Introduction to Quantum Q-Langlands Correspondence

Introduction to Quantum q-Langlands Correspondence

Ming Zhang∗ May 16–19, 2017

Abstract Quantum q-Langlands Correspondence was introduced by Aganagic, Frenkel and Okounkov in their recent work [AFO17]. Roughly speaking, it is a correspondence between q-deformed conformal blocks of the quantum aﬃne algebra and the deformed W-algebra. In this mini-course, I will deﬁne these objects and discuss some results in quantum K-theory of the Nakajima quiver varieties which is the key ingredient in the proof of the correspondence (in the simply-laced case). It will be helpful to know the basic concepts in the theory of Lie algebras (roots, weights, etc.).

Contents

1 Introduction 2 1.1 The geometric Langlands correspondence...... 2 1.2 A representation-theoretic statement and its generalization in [AFO17]...... 2

2 Lie algebras 2 2.1 Root systems...... 3 2.2 The Langlands dual...... 4

3 Aﬃne Kac–Moody algebras4 3.1 Via generators and relations...... 5 3.2 Via extensions...... 6

4 Representations of affine Kac–Moody algebras7 4.1 The extended algebra...... 7 4.2 Representations of finite-dimensional Lie algebras...... 8 4.2.1 Weights...... 8 4.2.2 Verma modules...... 8 4.3 Representations of affine Kac–Moody algebras...... 9 4.3.1 The category of representations...... 9 4.3.2 Verma modulesO...... 10 4.3.3 The evaluation representation...... 11 4.3.4 Level...... 11

∗Notes were taken by Takumi Murayama, who is responsible for any and all errors. Please e-mail [email protected] with any more corrections. Compiled on May 21, 2017. 1 1 Introduction

We will follow Aganagic, Frenkel, and Okounkov’s paper [AFO17], which came out in January of this year. Throughout, we work over the complex numbers C, although some things still work over an algebraically closed ﬁeld F of characteristic zero.

1.1 The geometric Langlands correspondence The geometric Langlands correspondence is the function ﬁeld analogue of the (number-theoretic) Langlands correspondence, conjectured by Beilinson and Drinﬁeld in the 1980s [BD]. The statement is as follows:

Conjecture 1.1 (Geometric Langlands correspondence [BD]). Let G be a complex reductive group, and let Σ be a projective algebraic curve (or a Riemann surface). Let LG denote the Langlands dual of G. Then, there is an equivalence of categories ∼ -Mod LocL (Σ) D-Mod Bun (Σ) , (1) O G −→ G L where -Mod LocLG(Σ) is the category of quasi-coherent sheaves on the G-moduli stack of local systems O on Σ, and D-Mod BunG(Σ) is the category of D-modules on the moduli stack of G-principal bundles on Σ. Example 1.2. Under this (proposed) correspondence, skyscraper sheaves correspond to Hecke eigensheaves. In [BD], Beilinson and Drinfeld show that there are subcategories on both sides of (1) that correspond to each other. On the other hand, the general statement of the conjecture is false: it fails in the case where G = SL(2) and Σ = P1 [Laf09]. However, there is a reﬁned conjecture due to [AG15] involving -categories that is still expected to hold. ∞

1.2 A representation-theoretic statement and its generalization in [AFO17] A key step in the proof of the correspondence in [BD] is reducing to the following representation-theoretic statement:

Key Step 1.3 [FF92]. There is an isomorphism between the center of the aﬃne Kac–Moody algebra Lcg at L L ∨ the critical level k = h and the classical -algebra ∞(g). − W W The goal of [AFO17] is to generalize this statement to consider deformations away from the critical level L for cg, which corresponds to a quantum deformation of ∞(g). More precisely, W Lcg is replaced by the quantum aﬃne algebra U (Lcg); • ~ β(g) is replaced by the deformed -algebra q,t(g). The•W simply-laced case (i.e., the ADE case)W of this generalizationW is proved in [AFO17].

2 Lie algebras

We start with some basic material on Lie algebras.

