Braid Group Action on Affine Yangian

We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $\hat{\mathfrak{gl}}_N$.

We can define the Yangian associated with a symmetrizable Kac-Moody Lie algebra g. Then we see that the braid group of g acts on it. In this paper we mainly focus on the affine Yangian Y (ŝl N ) of type A and the general case is studied in Appendix. The affine Yangian Y (ŝl N ) is a two-parameter deformation of the universal enveloping algebra of the universal central extension of sl N [s, t ±1 ], and is related to many interesting objects; symmetry of the spin Calogero-Sutherland model [U], [K1]; Schur-Weyl type duality for degenerate double affine Hecke algebra [G1]; quiver variety associated with the cyclic quiver [V], [K1].
The affine Yangian Y (ŝl N ) contains U (ŝl N ) as a subalgebra and hence it admits an action of the braid group associated withŝl N . We give a formula for the action on generators of degree one. We use it to show the compatibility of the braid group action and the coproduct on Y (ŝl N ) introduced by Guay [G2], Guay-Nakajima-Wendlandt [GNW]. This compatibility result holds for the Yangian of arbitrary finite or affine type except for A (1) 1 and A (2) 2 as discussed in Appendix. See Proposition 3.12 and A.4 for precise statements. We hope to apply the braid group action to the study of structure theory and representation theory of the affine Yangian. As a first step, we consider the following in the second half of the paper.
Guay [G2] introduced an evaluation map for the affine Yangian Y (ŝl N ) whose target space is a certain completion of U (ĝl N ). By its definition, the image contains U (ŝl N ), and we have expected that it contains U (ĝl N ). We give an affirmative answer to this question, under a certain assumption on the parameters, by constructing elements of Y (ŝl N ) whose images by the evaluation map coincide with generators of the diagonal Heisenberg algebra insideĝl N (Theorem 4.18). To construct such elements, we use the braid group action. Certainly it is desirable to lift the Heisenberg subalgebra inside the affine Yangian and we will come back to this problem in a future.
Our main result Theorem 4.18 implies that the pull-back of an irreducibleĝl N -module by the evaluation map is irreducible as a module of Y (ŝl N ). We determine the highest weights of the evaluation modules in [K2].
The plan of this paper is as follows. In Section 2, we define the affine Yangian Y (ŝl N ) and recall some automorphisms and the coproduct. In Section 3, we introduce the braid group action and study its properties. In particular, the compatibility with the coproduct is proved in 3.4. We construct elements of Y (ŝl N ) whose images by the evaluation map coincide with Heisenberg generators in Section 4. In Appendix, we consider the braid group action on the Yangian associated with a symmetrizable Kac-Moody Lie algebra. Then we give a proof of the compatibility with the coproduct when it is known to be well-defined.

Acknowledgments
The author would like to thank Yoshihisa Saito for suggesting him to study braid group action on affine Yangian. Discussions with him improved contents of this paper. He also thanks Nicolas Guay for telling him the reference [DK].
This work was supported by JSPS KAKENHI Grant Number 26287004, 17H06127, 18K13390, and The Kyoto University Foundation.

Affine Yangian
Fix an integer N ≥ 3 throughout the paper. We use the notation {x, y} = xy + yx.
Definition 2.1. The affine Yangian Y (ŝl N ) is the algebra over C generated by x + i,r , x − i,r , h i,r (i ∈ Z/N Z, r ∈ Z ≥0 ) with parameters ε 1 , ε 2 ∈ C subject to the relations: The subalgebra generated by x + i,0 , x − i,0 , h i,0 (i ∈ Z/N Z) is isomorphic to U (ŝl N ) (see [GNW,Section 2]). We write x ± i = x ± i,0 , h i = h i,0 and identify them with the standard Chevalley generators ofŝl N . Let {α i } i∈Z/N Z be the simple roots ofŝl N . The null root δ is given by δ = N −1 i=0 α i . Let θ = N −1 i=1 α i be the highest root of sl N and h −θ the coroot corresponding to −θ. We denote by∆,∆ + , and∆ re + the set of roots, positive roots, and positive real roots forŝl N , respectively. We need to considerŝl N ⊕ Cd with the degree operator to deal with the coproduct on Y (ŝl N ). Fix a nondegenerate invariant symmetric bilinear form ( , ) onŝl N ⊕ Cd such that (x + i , x − i ) = 1 and denote the induced bilinear form on the dual of the Cartan subalgebra by the same letter.
LetŴ be the Weyl group ofŝl N generated by the simple reflections s i (i ∈ Z/N Z). We denote by s θ the reflection corresponding to the highest root θ. For each α ∈∆, we assign the translation element t α inŴ . For example, we have s 0 s θ = t θ . The action of t α on the root lattice We can deduce the following identities directly from the defining relations of Y (ŝl N ).

