that a matrix A = [a ij ] is regarded as the element We will use the involutive matrix super-transposition t defined by for i = 1, . . . , n + 1, −1 for i = n + 2, . . . , 2n + 1.
This super-transposition is associated with the bilinear form on the space C 1|2n defined by the anti-diagonal matrix G = [δ ij ′ θ i ]. We will also regard t as the linear map In the case of multiple tensor products of the endomorphism algebras, we will indicate by t a the map (2.1) acting on the a-th copy of End C 1|2n .
A standard basis of the general linear Lie superalgebra gl 1|2n is formed by elements E ij of the parityī + mod 2 for 1 i, j 2n + 1 with the commutation relations We will regard the orthosymplectic Lie superalgebra osp 1|2n associated with the bilinear from defined by G as the subalgebra of gl 1|2n spanned by the elements Introduce the permutation operator P by The R-matrix associated with osp 1|2n is the rational function in u given by This is a super-version of the R-matrix originally found in [10]. Following [1], we define the extended Yangian X(osp 1|2n ) as a Z 2 -graded algebra with generators t (r) ij of parityī + mod 2, where 1 i, j 2n + 1 and r = 1, 2, . . . , satisfying certain quadratic relations. In order to write them down, introduce the formal series and combine them into the matrix Consider the algebra End C 1|2n ⊗End C 1|2n ⊗X(osp 1|2n )[[u −1 ]] and introduce its elements T 1 (u) and T 2 (u) by The defining relations for the algebra X(osp 1|2n ) take the form of the RT T -relation where c(u) is a series in u −1 . All its coefficients belong to the center ZX(osp 1|2n ) of X(osp 1|2n ) and generate the center. The Yangian Y(osp 1|2n ) is defined as the subalgebra of X(osp 1|2n ) which consists of the elements stable under the automorphisms We have the tensor product decomposition The Yangian Y(osp 1|2n ) can be equivalently defined as the quotient of X(osp 1|2n ) by the relation We will also use a more explicit form of the defining relations (2.3) written in terms of the series (2.2) as follows: For any a ∈ C the mapping t ij (u) → t ij (u + a) (2.8) defines an automorphism of the algebra X(osp 1|2n ).
The universal enveloping algebra U(osp 1|2n ) can be regarded as a subalgebra of X(osp 1|2n ) via the embedding This fact relies on the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian which was pointed out in [1] and [2]. It states that the associated graded algebra for Y(osp 1|2n ) is isomorphic to U(osp 1|2n [u]). A detailed proof of the theorem can be given by extending the arguments of [3,Sec. 3] to the super case with the use of the vector representation recalled below in (3.6). The extended Yangian X(osp 1|2n ) is a Hopf algebra with the coproduct defined by For the image of the series c(u) we have ∆ : c(u) → c(u) ⊗ c(u) and so the Yangian Y(osp 1|2n ) inherits the Hopf algebra structure from X(osp 1|2n ).

Highest weight representations
We will start by deriving a general reduction property for representations of the extended Yangians X(osp 1|2n ) analogous to [3,Lemma 5.13]. For an X(osp 1|2n )-module V set Proposition 3.1. The subspace V + is stable under the action of the operators t ij (u) subject to 2 i, j 2n. Moreover, the assignmentt ij (u) → t i+1,j+1 (u) for 1 i, j 2n − 1 defines a representation of the algebra X(osp 1|2n−2 ) on V + , where thet ij (u) denote the respective generating series for X(osp 1|2n−2 ).
Proof. Suppose that 2 k, l 2n and j > 1. For any η ∈ V + apply (2.7) to get Another application of (2.7) yields implying t 1j (u)t kl (u)η = 0. A similar calculation shows that t i 1 ′ (u)t kl (u)η = 0 for i < 1 ′ thus proving the first part of the proposition. Now suppose that 2 i, j, k, l 2n. By (2.7) the super-commutator [t ij (u), t kl (v)] of the operators in V + equals To transform these terms, use (2.7) again to get the relations Now combine the expressions together and observe that the actions of the operators t 11 (u) and Taking into account the change of the value κ → κ + 1 for the algebra X(osp 1|2n−2 ), we find that the formula for the super-commutator [t ij (u), t kl (v)] agrees with the defining relations of X(osp 1|2n−2 ). The arguments of that paper should apply to the super-case to lead to a Drinfeld-type presentation of the Yangians Y(osp 1|2n ) extending the work [2].