Deﬁnition 2.1. A Lie algebra is a vector space g over an algebraically closed ﬁeld F of characteristic zero, together with a bilinear operator [ , ] satisfying the following: [ax + by, z] = a[x, z] + b[y, z]− and− [z, ax + by] = a[z, x] + b[z, y], where a, b F and x, y, z g; • [x, x] = 0 for x g; ∈ ∈ • (Jacobi identity)∈ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. • The building blocks for any Lie algebra are the simple Lie algebras:

Deﬁnition 2.2. Let g be a Lie algebra. An ideal of g is a subspace i g such that [i, g] i. The Lie algebra g is simple if it has no non-trivial ideals, and is not abelian. ⊂ ⊂

There is a very famous classification of simple Lie algebras over F : they are classified by Dynkin diagrams. 2 Theorem 2.3. Finite-dimensional simple Lie algebras fall into four families An,Bn,Cn,Dn with five excep- tions E6,E7,E8,F4,G2, corresponding to the Dynkin diagrams below:

A n ···

B n ···

C n ···

D n ···

The simply-laced case corresponds to the Lie algebras with Dynkin diagrams without double-edges, i.e., the ADE case. These diagrams encode the data of a root system: nodes correspond to roots, and edges correspond to the angles between those roots.

2.1 Root systems We now define root systems and describe how they give rise to their corresponding Dynkin diagram. Definition 2.4. Fix a finite-dimensional R-vector space V with the standard Euclidean metric ( , ). A root system Φ is a finite collection of nonzero vectors called roots, such that − − (1) The roots span V ; (2) If x Φ and kx Φ for k R, then k = 1; (3) For any∈ two roots∈ x, y Φ,∈ the pairing ± ∈ 2(y, x) y, z := h i (x, x) is integer-valued; (4) For any two roots x, y Φ, ∈ 2(y, x) σ (y) = y x Φ. x − (x, x) ∈ Example 2.5. Let g be a Lie algebra, and let h denote the Cartan subalgebra (this correpsonds to the maximal torus under the correspondence between subgroups and subalgebras). Then, h is a nilpotent subalgebra (i.e., [h, [h,..., [h, h]]] = 0 for some finite number of iterations of the Lie bracket) such that if y g satisfies [x, y] h for all x h, then y h. Given x g, we can consider the adjoint action ∈ ∈ ∈ ∈ ∈ adx(y) = [x, y]

for y g. Consider the adjoint action h g. Then, g decomposes into eigenspaces for this action: ∈ M g = h g , ⊕ α α∈h∗ where gα = x g [t, x] = α(t)x for all t h . ∈ ∈ ∗ ∗ This collection α h gα = 0 gives a root system in h . ∈ 6 3 Definition 2.6. The positive roots Φ+ is a set of roots such that if α Φ, then either α Φ+ or α Φ+; • if α,∈ β Φ+ and α + β ∈ Φ, then−α +∈ β Φ+. A choice• of Weyl∈ chamber gives∈ a set of positive∈ roots. The subset ∆ Φ+ of simple roots consists of the positive roots that cannot be written as a sum of two other positive roots.⊂ Positive roots are non-negative combinations of simple roots. If α, β ∆, then the angle ∈ θα,β is defined by requiring (α, β) cos(θ ) = . α,β α β | || | Definition 2.7. Suppose that a root system Φ is irreducible, that is, it cannot be written as a union Φ Φ 1 ∪ 2 where (α, β) = 0 for all α Φ1, β Φ2. Let ∆ be a set of simple roots of Φ. Then, the Dynkin diagram for Φ has nodes corresponding to∈ the simple∈ roots ∆, and edges given by the angle between two roots: If θ = 2π/3, then α and β are connected by an undirected simple edge; • α,β If θα,β = 3π/4, then α and β are connected by a directed double edge; • If θ = 5π/6, then α and β are connected by a directed triple edge; • α,β If θα,β = π/2, then α and β are not connected by an edge. The• direction is given by pointing to the root with smaller Euclidean norm. It is a fact that a given irreducible root system has only two possible lengths.

2.2 The Langlands dual We now deﬁne the Langlands dual groups.

Deﬁnition 2.8. Let (Φ,V ) be a root system. If α Φ, we deﬁne the coroot ∈ 1 α∨ = α. (α, α)

This gives rise to another root system (Φ∨,V ), called the dual root or coroot system.

