Generators of affine W-algebras

Transcription

1 1 Generators of affine W-algebras Alexander Molev University of Sydney

2 2 The W-algebras first appeared as certain symmetry algebras in conformal field theory.

3 2 The W-algebras first appeared as certain symmetry algebras in conformal field theory. The W-algebra associated with sl 3 was constructed by A. Zamolodchikov, The construction relied on the seminal work of A. Belavin, A. Polyakov and A. Zamolodchikov, 1984.

4 2 The W-algebras first appeared as certain symmetry algebras in conformal field theory. The W-algebra associated with sl 3 was constructed by A. Zamolodchikov, The construction relied on the seminal work of A. Belavin, A. Polyakov and A. Zamolodchikov, The W-algebra W k (g) associated with any simple Lie algebra g was constructed by B. Feigin and E. Frenkel, 1990, via the quantum Drinfeld Sokolov reduction.

5 3 More recently, the W-algebras W k (g, f ) were introduced by V. Kac, S.-S. Roan and M. Wakimoto, Here f g is a nilpotent element. W k (g, f ) = W k (g) for the principal nilpotent f.

6 3 More recently, the W-algebras W k (g, f ) were introduced by V. Kac, S.-S. Roan and M. Wakimoto, Here f g is a nilpotent element. W k (g, f ) = W k (g) for the principal nilpotent f. The classical W-algebra W(g) is obtained as a limit of W k (g) when k. The algebra W(g) is commutative.

7 3 More recently, the W-algebras W k (g, f ) were introduced by V. Kac, S.-S. Roan and M. Wakimoto, Here f g is a nilpotent element. W k (g, f ) = W k (g) for the principal nilpotent f. The classical W-algebra W(g) is obtained as a limit of W k (g) when k. The algebra W(g) is commutative. It plays the role of the Weyl group invariants in the affine Harish-Chandra isomorphism z(ĝ) = W( L g) [Feigin Frenkel, 1992].

8 4 Moreover, the Feigin Frenkel duality provides an isomorphism W k (g) = W k ( L g) if (k + h )(k + L h )r = 1.

9 4 Moreover, the Feigin Frenkel duality provides an isomorphism W k (g) = W k ( L g) if (k + h )(k + L h )r = 1. In particular, z(ĝ) = W h (g) = W( L g).

10 4 Moreover, the Feigin Frenkel duality provides an isomorphism W k (g) = W k ( L g) if (k + h )(k + L h )r = 1. In particular, z(ĝ) = W h (g) = W( L g). Recent work: representation theory of W-algebras [T. Arakawa]; classical W-algebras and integrable Hamiltonian hierarchies [A. De Sole, V. Kac, D. Valeri].

11 Connection with finite W-algebras: 5

12 5 Connection with finite W-algebras: affine W-algebra W k (g, f ) W(g, f ) finite W-algebra Zhu algebra

13 5 Connection with finite W-algebras: affine W-algebra W k (g, f ) W(g, f ) finite W-algebra Zhu algebra [T. Arakawa, 2007, A. De Sole, V. Kac, 2006]

14 6 Connection with finite W-algebras: affine W-algebra W k (g, f ) W(g, f ) finite W-algebra Zhu algebra universal affine vertex algebra V k (g) f = 0 U(g) universal enveloping algebra [T. Arakawa, 2007, A. De Sole, V. Kac, 2006]

15 Vacuum modules 7

16 7 Vacuum modules Let b be a finite-dimensional Lie algebra with an invariant symmetric bilinear form,.

17 7 Vacuum modules Let b be a finite-dimensional Lie algebra with an invariant symmetric bilinear form,. Its Kac Moody affinization is the centrally extended Lie algebra b = b[t, t 1 ] C 1 with

18 7 Vacuum modules Let b be a finite-dimensional Lie algebra with an invariant symmetric bilinear form,. Its Kac Moody affinization is the centrally extended Lie algebra b = b[t, t 1 ] C 1 with [ X[r], Y[s] ] = [X, Y][r + s] + r δr, s X, Y 1, where X[r] = X t r for any X b and r Z.

19 8 The vacuum module V(b) over b is defined by V(b) = U( b) U(b[t] C 1) C, where C is regarded as the one-dimensional representation of b[t] C1 on which b[t] acts trivially and 1 acts as 1.

20 8 The vacuum module V(b) over b is defined by V(b) = U( b) U(b[t] C 1) C, where C is regarded as the one-dimensional representation of b[t] C1 on which b[t] acts trivially and 1 acts as 1. By the PBW theorem, V(b) = U ( t 1 b[t 1 ] ) as a vector space.

21 8 The vacuum module V(b) over b is defined by V(b) = U( b) U(b[t] C 1) C, where C is regarded as the one-dimensional representation of b[t] C1 on which b[t] acts trivially and 1 acts as 1. By the PBW theorem, V(b) = U ( t 1 b[t 1 ] ) as a vector space. V(b) is a vertex algebra with the vacuum vector 1, the translation operator τ : V(b) V(b) which is the derivation τ = t of the enveloping algebra X[ r] rx[ r 1], and

22 9 the following state-field correspondence map Y : a a(z), where a V(b) and a(z) = n Z a (n) z n 1, a (n) : V(b) V(b).

23 9 the following state-field correspondence map Y : a a(z), where a V(b) and a(z) = n Z a (n) z n 1, a (n) : V(b) V(b). By definition, for any X b the map Y acts by X[ 1] X(z) = n Z X[n]z n 1,

24 9 the following state-field correspondence map Y : a a(z), where a V(b) and a(z) = n Z a (n) z n 1, a (n) : V(b) V(b). By definition, for any X b the map Y acts by X[ 1] X(z) = n Z X[n]z n 1, X[ r 1] z r X(z), r 0. r!

