SCHUR-WEYL DUALITY FOR QUANTUM GROUPS

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1 SCHUR-WEYL DUALITY FOR QUANTUM GROUPS YI SUN Abstract. These are notes for a talk in the MIT-Northeastern Fall 2014 Graduate seminar on Hecke algebras and affine Hecke algebras. We formulate and sketch the proofs of Schur-Weyl duality for the pairs U qsl n), H qm)), Y sl n), Λ m), and U qŝln), Hqm)). We follow mainly [Ara99, Jim86, Dri86, CP96], drawing also on the presentation of [BGHP93, Mol07]. Contents 1. Introduction 1 2. Finite-type quantum groups and Hecke algebras Definition of the objects R-matrices and the Yang-Baxter equation From the Yang-Baxter equation to the Hecke relation Obtaining Schur-Weyl duality 3 3. Yangians and degenerate affine Hecke algebras Yang-Baxter equation with spectral parameter and Yangian The Yangian of sl n Degenerate affine Hecke algebra The Drinfeld functor Schur-Weyl duality for Yangians 6 4. Quantum affine algebras and affine Hecke algebras Definition of the objects Drinfeld functor and Schur-Weyl duality 9 References 9 1. Introduction Let V = C n be the fundamental representation of sl n. The vector space V m may be viewed as a Usl n ) and S m -representation, and the two representations commute. Classical Schur-Weyl duality gives a finer understanding of this representation. We first state the classifications of representations of S m and sl n. Theorem 1.1. The finite dimensional irreducible representations of S m are parametrized by partitions λ m. For each such λ, the corresponding representation S λ is called a Specht module. Theorem 1.2. The finite dimensional irreducible representations of sl n are parametrized by signatures λ with lλ) n and i λ i = 0. For any partition λ with lλ) n, there is a unique shift λ of λ so that i λ i = 0. We denote the irreducible with this highest weight by L λ. The key fact underlying classical Schur-Weyl duality is the following decomposition of a tensor power of the fundamental representation. Theorem 1.3. View V m as a representation of S m and Usl n ). We have the following: a) the images of C[S m ] and Usl n ) in EndW ) are commutants of each other, and Date: October 21,

2 2 YI SUN b) as a C[S m ] Usl n )-module, we have the decomposition V m = λ m lλ) n S λ L λ. We now reframe this result as a relation between categories of representations; this reformulation will be the one which generalizes to the affinized setting. Say that a representation of Usl n ) is of weight m if each of its irreducible components occurs in V m. In general, the weight of a representation is not well-defined; however, for small weight, we have the following characterization from the Pieri rule. Lemma 1.4. The irreducible L λ is of weight m n 1 if and only if λ = i c iω i with i ic i = m. Given a S m -representation W, define the Usl n )-representation FSW ) by FSW ) = Hom Sm W, V m ), where the Usl n )-action is inherited from the action on V m. RepUsl n )), and we may rephrase Theorem 1.3 as follows. Evidently, FS is a functor RepS m ) Theorem 1.5. For n > m, the functor FS is an equivalence of categories between RepS m ) and the subcategory of RepUsl n )) consisting of weight m representations. In this talk, we discuss generalizations of this duality to the quantum group setting. In each case, Usl n ) will be replaced with a quantization U q sl n ), Y sl n ), or U q ŝl n)), and C[S m ] will be replaced by a Hecke algebra H q m), Λ m, or H q m)). 