∗ ∨ Now let G be a complex connected reductive group. This gives the lattice data (X , ∆,X∗, ∆ ), where X∗ the lattice of characters of G (homomorphisms G C∗); • ∆ the set of simple roots; → • ∗ X∗ the lattice of one-parameter subgroups (homomorphisms C G); • ∆∨ the set of simple coroots. → • Deﬁnition 2.9. The Langlands dual LG of G is another complex connected reductive group, with lattice ∨ ∗ data (X∗, ∆ ,X , ∆).

∗ ∨ Remark 2.10. The lattice data (X , ∆,X∗, ∆ ) is more data than just the root system: GL(n) and SL(n) are both of type An, but the lattice for GL(n) is larger.

Example 2.11. Under this dualizing procedure, the families An,Dn,En are sent to themselves, while Cn and Dn get switched. For example,

GL(n) GL(n) A A ←→ n ←→ n SO(2n) SO(2n) D D ←→ n ←→ n SL(n) PGL(n) A A ←→ n ←→ n SO(2n + 1) Sp(n) B C ←→ n ←→ n 3 Aﬃne Kac–Moody algebras

We now discuss the basic deﬁnitions of aﬃne Kac–Moody algebras, following [Her06].

4 3.1 Via generators and relations

Recall that given a semisimple Lie algebra g, we can define a set ∆ = ei of simple roots, whose corresponding coroots are given by { } ∨ 2ei ei = . (ei, ej) Here, ( , ) is the standard Euclidean pairing on g when thought of as a vector space. − − Definition 3.1. The Cartan matrix for ∆ is the matrix ∨ 2(ei, ej) (cij) = (ei, ej ) = . (ei, ei) By the axioms for root systems (Definition 2.4), we deduce that the Cartan matrix satisfies the following: (1) cii = 2; (2) cij 0 for all i = j; (3) c ≤= 0 c 6 = 0; ij ⇐⇒ ji (4)( cij) = DS for a diagonal matrix D and a symmetric matrix S; (5) det (c ) ≤ ≤ > 0 for 1 R n 1, and det(C) = 0. ij 1 i,j R ≤ ≤ − If an arbitrary integer-valued matrix (cij) satisfies properties (1)–(5), it is called a generalized Cartan matrix. Given a generalized Cartan matrix, one can define a Kac–Moody algebra by generators and relations, where the relations are given by the Cartan matrix. Definition 3.2 (Affine Kac–Moody algebra via generators and relations). Let the following data be given: (1) A generalized Cartan matrix (cij) of size n n of rank r; (2)A C-vector space h of dimension 2n r; × − ∨ ∨ (3) A set of n linearly independent elements αi of h and a set n of linearly independent elements αi of h ∨ such that αi(αj ) = cij. Then, the associated affine Kac–Moody algebra g is the C-algebra generated by h and elements ei, fi for 1 i n, subject to the Chevalley–Serre relations: ≤ ≤[h, h0] = 0 for h, h0 h; • [h, e ] = α (h) e for∈h h; • i i i ∈ [h, fi] = αi(h) fi for h h; • [e , f ] =−δ α∨; ∈ • i j ij i (Serre relations) If i = j, then (ad )1−cij (e ) and (ad )1−cij (f ) = 0. • 6 ei j fi j For later use, we write hi = αi, in which case we can describe g as the algebra generated by ei, fi, hi subject to the Chevalley–Serre relations [hi, hj] = 0 for all i, j; • [h , e ] = α (h ) e = c e ; • i j j i j ji j [hi, fj] = αj(hi) fj = cjifj; • [e , f ] = δ− h ; − • i j ij i (Serre relations) If i = j, then (ad )1−cij (e ) and (ad )1−cij (f ) = 0. • 6 ei j fi j In the classical setting where h is a Cartan subalgebra and (cij) is a Cartan matrix, this gives a Lie algebra. More precisely, in the decomposition (cij) = DS of (4), If S is positive definite, then g is a simple Lie algebra (which are classified by Dynkin diagrams); • If S is positive semi-definite, then g is an affine Lie algebra (which are classified by generalized Dynkin • diagrams); If S is indefinite, then g is an affine Kac–Moody algebra of indefinite type (which are not easily • classified).