25 10 Furthermore, if a = X[ r 1] and b V(b) then ab : a(z)b(z) :,

26 10 Furthermore, if a = X[ r 1] and b V(b) then ab : a(z)b(z) :, where the normally ordered product is defined by : a(z)b(z) : = a(z) + b(z) + b(z) a(z)

27 10 Furthermore, if a = X[ r 1] and b V(b) then ab : a(z)b(z) :, where the normally ordered product is defined by : a(z)b(z) : = a(z) + b(z) + b(z) a(z) with a(z) + = a (n) z n 1, a(z) = a (n) z n 1. n<0 n 0

28 Vertex algebra axioms 11

29 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z,

30 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z, such that a (n) b = 0 for n 0.

31 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z, such that a (n) b = 0 for n 0. Furthermore, 1(z) = id V(b),

32 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z, such that a (n) b = 0 for n 0. Furthermore, 1(z) = id V(b), a(z)1 is a Taylor series in z and a(z)1 z=0 = a for a V(b),

33 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z, such that a (n) b = 0 for n 0. Furthermore, 1(z) = id V(b), a(z)1 is a Taylor series in z and a(z)1 z=0 = a for a V(b), τ 1 = 0 and [ τ, a(z) ] = z a(z) for a V(b),

34 11 Vertex algebra axioms The state-field correspondence map Y thus equips V(b) with a family of bilinear n-products: (a, b) a (n) b for n Z, such that a (n) b = 0 for n 0. Furthermore, 1(z) = id V(b), a(z)1 is a Taylor series in z and a(z)1 z=0 = a for a V(b), τ 1 = 0 and [ τ, a(z) ] = z a(z) for a V(b), for any a, b V(b) there exists N Z + such that (z w) N [a(z), b(w)] = 0.

35 12 We will regard V(b) as an algebra with respect to the ( 1)-product (a, b) a ( 1) b.

36 12 We will regard V(b) as an algebra with respect to the ( 1)-product (a, b) a ( 1) b. This product is quasi-associative: (a ( 1) b) ( 1) c = a ( 1) (b ( 1) c) + ( a ( j 2) b(j) c ) + ( b ( j 2) a(j) c ). j 0 j 0

37 We will regard V(b) as an algebra with respect to the ( 1)-product (a, b) a ( 1) b. This product is quasi-associative: (a ( 1) b) ( 1) c = a ( 1) (b ( 1) c) + ( a ( j 2) b(j) c ) + ( b ( j 2) a(j) c ). j 0 j 0 Note that if a = X[ r 1] and b V(b) then a ( 1) b = ab so we will omit the ( 1)-subscript in such cases. 12

38 W-algebra W k (g) for g = gl N 13

39 13 W-algebra W k (g) for g = gl N We use the definition of W k (g) via the BRST cohomology following E. Frenkel and D. Ben-Zvi, 2004.

40 13 W-algebra W k (g) for g = gl N We use the definition of W k (g) via the BRST cohomology following E. Frenkel and D. Ben-Zvi, Set b = span of {e ij i j}, m = span of {e ij i > j}.

41 13 W-algebra W k (g) for g = gl N We use the definition of W k (g) via the BRST cohomology following E. Frenkel and D. Ben-Zvi, Set b = span of {e ij i j}, m = span of {e ij i > j}. Given k C consider the affinization b of b with respect to the form: for i i and j j e i i, e j j = δ i i δ j j ( k + N ) ( δ i j 1 N ).

42 13 W-algebra W k (g) for g = gl N We use the definition of W k (g) via the BRST cohomology following E. Frenkel and D. Ben-Zvi, Set b = span of {e ij i j}, m = span of {e ij i > j}. Given k C consider the affinization b of b with respect to the form: for i i and j j ( ) ( e i i, e j j = δ i i δ j j k + N δ i j 1 ). N Let V k (b) be the corresponding vacuum module over b.

43 The W-algebra W k (g) is a vertex subalgebra of V k (b). 14

44 14 The W-algebra W k (g) is a vertex subalgebra of V k (b). To define it, introduce the Lie superalgebra â = â 0 â 1 with â 0 = b, â 1 = m[t, t 1 ], with the adjoint action of â 0 on â 1, whereas â 1 is regarded as a supercommutative Lie superalgebra.

45 14 The W-algebra W k (g) is a vertex subalgebra of V k (b). To define it, introduce the Lie superalgebra â = â 0 â 1 with â 0 = b, â 1 = m[t, t 1 ], with the adjoint action of â 0 on â 1, whereas â 1 is regarded as a supercommutative Lie superalgebra. We will write ψ j i [r] = e j i t r 1 for e j i t r 1 m[t, t 1 ] when it is considered as an element of â 1.

46 15 Let V k (a) be the vacuum module for â induced from the representation C of (b[t] C1) m[t] where b[t] â 0 and m[t] â 1 act trivially and 1 acts as 1.

47 15 Let V k (a) be the vacuum module for â induced from the representation C of (b[t] C1) m[t] where b[t] â 0 and m[t] â 1 act trivially and 1 acts as 1. Equip V k (a) with the ( 1)-product and introduce its derivation Q : V k (a) V k (a) determined by the following properties.

48 15 Let V k (a) be the vacuum module for â induced from the representation C of (b[t] C1) m[t] where b[t] â 0 and m[t] â 1 act trivially and 1 acts as 1. Equip V k (a) with the ( 1)-product and introduce its derivation Q : V k (a) V k (a) determined by the following properties. First, Q commutes with τ = t.