2. Finite-type quantum groups and Hecke algebras 2.1. Definition of the objects. Our first generalization of Schur-Weyl duality will be to the finite type quantum setting. In this case, U q sl n ) will replace Usl n ), and the Hecke algebra H q m) of type A m 1 will replace S m. We begin by defining these objects. Definition 2.1. Let g be a simple Kac-Moody Lie algebra of simply laced type with Cartan matrix A = a ij ). The Drinfeld-Jimbo quantum group U q g) is the Hopf algebra given as follows. As an algebra, it is generated by x ± i and q hi for i = 1,..., n 1 so that {q hi } are invertible and commute, and we have have the relations q hi x ± j q hi = q ±aij e j, [x + i, x j ] = δ q hi q hi ij q q 1, The coalgebra structure is given by the coproduct 1 a ij r=0 1) r [ 1 aij r ] x ± i )r x ± j x± i )1 aij r = 0. x + i ) = x+ i q hi + 1 x + i, x i ) = x i 1 + q hi x i, qhi ) = q hi q hi, and counit εx ± i ) = 0 and εqhi ) = 1, and the antipode is given by Sx + i ) = x+ i q hi, Sx i ) = qhi x i, Sqhi ) = q hi. Definition 2.2. The Hecke algebra H q m) of type A m 1 is the associative algebra given by H q m) = T 1,..., T m 1 T i q 1 )T i + q) = 0, T i T i+1 T i = T i+1 T i T i+1, [T i, T j ] = 0 for i j R-matrices and the Yang-Baxter equation. To obtain H q m)-representations from U q sl n )-representations, we use the construction of R-matrices. Proposition 2.3. There exists a unique universal R-matrix R U q sl n ) ˆ U q sl n ) such that: a) R q i xi xi 1 + U q n + ) ˆ U q n )) >0 ) for {x i } an orthonormal basis of, and b) R x) = 21 x)r, and c) 1)R = R 13 R 23 and 1 )R = R 13 R 12. We say that such an R defines a pseudotriangular structure on U q sl n ). Let P x y) = y x denote the flip map, and let R = P R. From Proposition 2.3, we may derive several additional properties of R and R. Corollary 2.4. The universal R-matrix of U q sl n ):

3 SCHUR-WEYL DUALITY FOR QUANTUM GROUPS 3 a) satisfies the Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 ; b) gives an isomorphism R : W V V W for any V, W RepU q sl n )); c) satisfies a different version of the Yang-Baxter equation R 23 R12 R23 = R 12 R23 R12 ; d) when evaluated in the tensor square V 2 of the fundamental representation of U q sl n ) is given by 1) R V V = q i E ii E ii + i j E ii E jj + q q 1 ) i>j E ij E ji From the Yang-Baxter equation to the Hecke relation. We wish to use Corollary 2.4 to define a H q m)-action on V m. Define the map σ m : H q m) EndV m ) by σ m : T i R i,i+1. Lemma 2.5. The map σ m defines a representation of H q m) on V m. Proof. The braid relation follows from Corollary 2.4c) and the commutativity of non-adjacent reflections from the definition of σ m. The Hecke relation follows from a direct check on the eigenvalues of the triangular matrix R V V from 1) Obtaining Schur-Weyl duality. We have analogues of Theorems 1.3 and 1.5 for V m. Theorem 2.6. If q is not a root of unity, we have: a) the images of U q sl n ) and H q m) in EndV m ) are commutants of each other; b) as a H q m) U q sl n )-module, we have the decomposition V m = S λ L λ, λ m lλ) n where S λ and L λ are quantum deformations of the classical representations of S m and Usl n ). Proof. We explain a proof for n > m, though the result holds in general. For a), we use a dimension count from the non-quantum case. By the definition of σ m in terms of R-matrices, each algebra lies inside the commutant of the other. We now claim that σ m H q m)) spans End Uqsl n)v m ). If q is not a root of unity, the decomposition of V m into U q sl n )-isotypic components is the same as in the classical case, meaning that its commutant has the same dimension as in the classical case. Similarly, H q m) is isomorphic to C[S m ]; because σ m is faithful, this means that σ m H q m)) has the same dimension as the classical case, and thus σ m H q m)) is the entire commutant of U q sl n ). Finally, because U q sl n ) is semisimple and V m is finite-dimensional, U q sl n ) is isomorphic to its double commutant, which is the commutant of H q m). For b), V m decomposes into such a sum by a), so it suffices to identify the multiplicity space of L λ with S λ ; this holds because it does under the specialization q 1. Corollary 2.7. For n > m, the functor FS q : RepH q m)) RepU q sl n )) defined by FS q W ) = Hom Hqm)W, V m ) with U q sl n )-module structure induced from V m is an equivalence of categories between RepH q m)) and the subcategory of weight m representations of U q sl n ). Proof. From semisimplicity and the explicit decomposition of V m provided by Theorem 2.6b).