Example 3.3. For sl2, we have

0 1 0 0 1 0 e = , f = , h = , c = (2). 0 0 1 0 0 1 −

5 3.2 Via extensions Another construction of an affine Kac–Moody algebra starts with a semi-simple (finite-dimensional) Lie algebra g. We can then construct bg, which is a central extension of g. Definition 3.4 (Affine Kac–Moody algebra via extensions). The loop algebra determined by g is the algebra

g C[t, t−1], ⊗ where C[t, t−1] is the ring of Laurent series. The aﬃne Kac–Moody algebra determined by g is an essential extension of the loop algebra, deﬁned by

g := g C[t, t−1] C c, b ⊗ ⊕ · where c is called the central element. The Lie bracket for both the the aﬃne Kac–Moody algebra is given by

[a tn + α c, b tm + β c] = [a, b]tm+n + (a, b) n δ c. ⊗ · ⊗ · m+n,0 Here, a, b g and α, β C, and ( , ) is the normalized Killing form ∈ ∈ − − Tr ada adb κ(a, b) (a, b) = ◦ = h∨ h∨ where h∨ is the dual Coxeter number of g. We have a new Cartan subalgebra

bh := h 1 C c g. ⊗ ⊕ · ⊂ b The algebra bg is an example of a nontwisted affine Lie algebra. We want to connect this to the previous description of an affine Kac–Moody algebra in Definition 3.2. Consider the decomposition M g = h g ⊕ α α∈∆ by eigenspaces for the adjoint representation of h on g, where gα = x g [h, x] = α(h) x for all h h , ∈ ∈ ∗ ∆ = α h r 0 gα = 0 ∈ { } 6 { }

Then, letting e g , f g− , and h such that α (h ) = c , we have the following: i ∈ αi j ∈ αj j i j ij Facts 3.5. There exists a unique θ ∆, called the longest root, such that θ + αi / ∆ 0 for all i; • There is an involution ω∈called the Killing reflection, defined by ∈ ∪ { } • ω(e ) = f , ω(f ) = e , ω( h ) = h ; i − i i − i − i − i There exists a pairing ( , ): h∗ h∗ C defined by • − − × →

(αi, αj) = cij/εi,

where εi are positive integers such that   ε1  ..  B =  .  C. εn

6 Given this data, we can write down a presentation for the affine Lie algebra g. For each 1 i n, let b ≤ ≤ E = e 1,F = f 1,H = h 1. i i ⊗ i i ⊗ i i ⊗ This will recover a finite-dimensional algebra inside the affine Kac–Moody algebra bg. We then need to define E ,F ,H . First, let f g such that 0 0 0 0 ∈ θ 2h∨ f , ω(f ) = . 0 0 −(θ, θ)

Then, e = ω(f ) g− . We then deﬁne 0 − 0 ∈ θ E = e t, F = f t−1,H = [E ,F ], 0 0 ⊗ 0 0 ⊗ 0 0 0 which are elements of bg Theorem 3.6. g is generated by E ,F ,H for 0 i n, and these generators satisfy the Chevalley–Serre b i i i ≤ ≤ relations as in Deﬁnition 3.2 with (Cij) a generalized Cartan matrix.

Example 3.7. Let g = sl2 and consider bg = slc2. Then, 0 1 0 0 1 0 e = , f = , h = , 0 0 1 0 0 1 − and E = e 1,F = f 1,H = h 1. 1 ⊗ 1 ⊗ 1 ⊗ Moreover, E = f t, F = e t−1,H = 2c H . 0 ⊗ 0 ⊗ 0 − 1 From this explicit formula and the description of the Lie bracket, we have that [H1,E0] = C10E0 = 2E0, for example. The generalized Cartan matrix is −

2 2 . 2− 2 − 2 The Ei,Fi,Hi generate slc2 since [E0[E0,E1]], for example, has a t term, and in this way all Laurent series can be generated. H0 gives the c terms.

4 Representations of aﬃne Kac–Moody algebras

We now want to study representations of the inﬁnite-dimensional algebra bg.

