On finite W -algebras for Lie superalgebras in the regular case. Elena Poletaeva. University of Texas-Pan American. Nordfjordeid, 7 June 2013

Transcription

1 On finite W -algebras for Lie superalgebras in the regular case Elena Poletaeva University of Texas-Pan American Nordfjordeid, 7 June

2 2 1. Introduction The finite W -algebras are certain associative algebras associated to a complex semi-simple Lie algebra g and a nilpotent element e g. They are quantizations of Poisson algebras of functions on the Slodowy slice at e to the orbit Ad(G)e, where g = Lie(G). Due to recent results of I. Losev, A. Premet and others, W -algebras play a very important role in description of primitive ideals. Finite W -algebras for semi-simple Lie algebras were introduced by A. Premet [13]. They have been studied by mathematicians and physicists: L. Fehér, I. Losev, V. Ginzburg, W. L. Gan, J. Brundan, S. Goodwin, W. Wang [1, 3, 6, 7, 8, 10, 15].

3 Let g be a finite-dimensional semi-simple or reductive Lie algebra over C, ( ) be a non-degenerate invariant symmetric bilinear form. 3 Definition 1.1. Adjoint representation of g: ad(x)(y) = [x, y], x, y g Definition 1.2. e g is nilpotent, if ad(e) is a nilpotent endomorphism of g. Example 1.3. g = gl(n) e gl(n) is nilpotent e is an n n-matrix with eigenvalues zero. Definition 1.4. A nilpotent element e g is regular nilpotent, if g e = Ker(ad(e)) attains the minimal dimension, which is equal to rank g. Theorem 1.5. Jacobson-Morozov [5]. Associated to a nonzero nilpotent element e g, there always exists an sl(2)-triple {e, h, f} which satisfies [e, f] = h, [h, e] = 2e, [h, f] = 2f. Proof. Induction on dim g.

4 4 2. Z-gradings Definition 2.1. A Dynkin Z-grading. Let sl(2) =< e, h, f >. The eigenspace decomposition of the adjoint action ad(h) : g g provides a Z-grading: g = j Z g j, g j = {x g ad(h)(x) = jx}. Properties: (1) e g 2, (2) ad(e) : g j g j+2 is injective for j 1, (3) ad(e) : g j g j+2 is surjective for j 1, (4) g e j 0 g j, (5) (g i g j ) = 0 unless i + j = 0, (6) dim g e = dim g 0 + dim g 1.

5 Definition 2.2. A good Z-grading. 5 A Z-grading g = j Z g j for a semi-simple g is called a good Z-grading for e, if it satisfies the conditions (1)-(3). For a reductive g, there is an additional condition: the center of g is in g 0. Remark 2.3. Properties (4)-(6) remain to be valid for every good Z-grading of g. Remark 2.4. Dynkin Z-grading = Good Z-grading. Good Z-grading Dynkin Z-grading.

6 6 3. Definition of finite W -algebras g is a reductive Lie algebra, ( ) is a non-degenerate invariant symmetric bilinear form, e is a nilpotent element, g = j Z g j is a good Z-grading for e, χ g χ(x) := (x e) x g. Define a bilinear form on g 1 as (x, y) := ([x, y] e) = χ([x, y]) x, y g 1 Remark 3.1. The bilinear form on g 1 is skew-symmetric and non-degenerate. Proof. The skew-symmetry follows by definition. The non-degeneracy follows from the bijection ad(e) : g 1 g 1 and the identity (x, y) = (x [y, e]). Hence dim g 1 is even.

7 Pick a Lagrangian (a maximal isotropic) subspace l of g 1 with respect to the form (, ). Then dim l = 1 2 dim g 1. 7 Let m = ( j 2 g j ) l. The restriction of χ to m χ : m C defines a one-dimensional representation C χ of m thanks to the Lagrangian condition on l. Let I χ be the left ideal of U(g) generated by a χ(a) for a m. Definition 3.2. The generalized Whittaker module is Q χ := U(g) U(m) C χ = U(g)/Iχ. Definition 3.3. The finite W -algebra associated to the nilpotent element e is W χ := End U(g) (Q χ ) op.