4 4 YI SUN 3. Yangians and degenerate affine Hecke algebras 3.1. Yang-Baxter equation with spectral parameter and Yangian. We extend the results of the previous section to the analogue of U q sl n ) given by the solution to the Yang-Baxter equation with spectral parameter. This object is known as the Yangian Y sl n ), and it will be Schur-Weyl dual to the degenerate affine Hecke algebra Λ m. We first introduce the Yang-Baxter equation with spectral parameter 2) R 12 u v)r 13 u)r 23 v) = R 23 v)r 13 u)r 12 u v). We may check that 2) has a solution in EndC n C n ) given by Ru) = 1 P u. This solution allows us to define the Yangian Y gl n ) via the RTT formalism. Definition 3.1. The Yangian Y gl n ) is the Hopf algebra with generators t k) ij 3) R 12 u v)t 1 u)t 2 v) = t 2 v)t 1 u)r 12 u v), and defining relation where tu) = i,j t iju) E ij Y sl n ) EndC n ), t ij u) = δ ij u 1 + k 1 tk) ij u k 1 Y sl n )[[u 1 ]], the superscripts denote action in a tensor coordinate, and the relation should be interpreted in Y sl n )v 1 ))[[u 1 ]] EndC n ) EndC n ). The coalgebra structure is given by n t ij u)) = t ia u) t aj u) and the antipode by Stu)) = tu) 1. Remark. There is an embedding of Hopf algebras Ugl n ) Y gl n ) given by t ij t 0) ij. Remark. Relation 3) is equivalent to the relations 4) [t r) ij, ts 1) kl for 1 i, j, k, l n and r, s 1 where t 1 ij a=1 ] [t r 1) ij, t s) kl ] = tr 1) kj 5) [t 0) aa, t s 1) ] = δ ka t s 1) kl t s 1) il t s 1) kj t r 1) il = δ ij ). For r = 0 and i = j = a, this implies that al δ al t s 1) ka, meaning that t k) ij and t 0) ij map between the same Ugl n )-weight spaces. Remark. For any a, the map ev a : Y gl n ) Ugl n ) given by ev a : t ij u) 1 + E ij u a is an algebra homomorphism but not a Hopf algebra homomorphism. Pulling back Ugl n )-representations through this map gives the evaluation representations of Y gl n ) The Yangian of sl n. For any formal power series fu) = 1 + f 1 u 1 + f 2 u 2 + C[[u 1 ]], the map tu) fu)tu) defines an automorphism µ f of Y gl n ). One can check that the elements of Y gl n ) fixed under µ f form a Hopf subalgebra. Definition 3.2. The Yangian Y sl n ) of sl n is Y sl n ) = {x Y gl n ) µ f x) = x}. We may realize Y sl n ) as a quotient of Y gl n ). Define the quantum determinant of Y gl n ) by 6) qdet tu) = σ S n 1) σ t σ1),1 u)t σ2),2 u 1) t σn),n u n + 1) Proposition 3.3. We have the following: a) the coefficients of qdet tu) generate ZY gl n )); b) Y gl n ) admits the tensor decomposition ZY gl n )) Y sl n ); c) Y sl n ) = Y gl n )/qdet tu) 1).

5 SCHUR-WEYL DUALITY FOR QUANTUM GROUPS 5 Observe that any representation of Y gl n ) pulls back to a representation of Y sl n ) under the embedding Y sl n ) Y gl n ). Further, the image of Usl n ) under the previous embedding Ugl n ) Y gl n ) lies in Y sl n ), so we may consider any Y sl n )-representation as a Usl n )-representation. We say that a representation of Y sl n ) is of weight m if it is of weight m as a representation of Usl n ) Degenerate affine Hecke algebra. The Yangian will be Schur-Weyl dual to the degenerate affine Hecke algebra Λ m, which may be viewed as a q 1 limit of the affine Hecke algebra. Definition 3.4. The degenerate affine Hecke algebra Λ m is the associative algebra given by Λ m = s 1,..., s m 1, x 1,..., x m s 2 i = 1, s i s i+1 s i = s i+1 s i s i+1, [x i, x j ] = 0, s i x i x i+1 s i = 1, [s i, s j ] = [s