49 Furthermore, [Q, E] = (ΨE) op (E Ψ) op and [Q, Ψ] = Ψ 2 with 16

50 16 Furthermore, [Q, E] = (ΨE) op (E Ψ) op and [Q, Ψ] = Ψ 2 with ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] E = e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] for α = k + N 1 and

51 16 Furthermore, [Q, E] = (ΨE) op (E Ψ) op and [Q, Ψ] = Ψ 2 with ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] E = e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] for α = k + N 1 and ψ 21 [0] Ψ = ψ N 1 [0] ψ N 2 [0]... ψ N N 1 [0] 0

52 17 Explicitly, [Q, E] = (ΨE) op (E Ψ) op reads j 1 [ Q, ej i [ 1] ] = e a i [ 1] ψ j a [0] a=i j ψ a i [0] e j a [ 1] + α ψ j i [ 1] + ψ j+1 i [0] ψ j i 1 [0]. a=i+1

53 17 Explicitly, [Q, E] = (ΨE) op (E Ψ) op reads j 1 [ Q, ej i [ 1] ] = e a i [ 1] ψ j a [0] a=i+1 a=i j ψ a i [0] e j a [ 1] + α ψ j i [ 1] + ψ j+1 i [0] ψ j i 1 [0]. The derivation Q is the differential of the BRST complex of the quantum Drinfeld Sokolov reduction.

54 17 Explicitly, [Q, E] = (ΨE) op (E Ψ) op reads j 1 [ Q, ej i [ 1] ] = e a i [ 1] ψ j a [0] a=i+1 a=i j ψ a i [0] e j a [ 1] + α ψ j i [ 1] + ψ j+1 i [0] ψ j i 1 [0]. The derivation Q is the differential of the BRST complex of the quantum Drinfeld Sokolov reduction. The definition of the W-algebra can be stated in the form W k (g) = {v V k (b) Q v = 0}.

55 Generators of W k (g) 18

56 18 Generators of W k (g) Recall that ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] E = e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1]

57 18 Generators of W k (g) Recall that ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] E = e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] and write cdet E = (ατ) N + W (1) (ατ) N W (N), W (i) V k (b).

58 19 Explicitly, cdet E = ( ατ + e[ 1] ) k 1 k 0 +1 ( ατ + e[ 1] ) k 2 k ( ατ + e[ 1] ) k m k m 1 +1,

59 19 Explicitly, cdet E = ( ατ + e[ 1] ) k 1 k 0 +1 ( ατ + e[ 1] ) k 2 k ( ατ + e[ 1] ) k m k m 1 +1, summed over m = 1,..., N and 0 = k 0 < k 1 < < k m = N.

60 19 Explicitly, cdet E = ( ατ + e[ 1] ) k 1 k 0 +1 ( ατ + e[ 1] ) k 2 k ( ατ + e[ 1] ) k m k m 1 +1, summed over m = 1,..., N and 0 = k 0 < k 1 < < k m = N. Theorem [T. Arakawa A. M., 2014] All coefficients W (1),..., W (N) of cdet E belong to W k (g).

61 19 Explicitly, cdet E = ( ατ + e[ 1] ) k 1 k 0 +1 ( ατ + e[ 1] ) k 2 k ( ατ + e[ 1] ) k m k m 1 +1, summed over m = 1,..., N and 0 = k 0 < k 1 < < k m = N. Theorem [T. Arakawa A. M., 2014] All coefficients W (1),..., W (N) of cdet E belong to W k (g). Moreover, they generate the W-algebra W k (g) V k (b).

62 20 Take the reverse determinant of the matrix Ẽ obtained from E by replacing α with β = α = (k + N 1).

63 20 Take the reverse determinant of the matrix Ẽ obtained from E by replacing α with β = α = (k + N 1). We have rev-det Ẽ = ( β τ + e[ 1] ) l 0 l 1 +1 ( β τ + e[ 1] ) l 1 l ( β τ + e[ 1] ) l m 1 l m+1,

64 20 Take the reverse determinant of the matrix Ẽ obtained from E by replacing α with β = α = (k + N 1). We have rev-det Ẽ = ( β τ + e[ 1] ) l 0 l 1 +1 ( β τ + e[ 1] ) l 1 l ( β τ + e[ 1] ) l m 1 l m+1, summed over m = 1,..., N and N = l 0 > l 1 > > l m = 0.

65 20 Take the reverse determinant of the matrix Ẽ obtained from E by replacing α with β = α = (k + N 1). We have rev-det Ẽ = ( β τ + e[ 1] ) l 0 l 1 +1 ( β τ + e[ 1] ) l 1 l ( β τ + e[ 1] ) l m 1 l m+1, summed over m = 1,..., N and N = l 0 > l 1 > > l m = 0. The coefficients U (1),..., U (N) defined by rev-det Ẽ = (β τ)n + U (1) (β τ) N U (N), are generators of the W-algebra W k (g).

66 21 Example. W k (sl 2 ) = W k (gl 2 )/(W (1) = 0). W (1) = e 11 [ 1] + e 22 [ 1], W (2) = e 11 [ 1] e 22 [ 1] + (k + 1) e 22 [ 2] + e 21 [ 1].

67 21 Example. W k (sl 2 ) = W k (gl 2 )/(W (1) = 0). W (1) = e 11 [ 1] + e 22 [ 1], W (2) = e 11 [ 1] e 22 [ 1] + (k + 1) e 22 [ 2] + e 21 [ 1]. The coefficients L n of the series L(z) = Y(ω) given by L(z) = n Z L n z n 2, ω = W(2) k + 2, k 2, generate the Virasoro algebra.

68 Miura map 22

69 22 Miura map Introduce the abelian subalgebra l gl N by l = span of {e ii i = 1,..., N}.

70 22 Miura map Introduce the abelian subalgebra l gl N by l = span of {e ii i = 1,..., N}. For k C consider the affinization l of l with respect to the form e i i, e j j = ( k + N )( δ i j 1 N ).

71 22 Miura map Introduce the abelian subalgebra l gl N by l = span of {e ii i = 1,..., N}. For k C consider the affinization l of l with respect to the form e i i, e j j = ( k + N )( δ i j 1 N ). Let V k (l) be the corresponding vacuum module.

72 22 Miura map Introduce the abelian subalgebra l gl N by l = span of {e ii i = 1,..., N}. For k C consider the affinization l of l with respect to the form e i i, e j j = ( k + N )( δ i j 1 N ). Let V k (l) be the corresponding vacuum module. The projection b l induces the vertex algebra homomorphism V k (b) V k (l).