Automorphisms
We introduce algebra (anti-)automorphisms of Y (ŝl N ) which will be used later.
Let ω be the algebra anti-automorphism of Y (ŝl N ) defined by It is easy to see that the assignment preserves the defining relations of Y (ŝl N ). The following algebra automorphism corresponds to the rotation of the Dynkin diagram.
The assignment gives an algebra automorphism ρ of Y (ŝl N ).
In particular we have ρ( Remark 2.4. The relation between generators X ± i,r , H i,r with parameters λ, β used in [G1, G2] and ours are as follows:

Coproduct
A formula for the coproduct on the affine Yangian Y (ŝl N ) is stated in [G2]. Guay-Nakajima-Wendlandt [GNW] give a detailed proof of the well-definedness. To recall it, we consider a bigger algebra Y (ŝl N ⊕ Cd), which is generated by x ± i,r , h i,r , d with modified defining relations given in [GNW,(2.8)]. Moreover we need certain completions Y (ŝl N ⊕ Cd) ⊗Y (ŝl N ⊕ Cd) and Y (ŝl N ) ⊗Y (ŝl N ) of the tensor products since the coproduct involves infinite sums. See [GNW,Section 5] for the precise definition of the completion.
We define the half Casimir operator Ω + as follows. Let {u k } be a C-basis of the Cartan subalgebra ofŝl N ⊕ Cd and {u k } its dual basis with respect to the nondegenerate bilinear form ( , ). Let {x Here mult α denotes the dimension of the root space corresponding to α. We take simple root vectors as x (1) Define a C-linear operator on Y (ŝl N ) by (X) = X ⊗ 1 + 1 ⊗ X. Note that is not an algebra homomorphism, but satisfies ([X, Y ]) = [ (X), (Y )].

Braid group action
We define automorphisms T i (i ∈ Z/N Z) of the affine Yangian and study their properties.

Definition
Since the adjoint actions of x ± i on Y (ŝl N ) are locally nilpotent, the operators exp ad x ± i are well defined by These are algebra automorphisms of Y (ŝl N ) as ad x ± i are derivations. We define an algebra automorphism T i of Y (ŝl N ) for each i ∈ Z/N Z by This operator appears in [GNW] and is used to construct real root vectors of Yangians.

Braid relations
A proof of the following proposition is exactly the same as one for the fact that {T i } satisfy the braid relations as automorphisms of U (ŝl N ). We give a proof for the sake of completeness.
Proposition 3.1. The operators {T i } satisfy the braid relations. That is, we have For each w ∈Ŵ with a reduced expression w = s i 1 · · · s i l , we can define an algebra automorphism T w of Y (ŝl N ) by T w = T i 1 · · · T i l thanks to the braid relations.
Let us start the proof with some preparations. A proof of the following formulas are straightforward.
Lemma 3.2. Assume a ij = −1. Then (i) exp ad x + i sends: (ii) exp ad(−x − i ) sends: The following two propositions are well known. Proposition 3.3 follows from Lemma 3.2. Then Proposition 3.4 follows from Proposition 3.3. Proposition 3.3. We have Proposition 3.4. Assume a ij = −1. Then we have Proof of Proposition 3.1. We use that ϕ • exp ad X • ϕ −1 = exp ad ϕ(X) holds for any algebra automorphism ϕ. If a ij = 0, then we have by the formulas in Proposition 3.3. If a ij = −1, then we have by the formulas in Proposition 3.4. The proof is complete.
We give an alternative definition of T i .
by the formulas in Proposition 3.3. Hence the assertion is proved.
Let M be a Y (ŝl N )-module and assume that x ± i act on M locally nilpotently. Then an automorphism T M i of M is defined similarly. The following property is immediate.
Proposition 3.7. Let M be a Y (ŝl N )-module and assume that x ± i act on M locally nilpotently. Then for any X ∈ Y (ŝl N ) and m ∈ M , holds.
In fact, this is a general property for any associative algebra such that it contains U (ŝl N ) as a subalgebra and the operators exp ad x ± i are well-defined.