A representation V of the algebra X(osp 1|2n ) is called a highest weight representation if there exists a nonzero vector ξ ∈ V such that V is generated by ξ, for some formal series The vector ξ is called the highest vector of V .
Proposition 3.3. The series λ i (u) associated with a highest weight representation V satisfy the consistency conditions for i = 1, . . . , n. Moreover, the coefficients of the series c(u) act in the representation V as the multiplications by scalars determined by Proof. To derive the consistency conditions, we will use the induction on n with the base case n = 1 already considered in [8]. Suppose that n 2 and introduce the subspace V + by (3.1). The vector ξ belongs to V + , and applying Proposition 3.1 we find that the cyclic span X(osp 1|2n−2 ) ξ is a highest weight submodule with the highest weight (λ 2 (u), . . . , λ 2 ′ (u)). By the induction hypothesis, this implies conditions (3.4) with i = 2, . . . , n. Furthermore, using the defining relations (2.7), we get Setting v = u − κ − 1 = u + n − 1/2 we obtain (3.4) for i = 1. Finally, the last part of the proposition is obtained by using the expression for c(u) implied by taking the (1 ′ , 1 ′ ) entry in the matrix relation (2.4).
Proof. The argument is essentially the same as for the proof of the corresponding counterparts of the property for the Yangians associated with Lie algebras; cf. [3, Thm 5.1], [7,Sec. 3.2]. We online some key steps.
Suppose that V is a finite-dimensional irreducible representation of the algebra X(osp 1|2n ) and introduce its subspace V 0 by First we note that V 0 is nonzero, which follows by considering the set of weights of V , regarded as an osp 1|2n -module defined via the embedding (2.9). This set is finite and hence contains a maximal weight with respect to the standard partial ordering on the set of weights of V . A weight vector with this weight belongs to V 0 . Furthermore, we show that V 0 is stable under the action of all operators t ii (u). This follows by straightforward calculations similar to those used in the proof of Proposition 3.1, relying on the defining relations (2.7). In a similar way, we verify that all the operators t ii (u) with i = 1, . . . , 2n + 1 form a commuting family of operators on V 0 . Hence they have a simultaneous eigenvector ξ ∈ V 0 . Since the representation V is irreducible, the submodule X(osp 1|2n )ξ must coincide with V thus proving that V is a highest weight module.
By considering the osp 1|2n -weights of V we can also conclude that the highest vector ξ of V is determined uniquely, up to a constant factor. Proposition 3.4 yields the first part of the Main Theorem. Our next step is to show that the conditions in the theorem are necessary for the representation L(λ(u)) to be finite-dimensional. So we now suppose that dim L(λ(u)) < ∞ and argue by induction on n. The conditions (1.1) in the base case n = 1 are implied by the main result of [8]. Suppose further that n 2.