4.1 The extended algebra

To study the representations of bg, we note that bg does not have a Killing form, that is, there is no way to define a nondegenerate form on bg. To fix this, we introduce the following: Definition 4.1. The extended algebra is g := g C d, e b ⊕ · where d is called the derivation element, and the Lie bracket is given by dP (t) [d, P (t) x] = t x and [d, c] = 0, ⊗ dt ⊗ for all P (t) C[t, t−1] and x g. For example, [d, tm x] = mtm x. There is an extended Cartan ∈ ∈ ⊗ ⊗ subalgebra eh = bh C d. ⊕ · This extended algebra does have a Killing form. 7 4.2 Representations of finite-dimensional Lie algebras Consider a finite-dimensional semi-simple (complex) Lie algebra g, together with its weight space decomposition M g = h g . ⊕ λ λ

g then has a presentation with generators e , f , h for 1 i n. { i} { i} { i} ≤ ≤ 4.2.1 Weights Definition 4.2. A weight λ is an element of h∗ such that h(v) = λ(h) v for some v V and for all elements ∈ h h. The fundamental weights are the elements of the dual basis corresponding to hi h. The dominant weights∈ are nonnegative linear combinations of fundamental weights. There is a partial{ } ⊂ order on the set of weights: There is a partial order on the set of dominant weights, given by X λ µ λ = µ + m α , m Z≥ , ≤ ⇐⇒ i i i ∈ 0 where the αi are the simple roots. The highest weight modules are those generated by a weight vector v such that e (v) = 0 for all i j n. The weight of v is the highest weight. i ≤ ≤ Note that while the weight of a highest weight v is maximal with respect to , the converse is not necessarily true. ≤ Theorem 4.3. Every finite-dimensional irreducible representation of a finite-dimensional complex semi-simple Lie algebra g is a highest weight module.

4.2.2 Verma modules Now let + ∗ X = λ h λ(hi) Z≥0 . ∈ ∈ Then, there is a one-to-one correspondence

ﬁnite-dimensional 1−1 + irreducible representations of g iso X V unique highest weight of V

Note that while there is a unique highest weight, the corresponding weight space may be larger than one-dimensional. To go in the other direction, we will introduce Verma modules. Consider a homomorphism g0 g of finite-dimensional Lie algebras. Then, a representation V of g gives g 0 → 0 a representation Resg0 V of g . There is an adjoint to this functor: a representation M of g gives an induced g representation Indg0 M, which is g Ind 0 M := (g) 0 M, g U ⊗U(g ) where ( ) denotes the universal enveloping algebra, defined by U − C g g g g⊗n (g) = T (g) = ⊕ ⊕ ⊗ ⊕ · · · ⊕ ⊕ · · ·. U ab ba [a, b]( a, b g) − − ∀ ∈ The homomorphism g0 g gives a homomorphism (g0) (g), and so the tensor product in the definition g → 0 U → U of Indg0 M makes sense: (g) has a right- (g )-action. Now take U U M g0 = b = h L . ⊕ α α∈R+

8 to be the Borel subalgebra. This algebra is a solvable Lie algebra, that is, [[[b, b], [b, b]], ] = 0. We then have another one-to-one correspondence ···

ﬁnite-dimensional 1−1 h∗ simple representations of b iso

Cλ λ The representations on the left-hand side are one-dimensional by a theorem of Cartan. Definition 4.4. Choose an arbitrary weight λ h∗. Then, the Verma module M(λ) is the induced representation ∈ M(λ) := Indg C = (g) C . b λ U ⊗U(b) λ A theorem of Poincaré–Birkhoff–Wittsays that the map (b) (g) is an injection. U → U Fact 4.5. M(λ) is a highest weight module of highest weight λ, where the highest weight vector is vλ = 1 1. M(λ) is not simple, but still is indecomposable (Poincaré–Birkhoff–Witt). ⊗ Proposition 4.6. Let g be a complex semi-simple Lie algebra. Then, (1) For all λ h∗, the Verma module M(λ) has a unique maximal submodule N(λ). (2) The quotient∈ L(λ) := M(λ)/N(λ) is a simple module, and the assignment λ L(λ) gives a one-to-one correspondence 7→ simple highest-weight 1−1 h∗ g-modules iso ⊃ ⊃ finite-dimensional 1−1 dominant weights simple representations of g iso

This gives a nice classiﬁcation for representations of ﬁnite-dimensional Lie algebras g.

4.3 Representations of aﬃne Kac–Moody algebras

We now turn to the inﬁnite-dimensional case, that is, representations of eg. Note that the reason to consider eg over bg is that the simple roots of bg may not be linearly independent (the generalized Cartan matrix is singular).