i, x j ] = 0 if i j 1. Remark. We have the following facts about Λ m : s i and x i generate copies of C[S m ] and C[x 1,..., x m ] inside Λ m ; the center of Λ m is C[x 1,..., x m ] Sm ; the elements y i = x i j<i s ij in Λ m give an alternate presentation via Λ = s 1,..., s m 1, y 1,..., y m sy i = y si) s, [y i, y j ] = y i y j )s ij The Drinfeld functor. We now upgrade FS to a functor between RepΛ m ) and RepY sl n )). For a Λ m -representation W, define the linear map ρ W : Y gl n ) EndFSW )) by where ρ W : tu) T 1, u x 1 )T 2, u x 2 ) T m, u x m ), T u x l ) = E ab E ab EndW V V ) u x l ab should be thought of as the image of the evaluation map ev a : Y gl n ) Ugl n ) given by t ij u) 1 + Eij u a at a = x l. Proposition 3.5. The map ρ W gives a representation of Y gl n ) on FSW ). Proof. Define S = ab E ab E ab. We first check the image of ρ W lies in Hom Sm W, V m ). For any f : W V m, we must check that ρ W f)s i w) = P i,i+1 ρ W f)w). Because all coefficients of l u x l) are central in Λ m, it suffices to check this for ρ W : tu) l u x l )ρ W tu)) = l u x l + S l, ). Notice that u x j + S j, ) commutes with the action of s i and P i,i+1 unless j = i, i + 1, so it suffices to check that u x i + S i, )u x i+1 + S i+1, )fs i w) = P i,i+1 u x i + S i, )u x i+1 + S i+1, )fw). We compute the first term as u x i + S i, )u x i+1 + S i+1, )fs i w) Now notice that and = u + S i, )u + S i+1, )fs i w) u + S i+1, )fx i s i w) u + S i, )fx i+1 s i w) + fx i x i+1 s i w). u + S i, )u + S i+1, )fs i w) = u + S i, )u + S i+1, )P i,i+1 fw) = P i,i+1 u + S i, )u + S i+1, )fw) + P i,i+1 [S i+1,, S i, ]fw) u + S i+1, )fx i s i w) = u + S i+1, )fs i x i+1 + 1)w) = P i,i+1 u + S i, )fx i+1 w) u + S i+1, )fw)

6 6 YI SUN and and Putting these together, we find that u + S i, )fx i+1 s i w) = u + S i, )fs i x i 1)w) = P i,i+1 u + S i+1, )fx i w) + u + S i, )fw) fx i x i+1 s i w) = P i,i+1 fx i x i+1 w). u x i + S i, )u x i+1 + S i+1, )fs i w) = P i,i+1 u x i + S i, )u x i+1 + S i+1, )fw) + P i,i+1 [S i+1,, S i, ] + S i, S i+1, ) fw). We may check in coordinates that [S i,, S i+1, ] = [P i,i+1, S i, ] so that P i,i+1 [S i+1,, S i, ] = P i,i+1 S i, P i,i+1 S i, = S i+1, S i,, which yields the desired. To check that ρ W is a valid Y gl n )-representation, we note that the x l form a commutative subalgebra of Λ m, hence the same proof that ev a is a valid map of algebras shows that ρ W is a representation, since the action of the x i commutes with the action of Ugl n ). Lemma 3.6. We may reformulate the action of Y gl n ) on EndFSW )) via the equality m 1 ρ W tu)) = 1 + S l, u y l In particular, in terms of the generators y l, we have ρ W t k) ij ) = δ ij + Proof. We claim by induction on k that k T l, u x l ) = 1 + m yl k Eji. l k 1 u y l S l,. The base case k = 1 is trivial. For the inductive step, noting that S l, S k+1, = P l,k+1 S k+1,, we have ) k ) ) 1 k 1 + S l, 1 + Sk+1, 1 = 1 + S l, 1 k S l, S k+1, u y l u x k+1 u y l u x k+1 u y l ) k 1 = 1 + S l, 1 k P l,k+1 u y l u x k+1 u y l ) k 1 = 1 + S l, 1 k P l,k+1 1 u y l u x k+1 u y k+1 k+1 1 = 1 + S l,. u y l S k+1, S k+1, 3.5. Schur-Weyl duality for Yangians. The upgraded functor FS is known as the Drinfeld functor, and an analogue of Theorem 1.5 holds for it. Theorem 3.7. For n > m, the functor FS : RepΛ m ) RepY sl n )) is an equivalence of categories onto the subcategory of RepY sl n )) generated by representations of weight m. Proof. We first show essential surjectivity. Viewing any representation W of Y sl n ) of weight m as a representation of Usl n ), we have by Theorem 1.5 that W = FSW ) for some S m -representation W. We must now extend the S m -action to an action of Λ m by defining the action of the y l. For this, we use that W is also a representation of Y gl n ) via the quotient map Y gl n ) Y sl n ). Lemma 3.8. We have the following:

7 SCHUR-WEYL DUALITY FOR QUANTUM GROUPS 7 a) if v V m is a vector with non-zero component in each isotypic component of V m viewed as a Usl n )-representation, the linear map W FSW ) given by w v w is injective, where w W is the image of w under the canonical isomorphism W W ; b) if e 1,..., e n is the standard basis for V, then v = e i1 e im V m is such a vector for i 1,..., i m distinct. Proof. Theorem 1.3 and reduction to isotypic components of W gives a), and b) follows because v is a cyclic vector for Usl n ) in V m. Define the special vectors v j) = e 2 e j e n e j+1 e m and w j) = e 2 e j e 1 e j+1 e m. For w W, the action of 1n on vj) w lies in w j) W by Usl n )-weight considerations via 5). By Lemma 3.8, we may define linear maps α j End C W ) by 1n vj) w ) = w j) α j w). Similarly, we may define maps β j, γ j End C W ) so that and 11 wj) w ) = w j) β j w) t 2) 1n vj) w ) = w j) γ j w). Evaluate the relation [ 1n, t0) 11 ] [t0) 1n, t1) 11 ] = 0 on vj) w to find that α j w) β j w) = 0. Now, combining the relations [t 2) 1n, t0) 11 ] = t2) 1n and [t2) 1n, t0) 11 ] [t1) 1n, t1) 11 ] = t1) 1n t0) 11 t0) 1n t1) 11, we find that t 2) 1n [t1) 1n, t1) 11 ] = t1) 1n t0) 11 t0) 1n t1) 11. Evaluating this on v j) w implies that γ j w) + αj 2 w) = 0. Lemma 3.9. The formulas for the action of the following Yangian elements hold on all of FSW ). Proof. For 1n 1n = l α l E l) 1n, t1) 11 = l, because t0) ij commutes with 1n α l E l) 11, t2) 1n = l α 2 l E l) 1n for i, j / {1, n}, it suffices by Lemma 3.8b) to verify the claim on basis vectors v V containing e 2,..., e n 1 at most once as tensor factors. In fact, for each configuration of e 1 s and e n s which occur, it suffices to verify the claim for a single such basis vector. Similar claims hold for 11 and basis vectors containing e 2,..., e n at most once. Call basis vectors containing r copies of e 1 and s of copies of e n vectors of type r, s). The claim holds for 11 for 0, ) trivially and for 1, ) because it holds for wj). Now, we have [ 11, t0) 1n ] =, so this implies that the claim holds for t1) 1n for 0, ). Now, observe that [t1) 1n, t0) 12 ] = 0, so replacing any v of type r, s) which does not contain e 2 with v which has e 2 instead of e 1 in a single tensor coordinate yields 1n 1n v = t1) 1n t0) 12 v = t 0) 12 t1) 1n v, whence the claim holds for 1n on v if it holds for v. Induction on r yields the claim for all 1n. Now, for t1) 11, suppose the claim holds for type r 1, 0), and choose a v of type r, 0) with e 1 in coordinates i 1,..., i r, and let v be the vector containing e n instead of e 1 in the single tensor coordinate i r. Then we have v = t 0) 1n v, so 11 v = t1) 11 t0) 1n v = t 0) 1n t1) 11 v + [t 0) 11, r 1 r 1 t1) 1n ]v = t 0) 1n α ij E ij) 11 v + α ir v = α ij + α ir )v, which yields the claim for 11 j=1 by induction on r. The claim for t2) 1n j=1 follows from the relation t 2) 1n = t0) 1n t1) 11 t1) 1n t0) 11 [t1) 1n, t1) 11 ].