Action on generators
We compute the action of T i on x ± j,1 ,h j,1 . We will use the following formulas. Lemma 3.8. Assume a ij = −1. Then Proof. By a direct computation using the defining relations of Y (ŝl N ) with Lemma 2.2. For example, by Proposition 3.9. We have Proof. The formulas for a ij = 0 trivially hold. Let us consider the other cases.
Then we have Next assume a ij = −1. Then we have The formulas for T i (x + j,1 ) are obtained by applying ω to these results. We compute T i (h j,1 ). First assume i = j. Then we have Next assume a ij = −1. Then we have Since we have We can easily obtain the formulas for T i (h j,1 ) from these identities.
Proposition 3.10. Assume a ij = −1. Then we have Proof. We show the first identity. Since we have The second identity is obtained by applying ω to the first one. The third identity follows from We give formulas for T 2 i . Proofs are straightforward. Proposition 3.11. We have T 2

Compatibility with the coproduct
The goal of this subsection is to prove the following proposition.
Proof of Proposition 3.12. We use First assume i = j. We claim that the both sides coincide with The left-hand side is hence the claim holds. Next assume a ij = 0. Then the left-hand side is ∆(x + j,1 ). The right-hand side is hence the claim holds.
Finally assume a ij = −1. We claim that the both sides coincide with The left-hand side is Then we see the claim by The right-hand side is hence the claim holds.
4 Heisenberg generators 4.1 Affine Lie algebraĝl N Let gl N be the complex general linear Lie algebra consisting of N ×N matrices. We denote by E i,j the matrix unit with (i, j)-th entry 1, and set 1 = N i=1 E i,i . The indices i, j of E i,j are regarded as elements of Z/N Z. The transpose of an element X of gl N is denoted by t X.
Letĝl N = gl N ⊗ C[t, t −1 ] ⊕ Cc be the affine Lie algebra whose Lie bracket is given by We denote the element X ⊗ t s by X(s). We identify the generators as usual: We define automorphisms analogous to ω and ρ forĝl N . Let ω be the algebra antiautomorphism of U (ĝl N ) defined by ω(X(s)) = t X(−s) and ω(c) = c. The assignment gives an algebra automorphism ρ of U (ĝl N ).
Proof. We can show the identities for i = j inductively from those for the Chevalley generators. Then we can show The other cases are similarly proved. The identity (4.1) will be used later. Let us consider the case i = j. The case s = 0 is proved as follows. Note that the identity 1 = N −1 i=1 ih i + N E N,N holds. Applying ρ to the both sides, we obtain The right-hand side is equal to The case s = 0 is similarly proved by considering and (4.1).
We similarly define an algebra automorphism Proof. Obvious from the definition of T i and the Lie bracket ofĝl N .
Lemma 4.3. We have otherwise.
(i = 0) Proof. We show the case i = 0. By Lemma 4.2, it is enough to consider the case s = 0.
Apply T i to the identity 1 = N −1 j=1 jh j + N E N,N . Then the left-hand side is T i (1) = 1 and the right-hand side is This shows Now let i = 1. We can inductively show that E N −1,N −1 , E N −2,N −2 , . . . , E 3,3 are invariant under T 1 and T 1 (E 2,2 ) = T 1 (h 2 + E 3,3 ) = h 2 + h 1 + E 3,3 = E 1,1 , Thus we have shown the assertion for i = 1. Similarly we can show the other cases. The case i = 0 is obtained from the case i = 1 by applying ρ and using Lemma 4.1.
Proof. First we prove the assertion for 0 ≤ k ≤ N − 2 by induction on k. The case k = 0 is trivial. Assume that it holds for k, then Next we prove the assertion for k = N − 1. We have The proof is complete.
Lemma 4.5. Let i ≤ j. We have Proof. The assertion is easily proved by induction.
Lemma 4.6. We have Proof. The identity (4.5) is deduced from (4.4) by applying ω. We use (4.2) to show the other identities as:

Evaluation map
The evaluation map for the affine Yangian Y (ŝl N ) is introduced by Guay [G2]. Let U (ĝl N ) comp,− be the completion of U (ĝl N ) defined in [K2,Definition 2.5]. From now on, we regard the central element c ofĝl N as a complex number.
Theorem 4.7. Assume c = N ε 2 . Then there exists an algebra homomorphism ev : Y (ŝl N ) → U (ĝl N ) comp,− uniquely determined by We will use the following property.
Proposition 4.8. We have Proof. The assertion (i) is immediate from the definition of ev. The assertion (ii) is stated in [G2, Section 6, p. 463] and a proof is given in [K2,Proposition 3.6]. Since ev is the identity on the subalgebra U (ŝl N ) and T i is defined via the generators of U (ŝl N ), the assertion (iii) holds.