73 23 By restricting to the subalgebra W k (g) V k (b) we get the map Υ : W k (g) V k (l), called the Miura map (or Miura transformation).

74 23 By restricting to the subalgebra W k (g) V k (b) we get the map Υ : W k (g) V k (l), called the Miura map (or Miura transformation). This is an injective vertex algebra homomorphism [T. Arakawa].

75 23 By restricting to the subalgebra W k (g) V k (b) we get the map Υ : W k (g) V k (l), called the Miura map (or Miura transformation). This is an injective vertex algebra homomorphism [T. Arakawa]. For generic k we have [B. Feigin and E. Frenkel]: im Υ = N 1 i=1 ker V i, where V i are the screening operators acting on V k (l).

76 To define the V i, for i = 1,..., N 1 set ( ) ( b i [r] V i (z) = exp z r exp r r<0 r>0 b i [r] r z r ), 24

77 To define the V i, for i = 1,..., N 1 set ( ) ( b i [r] V i (z) = exp z r exp r r<0 r>0 b i [r] r z r ), where b i [r] = 1 ( ) e i i [r] e i+1 i+1 [r]. k + N 24

78 To define the V i, for i = 1,..., N 1 set ( ) ( b i [r] V i (z) = exp z r exp r r<0 r>0 b i [r] r z r ), where b i [r] = 1 ( ) e i i [r] e i+1 i+1 [r]. k + N For the screening operator we have V i = V (1) i, where V i (z) = n Z V (n) i z n. 24

79 25 Under the Miura map we have Υ : cdet E ( ατ + e 1 1 [ 1] )... ( ατ + e N N [ 1] ).

80 25 Under the Miura map we have Υ : cdet E ( ατ + e 1 1 [ 1] )... ( ατ + e N N [ 1] ). Expand the product as (ατ) N + w (1) (ατ) N w (N), w (i) V k (l).

81 25 Under the Miura map we have Υ : cdet E ( ατ + e 1 1 [ 1] )... ( ατ + e N N [ 1] ). Expand the product as (ατ) N + w (1) (ατ) N w (N), w (i) V k (l). Corollary [Fateev and Lukyanov, 1988]. The coefficients w (1),..., w (N) generate the W-algebra W k (g) V k (l).

82 Suppose that (k + N)(k + N) = 1. 26

83 26 Suppose that (k + N)(k + N) = 1. We have ( (k + N 1)τ + e1 1 [ 1] )... ( (k + N 1)τ + e N N [ 1] ) = ( k N) N( α τ + e 1 1[ 1] )... ( α τ + e N N[ 1] )

84 26 Suppose that (k + N)(k + N) = 1. We have ( (k + N 1)τ + e1 1 [ 1] )... ( (k + N 1)τ + e N N [ 1] ) = ( k N) N( α τ + e 1 1[ 1] )... ( α τ + e N N[ 1] ) for α = k + N 1 and e i i[ 1] = e i i[ 1] k + N.

85 26 Suppose that (k + N)(k + N) = 1. We have ( (k + N 1)τ + e1 1 [ 1] )... ( (k + N 1)τ + e N N [ 1] ) = ( k N) N( α τ + e 1 1[ 1] )... ( α τ + e N N[ 1] ) for α = k + N 1 and e i i[ 1] = e i i[ 1] k + N. Corollary [Feigin Frenkel duality]. W k (g) = W k (g).

86 The W-algebra W k (gl N, f ) 27

87 27 The W-algebra W k (gl N, f ) Fix a partition of N and depict it as the right justified pyramid

88 27 The W-algebra W k (gl N, f ) Fix a partition of N and depict it as the right justified pyramid Let p 1 p 2 p n and q 1 q 2 q l be the lengths of the rows be the lengths of the columns.

89 28 Number the bricks of a pyramid π by the rule

90 28 Number the bricks of a pyramid π by the rule Define the corresponding nilpotent element f gl N by f = e ji summed over all dominoes i j occurring in π:

91 28 Number the bricks of a pyramid π by the rule Define the corresponding nilpotent element f gl N by f = e ji summed over all dominoes i j occurring in π: f = e 21 + e 42 + e 64 + e 53 + e 75.

92 28 Number the bricks of a pyramid π by the rule Define the corresponding nilpotent element f gl N by f = e ji summed over all dominoes i j occurring in π: f = e 21 + e 42 + e 64 + e 53 + e 75. Introduce a grading on gl N by deg e ij = col(j) col(i).

93 29 We have gl N = i Z g i, f g 1.

94 29 We have gl N = i Z g i, f g 1. In particular, g 0 = glq1 gl q2 gl ql.

95 29 We have gl N = i Z g i, f g 1. In particular, g 0 = glq1 gl q2 gl ql. Set b = p 0 g p and m = p<0 g p.

96 29 We have gl N = i Z g i, f g 1. In particular, g 0 = glq1 gl q2 gl ql. Set b = p 0 g p and m = p<0 g p. Equip b with the symmetric invariant bilinear form X, Y = k + N 2N tr b(ad X ad Y) 1 2 tr g 0 (ad X ad Y).

97 30 Consider the affinization b of b with respect to this form and let V k (b) be the corresponding vacuum module over b.

98 30 Consider the affinization b of b with respect to this form and let V k (b) be the corresponding vacuum module over b. Introduce the Lie superalgebra â = â 0 â 1 with â 0 = b, â 1 = m[t, t 1 ], with the adjoint action of â 0 on â 1, whereas â 1 is regarded as a supercommutative Lie superalgebra.

99 30 Consider the affinization b of b with respect to this form and let V k (b) be the corresponding vacuum module over b. Introduce the Lie superalgebra â = â 0 â 1 with â 0 = b, â 1 = m[t, t 1 ], with the adjoint action of â 0 on â 1, whereas â 1 is regarded as a supercommutative Lie superalgebra. We will write ψ j i [r] = e j i t r 1 for e j i t r 1 m[t, t 1 ].