8 8 YI SUN To conclude, we claim that the assignment y l α l extends the S m -action on FSW ) to a Λ m -action. For this, we evaluate relations from Y sl n ) on carefully chosen vectors in FSW ). To check that s i y i = y i+1 s i, note that v i) w = v i+1) s i w), so acting by 1n on both sides gives the desired For the second relation, we evaluate on s i w i+1) α i w) = w i) α i w) = w i+1) α i+1 s i w)). t 2) 1n [t1) 1n, t1) 11 ] = t1) 1n t0) 11 t0) 1n t1) 11 e 2 e i e n e i+1 e j 1 e 1 e j e m w = e 2 e i e 1 e i+1 e j 1 e n e j e m s ij w), we find that α j α i )s ij w = α i α j w)) α j α i w)), which shows that [α i, α j ] = α i α j )s ij. It remains to show that FS is fully faithful. Injectivity on morphisms follows because FS is fully faithful in the classical case. For surjectivity, any map F : FSW ) FSW ) of Y sl n )-modules is of the form F = FSf) for a map f : W W of S m -modules. Further, viewing W and W as Y gl n )-modules via the quotient map, F commutes with the full Y gl n )-action because the center acts trivially on both W and W. Now, because F commutes with the action of 1n, we see for all w W and v V m that m E l) 1n v fy lw) = m E l) 1n v y lfw)). Taking v = w j) shows that fy j w) = y j fw), so that f is a map of Λ m -modules, as needed. 4. Quantum affine algebras and affine Hecke algebras 4.1. Definition of the objects. Our goal in this section will be to extend Corollary 2.7 to the case of U q ŝl n) and H q m). We first define these objects. Definition 4.1. The quantum affine algebra U q ŝl n) is the quantum group of the Kac-Moody algebra associated to type A 1) n 1, meaning that the Cartan matrix A is given by A = Remark. The obvious embedding x ± i x ± i and q hi/2 q hi/2 realizes U q sl n ) as a Hopf subalgebra of U q ŝl n). We say that a U q ŝl n)-representation is of weight m if it is of weight m as a U q sl n )-representation. Definition 4.2. The affine Hecke algebra H q m) is the associative algebra given by H q m) = T 1 ±,..., T m 1 ±, X± 1,..., X± m [X i, X j ] = 0, T i q 1 )T i + q) = 0, T i T i+1 T i = T i+1 T i T i+1, T i X i T i = q 2 X i+1, [T i, T j ] = [T i, X j ] = 0 for i j 1.

9 SCHUR-WEYL DUALITY FOR QUANTUM GROUPS Drinfeld functor and Schur-Weyl duality. We now give an extension of Corollary 2.7 to the affine setting. The strategy is the analogue of the one we took for Yangians. For variety, we present a construction directly in the Kac-Moody presentation in this case. For a H q m)-representation W, define the linear map ρ q,w : U q ŝl n) EndFS q W )) by ρ q,w x ± 0 ) = m X ± l ρ q,w q h0 ) = 1 q h θ ) m, q h θ/2 ) l 1 x θ q h θ/2 ) m l, and where x + θ = E 1n and x θ = E n1 as operators in EndV ), and q h θ = q h1+ +hn 1. Theorem 4.3. The map ρ q,w defines a representation of U q ŝl n) on FS q W ). Proof. By a direct computation of the relations of U q ŝl n). For details, the reader may consult [CP96, Theorem 4.2]; note that the coproduct used there differs from our convention, which follows [Jim86]. Theorem 4.4. For n > m, the functor FS q : RepH q m)) RepU q ŝl n)) is an equivalence of categories onto the subcategory of RepU q ŝl n)) generated by representations of weight m. Proof. The proof of essential surjectivity is analogous to that of Theorem 3.7. The action of X ± i is obtained by evaluation on some special basis vectors in V m and the relations of H q m) are shown to be satisfied for them from the relations of U q ŝl n). For details, see [CP96, Sections ]. The check that FS q is fully faithful is again essentially the same as in Theorem 3.7. References [Ara99] Tomoyuki Arakawa. Drinfeld functor and finite-dimensional representations of Yangian. Comm. Math. Phys., 2051):1 18, [BGHP93] D. Bernard, M. Gaudin, F. D. M. Haldane, and V. Pasquier. Yang-Baxter equation in long-range interacting systems. J. Phys. A, 2620): , [CP96] Vyjayanthi Chari and Andrew Pressley. Quantum affine algebras and affine Hecke algebras. Pacific J. Math., 1742): , [Dri86] V. G. Drinfel d. Degenerate affine Hecke algebras and Yangians. Funktsional. Anal. i Prilozhen., 201):69 70, [Jim86] Michio Jimbo. A q-analogue of UglN + 1)), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys., 113): , [Mol07] Alexander Molev. Yangians and classical Lie algebras, volume 143 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, address: yisun@math.mit.edu