4.3.1 The category of representations O We want to consider infinite-dimensional representations, but this class is too large. To define the category O of representations of eg, we first define the following: Definition 4.7. A representation V of eg is eh-diagonalizable if there exists a weight decomposition M V = Vλ, h∈bh∗ where ∗ Vλ = v V hv = λ(h) v for all h bh and wt(V ) = λ eh Vλ = 0 . ∈ ∈ ∈ 6 { } There is a partial order on wt(V ), defined as ≤ n X λ µ λ + m α for some m Z≥ . ≤ ⇐⇒ i i i ∈ 0 i=0

For λ eh∗, we also define ∈ ∗ D(λ) := µ eh µ λ . ∈ ≤ 9 Definition 4.8. The category is the category of g-modules V , such that O e (1) V is eh-diagonalizable; (2) Vλ is finite-dimensional; ∗ S (3) There exists a finite number of weights λ1, . . . , λs eh such that wt(V ) D(λi). ∈ ⊂ 1≤i≤s This will end up being the right category of representations to consider for an affine Kac–Moody algebra.

4.3.2 Verma modules We can now deﬁne Verma modules, which will be the simple elements in the category . Recall that for the Lie algebra g, we have the triangular decomposition O g = n+ h+ n−. ⊕ ⊕ This gives an induced decomposition g =n ˆ+ bh nˆ−, b ⊕ ⊕ where n+ = t C[t] (n− h) C[t] n+ b · ⊗ ⊕ ⊕ ⊗ n− = t−1 C[t−1] (n+ h) C[t] n− b · ⊗ ⊕ ⊕ ⊗ are the positive and negative nilpotent parts of bg, respectively. Similarly, g = n+ eh n− e b ⊕ ⊕ b where eh = 1 g C c C d. ⊗ ⊕ · ⊕ · + Also, eb = nb eg. We give two⊕ equivalent deﬁnitions of Verma modules:

Definition 4.9. Given an affine Kac–Moody algebra bg and its associated generalized Cartan matrix (Cij), we can define the enveloping algebra (bg) of bg by having generators Ei,Fi,Hi, and relations as in Definition 3.2 (interpreting the bracket as [a, b] =Uab ba). Similarly, we can define the enveloping algebra (g) for g. Then, − U e e given λ eh∗, we define a left ideal ∈ X J(λ) = (g)n ˆ+ + (g)h λ(h) (g). U e U e − ⊂ U e h∈eh The quotient M(λ) = (g)/J(λ) U e has a natural structure of a left (g)-module, and is called a Verma module. U e Definition 4.10. For each λ eh∗, we define ∈ eg M(λ) = Ind Cλ = (g) Cλ. eb U e ⊗U(eb) Proposition 4.11. M(λ) is an object of the category . O Proposition 4.12. There exists a unique maximal proper submodule N(λ) M(λ). The modules ⊂ L(λ) := M(λ)/N(λ)

for λ eh∗ are the simple modules in the category . ∈ O Fact 4.13. L(λ) is not finite-dimensional in general (in the classical case, it is finite-dimensional if and only if λ is a dominant weight), but it is a integrable representation in the category . O Here, integrability is a sort of finiteness condition. This theory is almost parallel to that for finite- dimensional Lie algebra. For applications, one often needs to restrict further to this subcategory of consisting of integrable representations. O 10 4.3.3 The evaluation representation Another important representation, especially for mathematical physics, is the evaluation representation.

Definition 4.14. Fix a C∗, and choose a finite-dimensional representation V of g. Then, we define the ∈ action of bg on V as follows: P (t) x(v) = P (a) x(v), c(v) = 0 ⊗ for all x g,P (t) C[t, t−1]. This gives a g-structure on the representation V . ∈ ∈ b Remark 4.15. In general, this bg-action cannot be lifted to eg.

4.3.4 Level Another important concept for representations is the level. In the ﬁnite-dimensional case, Schur’s lemma implies the following:

Fact 4.16. If L is a finite-dimensional simple representation of g, and c is a central element ([c, g] = 0 for all g g), then c acts on L as a scalar. ∈ Proof. Consider the kernel of c λ Id, and apply Schur’s lemma. − · This holds also for simple representations in the category . O Definition 4.17. If V is in the category , then the level of V is k C if c acts on V as k Id. O ∈ · This gives a grading on the category : for example, k consists of the representations of level k. You can define a product structure O O

0 0 Ok × Ok −→ Ok+k using what is called the fusion product.

References

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