Construction of Heisenberg generators
We construct elements a m (m ∈ Z) of the affine Yangian Y (ŝl N ) such that ev(a m ) = 1(m) under the assumption ε 2 = 0. First consider the case m = 0.
Proposition 4.9. We have ev Proof. Put By the assumption c = N ε 2 , we have The assertion holds since we have Assume c = N ε 2 and ε 2 = 0. Put Then we have ev(a 0 ) = 1 by Proposition 4.9.
Next consider the case m ≥ 1. For each i ∈ Z/N Z and a fixed m, define an element w(i, m) of the affine Weyl groupŴ by This element has the property w(i, m)(α i−1 ) = −α i + mδ. Hence the elements have weight mδ. We will see in Lemma 4.13, and will compute the value of in Proposition 4.15. Then we will take the summation over i in Proposition 4.16. The result will yield a construction of the elements a m in Theorem 4.18. By Lemma 3.6 (ii), ρ • T w(i,m) = T w(i−1,m) • ρ holds. The case i = 1 will be important. Note that w(1, 1) = s 2 s 3 · · · s N −1 and t −α 2 = s 2 s 3 · · · s 0 s 1 s 0 · · · s 3 . Lemma 4.10. We have Proof. The assertion is easily proved by Lemma 4.3.
Proof. We prove the assertion for i = 1. The other cases are deduced from this case by applying ρ and Lemma 4.1. First assume m = 1. Recall T w(1,1) = T 2 T 3 · · · T N −1 . We have hence the case m = 1 is proved. Next consider the case m ≥ 2. Since the case m = 2 yields the equality for general m ≥ 2 inductively, it is enough to prove (4.8) We have (2). Here the second equality follows from the braid relations and Proposition 3.4, and the last from Lemma 4.4. Then the right-hand side of (4.8) is T 2 T 3 · · · T N −1 (−E N,1 (2)) = −(T 2 T 3 · · · T N −1 (E N,1 ))(2) by Lemma 4.2 = −E 2,1 (2) by (4.4), hence the assertion is proved.
Lemma 4.14. We have Proof. We prove: and (4.10) The equalities (4.9) for k = 1 and (4.10) for k = N follow from Lemma 4.11. Consider (4.9) for k = N and (4.10) for k = 1. Note that Lemma 4.12 for i = 1 is nothing but (4.9) for k = N and s = 0. We can prove the other cases by applying [−, E 1,1 (±s)] to this case as [T w(1,m) Here we use the fact that E 1,1 (±s) is invariant under T w(1,m) proved in Lemma 4.11. In the sequel, we prove (4.9) and (4.10) for 2 ≤ k ≤ N − 1.
Thus we have proved the case m = 1. Next we consider the case m ≥ 2. Since the case m = 2 yields the equality for general m ≥ 2 inductively, it is enough to prove: If we prove the assertion for s = 0, we can prove the other cases by applying [−, E 1,1 (s)] to this case by using the fact that E 1,1 (s) is invariant under T t −α 2 proved in Lemma 4.10. We prove (4.11) for s = 0. Since we have E 3,1 = −T 2 (x − 1 ), the left-hand side of (4.11) for We prove (4.12) for s = 0 by backward induction on k. The case k = N − 1 is proved as Assume that the assertion holds for k. Since we have E k,1 = T k (E k+1,1 ) by (4.4), Here the second equality holds by k ≥ 3. Thus we have proved (4.9). We prove (4.10). First we consider the case m = 1. We prove T w(1,1) (E N,k (−s)) = −E N,k+1 (−s) for 2 ≤ k ≤ N − 1 by backward induction on k. By Lemma 4.2, it is enough to prove T w(1,1) (E N,k ) = −E 2,k+1 . When k = N − 1, we have by (4.2). Assume that the assertion holds for k. Since we have E N,k−1 = T k−1 (E N,k ) by (4.5), = −T k (E 2,k+1 ) = −E 2,k by (4.5).
Thus we have proved the case m = 1. Next we consider the case m ≥ 2. Since the case m = 2 yields the equality for general m ≥ 2 inductively, it is enough to prove: T t −α 2 (E 2,3 (−s)) = (−1) N E 2,3 (−s + 2), (4.13) (4.14) If we prove the assertion for s = 0, we can prove the other cases by applying [E 2,2 (−s), −] to this case by using the fact that E 2,2 (−s) is invariant under T t −α 2 for s ≥ 1 proved in Lemma 4.10. We prove (4.13) for s = 0. Since we have
Hence the assertion holds for i = 0. Then apply ρ to (4.15) for i = 0. Similarly the left-hand side is , and for the right-hand side, we can see Note that c never appears in the last equality as −s + m − 1 cannot be 0 for 0 ≤ s ≤ m − 2. Hence the assertion holds for i = N − 1. Continuing this process we prove the assertions for i = N − 2, N − 3, . . . , 2 since we have Proposition 4.16. We have Remark 4.17. The point of the above statement is as follows: although each term ev([x + i , T w(i,m) (x + i−1,1 )]) lies in the completion of U (ĝl N ), we obtain R m an element of U (ŝl N ) as a remainder term after cancellation.
Proof. We use the notation in Proposition 4.15. We have In the second equality we use the condition c = N ε 2 . We compute N k=1 (P i+1,k − Q i,k ) as follows: Hence the assertion holds by Applying ω to (4.17), we obtain Now the following theorem has been proved.
Theorem 4.18. Assume c = N ε 2 and ε 2 = 0. Let R m (m ≥ 1) be the element of U (ŝl N ) ⊂ Y (ŝl N ) as in Proposition 4.16, and define for m ∈ Z, Then we have ev(a m ) = 1(m). In particular, the image of the evaluation map ev contains U (ĝl N ).
Corollary 4.19. Assume c = N ε 2 and ε 2 = 0. Then the pull-back of an irreduciblê gl N -module by the evaluation map ev is irreducible as a module of Y (ŝl N ).
Remark 4.20. If ε 2 = 0, we cannot apply our construction. In fact, if we assume c = 0, the condition c = N ε 2 and ε 2 = 0 implies ε 1 = 0. The affine Yangian at ε 1 = ε 2 = 0 is isomorphic to the universal enveloping algebra of the universal central extension of the Lie algebra sl N [s, t ±1 ]. Moreover the evaluation map becomes the genuine evaluation at s = 0. In this situation, the image of the evaluation map is U (ŝl N ), and hence it does not contain the Heisenberg algebra generated by 1(m) (m ∈ Z).