Recall that the Yangian Y(gl n ) for the general linear Lie algebra gl n is defined as a unital associative algebra with countably many generators t ij , . . . where 1 i, j n, and the defining relations written in terms of the series see [7] for a detailed exposition of the algebraic structure and representation of the Yangians associated with gl n . The Yangian Y(gl n ) can be regarded as a subalgebra of X(osp 1|2n ) via the embedding The cyclic span Y(gl n )ξ ⊂ L(λ(u)) is a highest weight module over Y(gl n ). Its highest weight is the n-tuple (λ 1 (−u), . . . , λ n (−u)). Since dim L(λ(u)) < ∞, the corresponding conditions for finite-dimensional highest weight representations of Y(gl n ) must be satisfied; see [7,Sec. 3.4]. This implies conditions (1.1) of the Main Theorem for i = 1, . . . , n − 1. Furthermore, by Proposition 3.1, the subspace L(λ(u)) + is a module over the extended Yangian X(osp 1|2n−2 ). The vector ξ generates a highest weight X(osp 1|2n−2 )-module with the highest weight (λ 2 (u), . . . , λ n+1 (u)). Since this module is finite-dimensional, conditions (1.1) hold for i = 2, . . . , n by the induction hypothesis. This completes the proof of the necessity of the conditions. Now suppose that conditions (1.1) hold and derive that the corresponding module L(λ(u)) is finite-dimensional. The n-tuple of Drinfeld polynomials (P 1 (u), . . . , P n (u)) determines the highest weight λ(u) up to a simultaneous multiplication of all components λ i (u) by a series f (u) ∈ 1 + u −1 C[[u −1 ]]. This operation corresponds to twisting the action of the algebra X(osp 1|2n ) on L(λ(u)) by the automorphism (2.5). Hence, it suffices to prove that a particular module L(λ(u)) corresponding to a given set of Drinfeld polynomials is finite-dimensional.
This observation implies that the cyclic span corresponds to the set of Drinfeld polynomials (P 1 (u)Q 1 (u), . . . , P n (u)Q n (u)), where the P i (u) and Q i (u) are the Drinfeld polynomials for L(λ(u)) and L(µ(u)), respectively. Therefore, we only need to establish the sufficiency of conditions (1.1) for the fundamental representations of X(osp 1|2n ) associated with the n-tuples of Drinfeld polynomials such that P j (u) = 1 for all j = i and P i (u) = u + b for a certain i ∈ {1, . . . , n} and b ∈ C; cf. [4]. Moreover, it is sufficient to take one particular value of b ∈ C; the general case will then follow by twisting the action of the algebra X(osp 1|2n ) in such representations by automorphisms of the form (2.8).
Consider the vector representation of X(osp 1|2n ) on C 1|2n defined by The homomorphism property follows from (2.3) by applying the standard transposition to one copy of End C 1|2n in the Yang-Baxter equation satisfied by R(u). Now use the coproduct (2.10) and suitable automorphisms (2.8) to equip the tensor product space (C 1|2n ) ⊗k with the action of X(osp 1|2n ) by setting where the generators act in the respective copies of the vector space C 1|2n via the rule (3.6). For the values k = 1, . . . , n introduce the vectors ξ k = σ∈S k sgn σ · e σ(1) ⊗ · · · ⊗ e σ(k) ∈ (C 1|2n ) ⊗k .
Now verify that each vector ξ k has the properties t ij (u) ξ k = 0 for 1 i < j n + 1 (3.8) and ξ k for i = 1, . . . , k, ξ k for i = k + 1, . . . , n + 1. (3.9) The expression for the vector ξ k involves only tensor products of the basis vectors e i with i n. This implies that for the application of the operators t ij (u) with 1 i j n to ξ k we may restrict the sum in formula (3.7) to the values a p ∈ {1, . . . , n}.
By using the embedding (3.5), we may regard the cyclic span Y(gl n )ξ k as a Y(gl n )-module. Moreover, this module is isomorphic to A (k) (C n ) ⊗k , where A (k) is the anti-symmetrization operator. It is well-known that this Y(gl n )-module is isomorphic to the evaluation module L(1, . . . , 1, 0, . . . , 0) (with k ones) twisted by a shift automorphism u → u + k − 1; see e.g. [7,Sec. 6.5]. This yields formulas (3.8) and (3.9) with 1 i j n. They are easily verified directly for the remaining generators.
Formulas (3.9) show that the corresponding set of Drinfeld polynomials for the highest weight module X(osp 1|2n )ξ k has the form P i (u) = 1 for i = k, while P k (u) = u − k. This completes the proof of the second part of the Main Theorem concerning conditions (1.1). The last part is immediate from the decomposition (2.6); cf. [3,Sec. 5.3].