100 31 Equip the vacuum module V k (a) with the ( 1)-product and introduce its derivation Q : V k (a) V k (a)

101 31 Equip the vacuum module V k (a) with the ( 1)-product and introduce its derivation Q : V k (a) V k (a) for which we have [ Q, ej i [ 1] ] = j 1 col(a)=i ψ j a [0] e a i [ 1] j col(a)=i+1 e j a [ 1] ψ a i [0] + ( k + N q col(i) ) ψj i [ 1] + ψ j + i[0] ψ j i [0] for dominoes i i and j j + occurring in π.

102 31 Equip the vacuum module V k (a) with the ( 1)-product and introduce its derivation Q : V k (a) V k (a) for which we have [ Q, ej i [ 1] ] = j 1 col(a)=i ψ j a [0] e a i [ 1] j col(a)=i+1 e j a [ 1] ψ a i [0] + ( k + N q col(i) ) ψj i [ 1] + ψ j + i[0] ψ j i [0] for dominoes i i and j j + occurring in π. The W-algebra is defined by W k (g, f ) = {v V k (b) Q v = 0}.

103 32 By a theorem of [V. Kac and M. Wakimoto, 2004] (also [T. Arakawa 2005]), there exists a filtration F p W k (g, f ) such that gr F W k (g, f ) = V(g f ).

104 32 By a theorem of [V. Kac and M. Wakimoto, 2004] (also [T. Arakawa 2005]), there exists a filtration F p W k (g, f ) such that gr F W k (g, f ) = V(g f ). Hence, the W-algebra W k (g, f ) admits generators associated with basis elements of the centralizer g f.

105 32 By a theorem of [V. Kac and M. Wakimoto, 2004] (also [T. Arakawa 2005]), there exists a filtration F p W k (g, f ) such that gr F W k (g, f ) = V(g f ). Hence, the W-algebra W k (g, f ) admits generators associated with basis elements of the centralizer g f. In the principal nilpotent case, the generator W (i) is associated with the element e i 1 + e i e N N i+1 g f.

106 Rectangular pyramids 33

107 33 Rectangular pyramids Take a pyramid with n rows and l columns,

108 33 Rectangular pyramids Take a pyramid with n rows and l columns, so that p 1 = = p n = l are the lengths of the rows and q 1 = = q l = n are the lengths of the columns.

109 33 Rectangular pyramids Take a pyramid with n rows and l columns, so that p 1 = = p n = l are the lengths of the rows and q 1 = = q l = n are the lengths of the columns. We have dim g f = l n 2.

110 34 We will use the isomorphism gl l gl n = gln such that e 11 E... e 1l E [e i j ] N i,j=1 = , e l 1 E... e l l E where E = [e ab ] n a,b=1.

111 34 We will use the isomorphism gl l gl n = gln such that e 11 E... e 1l E [e i j ] N i,j=1 = , e l 1 E... e l l E where E = [e ab ] n a,b=1. Explicitly, e (i 1)n+a, (j 1)n+b = e i j e a b.

112 35 Define the homomorphism from the tensor algebra T : T ( gl l [t 1 ]t 1) End C n U ( gl N [t 1 ]t 1),

113 35 Define the homomorphism from the tensor algebra T : T ( gl l [t 1 ]t 1) End C n U ( gl N [t 1 ]t 1), n x T (x) = e a b T a b (x) a,b=1

114 35 Define the homomorphism from the tensor algebra T : T ( gl l [t 1 ]t 1) End C n U ( gl N [t 1 ]t 1), n x T (x) = e a b T a b (x) a,b=1 by setting for x gl l [t 1 ]t 1 : T a b (x) = x e b a gl l [t 1 ]t 1 gl n = gl N [t 1 ]t 1.

115 35 Define the homomorphism from the tensor algebra T : T ( gl l [t 1 ]t 1) End C n U ( gl N [t 1 ]t 1), n x T (x) = e a b T a b (x) a,b=1 by setting for x gl l [t 1 ]t 1 : T a b (x) = x e b a gl l [t 1 ]t 1 gl n = gl N [t 1 ]t 1. Hence, for any elements x, y of the tensor algebra, n T a b (xy) = T a c (x)t c b (y). c=1

116 36 Consider the matrix ατ + e 11 [ 1] e E = 21 [ 1] ατ + e 22 [ 1] e l 1 [ 1] e l 2 [ 1] ατ + e l l [ 1] with α = k + N n,

117 36 Consider the matrix ατ + e 11 [ 1] e E = 21 [ 1] ατ + e 22 [ 1] e l 1 [ 1] e l 2 [ 1] ατ + e l l [ 1] with α = k + N n, and introduce n 2 polynomials in τ, each of degree l, by setting

118 36 Consider the matrix ατ + e 11 [ 1] e E = 21 [ 1] ατ + e 22 [ 1] e l 1 [ 1] e l 2 [ 1] ατ + e l l [ 1] with α = k + N n, and introduce n 2 polynomials in τ, each of degree l, by setting W ab = T ab ( cdet E ), a, b = 1,..., n.

119 37 Write and regard W (r) ab W ab = δ ab (ατ) l + W (1) ab (ατ)l W (l) ab, as elements of V k (b).

120 37 Write and regard W (r) ab W ab = δ ab (ατ) l + W (1) ab (ατ)l W (l) ab, as elements of V k (b). Theorem [T. Arakawa A. M., 2014] The coefficients W (r) ab with a, b {1,..., n} and r = 1,..., l generate the W-algebra W k (g, f ).

121 38 The Miura map V k (b) V k (l) with l = gl n gl n is induced by the projection b l.