A General case
We use the notation (ad X) (n) (Y ) = (ad X) n (Y )/n! for divided power operators.

A.1 Yangian and braid group action
Let (a ij ) i,j∈I be a symmetrizable generalized Cartan matrix and fix integers (d i ) i∈I such that (d i a ij ) i,j∈I is symmetric. We denote by g the corresponding Kac-Moody Lie algebra. Then the Yangian Y (g) is defined to be generated by x + i,r , x − i,r , h i,r (i ∈ I, r ∈ Z ≥0 ) with a parameter ∈ C subject to the relations: .
Then the standard Chevalley generators of g are identified with d Following [GNW], we define In the sequel, we put e i = d Proposition A.1. The operators {T i } satisfy the braid relations. That is, we have Proof. The identities follow from as in the proof of Proposition 3.1.
Proposition A.2. We have Proof. The formulas for T i (x ± j ), T i (h j ) are deduced from well-known formulas for the Chevalley generators. We produce a computation of T i (x − j ) for a ij < 0 since the case T i (x − j,1 ) for a ij < 0 is verified in a very similar way. Put m = −a ij . We have Since we have and It is easy to see that this is equal to The formula for T i (x − i,1 ) is proved in a way similar to Proposition 3.9. We use: As we mentioned, T i (x − j,1 ) for a ij < 0 is computed by replacing x − j with x − j,1 in the argument for T i (x − j ). Then the formulas for T i (x + j,1 ) are obtained from those for T i (x − j,1 ) by applying ω.
Remark A.3. In this appendix we impose no further condition on (a ij ) i,j∈I , and hence on g, to study the braid group action. However the defining relations of Y (g) given here may not produce a correct definition of Yangian for some generalized Cartan matrix. For example, it is known that for Y (ŝl 2 ) the defining relations should be modified as in [K1,Definition 5.1] or [BT,1.2] (In [K1], some relations are missing). In [GRW,(2.1)], the authors suggest the following condition: for all i and j, if a ij ≤ −2 then a ji = −1 holds.

A.2 Compatibility with the coproduct
Assume that g is a Kac-Moody Lie algebra of finite or affine type except for A (1) 1 and A (2) 2 . Then the coproduct ∆ on Y (g) is well-defined by the same formula as in Theorem 2.5 [GNW,Definition 4.7,Theorem 4.11,Proposition 5.17]. Let us prove the main result of this appendix.
Proof. Lemma 3.13 holds in a general situation and we use it. It is enough to prove ∆T i (x + j,1 ) = (T i ⊗ T i )∆(x + j,1 ) for a ij < 0 since the proofs concerning the other generators are the same as in the proof of Proposition 3.12. We claim that the both sides coincide with [(ad (e i )) (−a ij −n) (1 ⊗ x + j ), (ad (e i )) (n) (Ω + )] [1 ⊗ (ad e i ) (−a ij −n) (x + j ), (ad (e i )) (n) (Ω + )]   . and