122 38 The Miura map V k (b) V k (l) with l = gl n gl n is induced by the projection b l. Corollary. Under the Miura map we have Υ : T ab ( cdet E ) Tab ( (ατ + e1 1 [ 1] )... ( ατ + e l l [ 1] )).

123 38 The Miura map V k (b) V k (l) with l = gl n gl n is induced by the projection b l. Corollary. Under the Miura map we have Υ : T ab ( cdet E ) Tab ( (ατ + e1 1 [ 1] )... ( ατ + e l l [ 1] )). By a general result of [N. Genra, 2016] the image of the Miura map coincides with the intersection of the kernels of screening operators (k is generic).

124 Classical W-algebras 39

125 39 Classical W-algebras Divide by k each row of the matrix ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] and set e i j [r] = e i j [r]/k.

126 39 Classical W-algebras Divide by k each row of the matrix ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] and set e i j [r] = e i j [r]/k. Since α = k + N 1, in the limit k the column-determinant equals

127 39 Classical W-algebras Divide by k each row of the matrix ατ + e 11 [ 1] e 21 [ 1] ατ + e 22 [ 1] e N 1 [ 1] e N 2 [ 1] ατ + e N N [ 1] and set e i j [r] = e i j [r]/k. Since α = k + N 1, in the limit k the column-determinant equals ( τ + e11 [ 1] )... ( τ + e N N [ 1] ).

128 40 We thus recover the (classical) Miura transformation ( τ + e11 [ 1] )... ( τ + e N N [ 1] ) = τ N + u (1) τ N u (N)

129 40 We thus recover the (classical) Miura transformation ( τ + e11 [ 1] )... ( τ + e N N [ 1] ) = τ N + u (1) τ N u (N) providing generators u (i) of the classical W-algebra W(gl N ) [M. Adler, 1979; I. M. Gelfand and L. A. Dickey, 1978].

130 40 We thus recover the (classical) Miura transformation ( τ + e11 [ 1] )... ( τ + e N N [ 1] ) = τ N + u (1) τ N u (N) providing generators u (i) of the classical W-algebra W(gl N ) [M. Adler, 1979; I. M. Gelfand and L. A. Dickey, 1978]. The elements u (i) and all their derivatives are algebraically independent generators of W(gl N ).

131 41 At the critical level k = N the expansion of the column-determinant yields a different presentation of the classical W-algebra W(gl N ).

132 41 At the critical level k = N the expansion of the column-determinant yields a different presentation of the classical W-algebra W(gl N ). Its elements are understood as differential polynomials in the variables E ij := e ij [ 1] with N i j 1.

133 42 Expand the column-determinant τ + E E cdet 21 τ + E E N 1 E N 2 E N τ + E N N = ( τ) N + w (1) ( τ) N w (N).

134 42 Expand the column-determinant τ + E E cdet 21 τ + E E N 1 E N 2 E N τ + E N N = ( τ) N + w (1) ( τ) N w (N). The elements w (1),..., w (N) together with all their derivatives are algebraically independent generators of W(gl N ).

135 43 By taking the Zhu algebra of W k (gl N, f ) for a rectangular pyramid, we recover the generators of the finite W-algebra W(gl N, f ). [E. Ragoucy and P. Sorba, 1999]; [J. Brundan and A. Kleshchev, 2006].

136 44 Affine Poisson vertex algebra V(g) Let g be a simple Lie algebra over C and let X 1,..., X d be a basis of g.

137 44 Affine Poisson vertex algebra V(g) Let g be a simple Lie algebra over C and let X 1,..., X d be a basis of g. Consider the differential algebra V = V(g), V = C[X (r) 1,..., X(r) d r = 0, 1, 2,... ] with X (0) i = X i,

138 44 Affine Poisson vertex algebra V(g) Let g be a simple Lie algebra over C and let X 1,..., X d be a basis of g. Consider the differential algebra V = V(g), V = C[X (r) 1,..., X(r) d r = 0, 1, 2,... ] with X (0) i = X i, equipped with the derivation, (X (r) i ) = X (r+1) i for all i = 1,..., d and r 0.

139 45 Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}.

140 45 Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + (X Y)λ for X, Y g,

141 45 Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + (X Y)λ for X, Y g, and extended to V by sesquilinearity (a, b V): { a λ b} = λ {a λ b},

142 45 Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + (X Y)λ for X, Y g, and extended to V by sesquilinearity (a, b V): { a λ b} = λ {a λ b}, skewsymmetry {a λ b} = {b λ a},

143 45 Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + (X Y)λ for X, Y g, and extended to V by sesquilinearity (a, b V): { a λ b} = λ {a λ b}, skewsymmetry {a λ b} = {b λ a}, and the Leibniz rule (a, b, c V): {a λ bc} = {a λ b}c + {a λ c}b.

144 Hamiltonian reduction 46

145 46 Hamiltonian reduction For a triangular decomposition g = n h n + set b = n h and define the projection π b : g b.

146 46 Hamiltonian reduction For a triangular decomposition g = n h n + set b = n h and define the projection π b : g b. Let f n be a principal nilpotent in g.

147 46 Hamiltonian reduction For a triangular decomposition g = n h n + set b = n h and define the projection π b : g b. Let f n be a principal nilpotent in g. Define the differential algebra homomorphism ρ : V V(b), ρ(x) = π b (X) + (f X), X g.

148 46 Hamiltonian reduction For a triangular decomposition g = n h n + set b = n h and define the projection π b : g b. Let f n be a principal nilpotent in g. Define the differential algebra homomorphism ρ : V V(b), ρ(x) = π b (X) + (f X), X g. The classical W-algebra W(g) is defined by W(g) = {P V(b) ρ{x λ P} = 0 for all X n + }.

149 47 The classical W-algebra W(g) is a Poisson vertex algebra equipped with the λ-bracket {a λ b} ρ = ρ{a λ b}, a, b W(g).

150 47 The classical W-algebra W(g) is a Poisson vertex algebra equipped with the λ-bracket {a λ b} ρ = ρ{a λ b}, a, b W(g). Motivation: integrable Hamiltonian hierarchies Drinfeld and Sokolov, 1985; De Sole, Kac and Valeri,

151 48 Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here b is the subalgebra of lower triangular matrices.

152 48 Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here b is the subalgebra of lower triangular matrices. Set f = E E E N N 1.

153 48 Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here b is the subalgebra of lower triangular matrices. Set f = E E E N N 1. We will work with the algebra V(b) C[ ], E (r) ij E (r) ij = E (r+1) ij.

154 48 Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here b is the subalgebra of lower triangular matrices. Set f = E E E N N 1. We will work with the algebra V(b) C[ ], E (r) ij E (r) ij = E (r+1) ij. The invariant symmetric bilinear form on gl N is defined by (X Y) = tr XY, X, Y gl N.

155 49 Expand the column-determinant with entries in V(b) C[ ], + E E 21 + E cdet E N 1 1 E N 1 2 E N E N 1 E N 2 E N E N N = N + w 1 N w N.

156 49 Expand the column-determinant with entries in V(b) C[ ], + E E 21 + E cdet E N 1 1 E N 1 2 E N E N 1 E N 2 E N E N N = N + w 1 N w N. Theorem [M. Ragoucy, 2015], [De Sole Kac Valeri, 2015]. All elements w 1,..., w N belong to W(gl N ). Moreover, W(gl N ) = C[w (r) 1,..., w(r) N r 0].

157 50 Generators of W(o 2n+1 ) Given a positive integer N = 2n, or N = 2n + 1 set i = N i + 1.

158 50 Generators of W(o 2n+1 ) Given a positive integer N = 2n, or N = 2n + 1 set i = N i + 1. The Lie subalgebra of gl N spanned by the elements F i j = E i j E j i, i, j = 1,..., N, is the orthogonal Lie algebra o N.

159 50 Generators of W(o 2n+1 ) Given a positive integer N = 2n, or N = 2n + 1 set i = N i + 1. The Lie subalgebra of gl N spanned by the elements F i j = E i j E j i, i, j = 1,..., N, is the orthogonal Lie algebra o N. Cartan subalgebra h = span of {F 11,..., F n n }.

160 50 Generators of W(o 2n+1 ) Given a positive integer N = 2n, or N = 2n + 1 set i = N i + 1. The Lie subalgebra of gl N spanned by the elements F i j = E i j E j i, i, j = 1,..., N, is the orthogonal Lie algebra o N. Cartan subalgebra h = span of {F 11,..., F n n }. The subalgebras n + and n are spanned by F ij with i < j and i > j, respectively, and b = n h.

161 50 Generators of W(o 2n+1 ) Given a positive integer N = 2n, or N = 2n + 1 set i = N i + 1. The Lie subalgebra of gl N spanned by the elements F i j = E i j E j i, i, j = 1,..., N, is the orthogonal Lie algebra o N. Cartan subalgebra h = span of {F 11,..., F n n }. The subalgebras n + and n are spanned by F ij with i < j and i > j, respectively, and b = n h. f = F F F n+1 n o 2n+1.

162 51 Expand the column-determinant of the matrix + F F F F n 1 F n F n n F n+1 1 F n F n+1 n F n 1 F n F n n+1 + F n n F F 2 n F F F 1 n F F 1 1

163 51 Expand the column-determinant of the matrix + F F F F n 1 F n F n n F n+1 1 F n F n+1 n F n 1 F n F n n+1 + F n n F F 2 n F F F 1 n F F 1 1 as a differential operator 2n+1 + w 2 2n 1 + w 3 2n w 2n+1, w i V(b).

164 52 Theorem [MR]. All elements w 2,..., w 2n+1 belong to W(o 2n+1 ). Moreover, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0].

165 52 Theorem [MR]. All elements w 2,..., w 2n+1 belong to W(o 2n+1 ). Moreover, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. One proof is based on the folding procedure. The subalgebra o 2n+1 gl 2n+1 is considered as the fixed point subalgebra for an involutive automorphism of gl 2n+1.

166 52 Theorem [MR]. All elements w 2,..., w 2n+1 belong to W(o 2n+1 ). Moreover, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. One proof is based on the folding procedure. The subalgebra o 2n+1 gl 2n+1 is considered as the fixed point subalgebra for an involutive automorphism of gl 2n+1. For the principal nilpotent we have f f = E E E n+1n E n+2n+1 E 2n+1 2n.

167 53 Generators of W(sp 2n ) The Lie subalgebra of gl 2n spanned by the elements F ij = E ij ε i ε j E j i, i, j = 1,..., 2n, is the symplectic Lie algebra sp 2n, where ε i = 1 for i = 1,..., n and ε i = 1 for i = n + 1,..., 2n.

168 53 Generators of W(sp 2n ) The Lie subalgebra of gl 2n spanned by the elements F ij = E ij ε i ε j E j i, i, j = 1,..., 2n, is the symplectic Lie algebra sp 2n, where ε i = 1 for i = 1,..., n and ε i = 1 for i = n + 1,..., 2n. Cartan subalgebra h = span of {F 11,..., F n n }.

169 53 Generators of W(sp 2n ) The Lie subalgebra of gl 2n spanned by the elements F ij = E ij ε i ε j E j i, i, j = 1,..., 2n, is the symplectic Lie algebra sp 2n, where ε i = 1 for i = 1,..., n and ε i = 1 for i = n + 1,..., 2n. Cartan subalgebra h = span of {F 11,..., F n n }. f = F F F n n F n n sp 2n.

170 54 Expand the column-determinant of the matrix + F F F F n 1 F n F n n F n 1 F n 2... F n n + F n n F 2 1 F F 2 n F 2 n F F 1 1 F F 1 n F 1 n F F 1 1

171 54 Expand the column-determinant of the matrix + F F F F n 1 F n F n n F n 1 F n 2... F n n + F n n F 2 1 F F 2 n F 2 n F F 1 1 F F 1 n F 1 n F F 1 1 as a differential operator 2n + w 2 2n 2 + w 3 2n w 2n, w i V(b).

172 55 Theorem [MR]. All elements w 2,..., w 2n belong to W(sp 2n ). Moreover, W(sp 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0].

173 55 Theorem [MR]. All elements w 2,..., w 2n belong to W(sp 2n ). Moreover, W(sp 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. This can be proved by using the folding procedure for the subalgebra sp 2n gl 2n.

174 55 Theorem [MR]. All elements w 2,..., w 2n belong to W(sp 2n ). Moreover, W(sp 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. This can be proved by using the folding procedure for the subalgebra sp 2n gl 2n. For the principal nilpotent we have f f = E E E n+1n E n+2n+1 E 2n2n 1.

175 56 Generators of W(o 2n ) Introduce the algebra of pseudo-differential operators V(b) C(( 1 )), 1 F (r) ij = s=0 ( 1) s F (r+s) ij s 1.

176 56 Generators of W(o 2n ) Introduce the algebra of pseudo-differential operators V(b) C(( 1 )), 1 F (r) ij = s=0 ( 1) s F (r+s) ij s 1. Take the principal nilpotent element f o 2n in the form f = F F F n n 1 + F n n 1.

177 57 Remark. Under the embedding o 2n gl 2n, f f, f is not a principal nilpotent in gl 2n :

178 57 Remark. Under the embedding o 2n gl 2n, f f, f is not a principal nilpotent in gl 2n : f =

179 58 Expand the column-determinant of the (2n + 1) (2n + 1) matrix + F F F F n 1 F n 1 F n 2 F n F n n F n 1 F n F n n F F 2 n F 2 n... + F F F 1 n F 1 n... F F 1 1

180 58 Expand the column-determinant of the (2n + 1) (2n + 1) matrix + F F F F n 1 F n 1 F n 2 F n F n n F n 1 F n F n n F F 2 n F 2 n... + F F F 1 n F 1 n... F F 1 1 as a pseudo-differential operator 2n 1 + w 2 2n 3 + w 3 2n w 2n 1 + ( 1) n y n 1 y n.

181 59 Theorem [MR]. All elements w 2, w 3,..., w 2n 1 and y n belong to W(o 2n ). Moreover, W(o 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n 2, y(r) n r 0 ].

182 59 Theorem [MR]. All elements w 2, w 3,..., w 2n 1 and y n belong to W(o 2n ). Moreover, W(o 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n 2, y(r) n r 0 ]. We have + F F 21 + F y n = cdet F n 1 1 F n 1 2 F n F n1 F n 1 F n 2 F n 2 F n 3 F n F nn

183 Chevalley-type theorem 60

184 60 Chevalley-type theorem Let φ : V(b) V(h) denote the homomorphism of differential algebras defined on the generators as the projection b h with the kernel n.

185 60 Chevalley-type theorem Let φ : V(b) V(h) denote the homomorphism of differential algebras defined on the generators as the projection b h with the kernel n. The restriction of φ to W(g) is injective. The embedding φ : W(g) V(h) is known as the Miura transformation.

186 61 For g = gl N, the image of the column-determinant + E E 21 + E cdet E N 1 1 E N 1 2 E N E N 1 E N 2 E N E N N

187 61 For g = gl N, the image of the column-determinant + E E 21 + E cdet E N 1 1 E N 1 2 E N E N 1 E N 2 E N E N N equals ( + E 11 )... ( + E N N ) = N + w 1 N w N.

188 61 For g = gl N, the image of the column-determinant + E E 21 + E cdet E N 1 1 E N 1 2 E N E N 1 E N 2 E N E N N equals ( + E 11 )... ( + E N N ) = N + w 1 N w N. Therefore, we recover the Adler Gelfand Dickey generators: W(gl N ) = C [ w (r) 1,..., w(r) N r 0].

189 62 Drinfeld Sokolov generators for o 2n+1 : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n+1 + w 2 2n 1 + w 3 2n w 2n+1,

190 62 Drinfeld Sokolov generators for o 2n+1 : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n+1 + w 2 2n 1 + w 3 2n w 2n+1, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0].

191 62 Drinfeld Sokolov generators for o 2n+1 : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n+1 + w 2 2n 1 + w 3 2n w 2n+1, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. Drinfeld Sokolov generators for sp 2n : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n + w 2 2n 2 + w 3 2n w 2n,

192 62 Drinfeld Sokolov generators for o 2n+1 : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n+1 + w 2 2n 1 + w 3 2n w 2n+1, W(o 2n+1 ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0]. Drinfeld Sokolov generators for sp 2n : ( + F 11 )... ( + F n n ) ( F n n )... ( F 11 ) = 2n + w 2 2n 2 + w 3 2n w 2n, W(sp 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n r 0].

193 63 Drinfeld Sokolov generators for o 2n : ( + F 11 )... ( + F n n ) 1 ( F n n )... ( F 11 ) = 2n 1 + w 2 2n 3 + w 3 2n w 2n 1 + ( 1) n y n 1 y n.

194 63 Drinfeld Sokolov generators for o 2n : ( + F 11 )... ( + F n n ) 1 ( F n n )... ( F 11 ) = 2n 1 + w 2 2n 3 + w 3 2n w 2n 1 + ( 1) n y n 1 y n. In particular, y n = ( + F 11 )... ( + F n n ) 1.

195 63 Drinfeld Sokolov generators for o 2n : ( + F 11 )... ( + F n n ) 1 ( F n n )... ( F 11 ) = 2n 1 + w 2 2n 3 + w 3 2n w 2n 1 + ( 1) n y n 1 y n. In particular, y n = ( + F 11 )... ( + F n n ) 1. Then W(o 2n ) = C [ w (r) 2, w(r) 4,..., w(r) 2n 2, y(r) n r 0 ].