8 8 Remark 3.4. W χ can be identified as the space of Whittaker vectors in U(g)/I χ. Let π : U(g) U(g)/I χ be the natural projection, and let y U(g). W χ = (Q χ ) adm = {π(y) U(g)/I χ [a, y] I χ a m}. The multiplication is π(y 1 )π(y 2 ) = π(y 1 y 2 ) for y i U(g) such that [a, y i ] I χ a m and i = 1, 2. Remark 3.5. The isoclasses of finite W -algebras do not depend on good Z-grading and Lagrangian subspace l [3, 6]. Example 3.6. Let e = 0. Then χ = 0, g 0 = g, m = 0, Q χ = U(g), W χ = U(g). Theorem 3.7. B. Kostant, 1978 [9]. For a regular nilpotent element e g, W χ = Z(g), the center of U(g).

9 Definition 3.8. Kazhdan filtration on W χ. 9 Let g be a reductive Lie algebra with a Dynkin Z-grading: g = j Z g j, m = ( j 2 g j ) l. Let n g be an ad(h)-invariant subspace such that g = m n, Cχ =< v > be one-dimensional representation of m. Then W χ = {X U(g)/U(g)m = S(n) axv = χ(a)xv, a m}. For any y n, let wt(y) be the weight of y with respect to ad(h) and deg(y) = wt(y) + 2. The degree function deg induces a Z-grading on S(n). This grading defines a filtration on W χ.

10 10 Theorem 3.9. A. Premet [13]. The associated graded algebra Gr(W χ ) is isomorphic to S(g e ). Idea of Proof. Introduce the map P : W χ S(g e ). For X W χ S(n) the term P (X) of highest degree and highest weight belongs to S(g e ). Example If e = 0, this is the PBW Theorem: Gr(U(g)) = S(g).

11 4. The case of g = gl(n) Form: (a b) = tr(ab) e = , f = h = diag(n 1, n 3,..., 3 n, 1 n). e is a regular nilpotent element n (n 2) n 1 0 h defines an even Dynkin Z-grading of g whose degrees on the elementary matrices E ij are n n n n 8 2 2n

12 12 z = diag(1,..., 1) is the center of gl(n). g e =< z, e, e 2, e 3,..., e n 1 >, dim g e = n n m = g 2 2j, χ(e i+1,i ) = 1, χ(e i+k,i ) = 0 if k 2 j 2 W χ is a polynomial algebra generated by n elements: Ω k is the k-th Casimir element of gl(n): π(z), π(ω 2 ), π(ω 3 ),..., π(ω n ) Ω k = i 1,i 2,...,i k E i1 i 2 E i2 i 3... E ik i 1 The generators of W χ can be identified with elements of g e : 1 k π(ω k) π(z) P z, P e k 1, for k = 2,..., n.

13 5. Lie superalgebras 13 Definition 5.1. A Lie superalgebra is a Z 2 -graded algebra g = g 0 g 1 with an operation [, ] satisfying the following axioms: 1. super-anticommutativity: [x, y] = ( 1) p(x)p(y) [y, x] for x, y g, 2. super-jacobi identity: [x, [y, z]] = [[x, y], z] + ( 1) p(x)p(y) [y, [x, z]] for x, y, z g. Remark 5.2. g 0 is a Lie algebra, [g 0, g 1] g 1 = g 1 is a module over g 0, [g 1, g 1] g 0.

14 14 General Linear Lie superalgebra Super-bracket: g = gl(m n) = g 0 g 1, ( A 0 g 0 = { 0 B ( 0 C g 1 = { D 0 ) ) A is a m m matrix, B is a n n matrix }, C is a m n matrix, D is a n m matrix }. [X, Y ] = XY ( 1) p(x)p(y ) Y X for X, Y g. Super-trace: str ( A C D B ) = tr(a) tr(b).

15 Definition 5.3. A Lie superalgebra g is simple, if it is not abelian and the only Z 2 -graded ideals of g are {0} and g. 15 Special Linear Lie superalgebra sl(m n) = {X gl(m n) strx = 0}. sl(m n) is simple m n. If m = n, then sl(n n)/ < 1 2n > is simple.

16 16 6. Finite W -algebras for Lie superalgebras joint work with V. Serganova (UC Berkeley) Finite W -algebras for Lie superalgebras were studied by mathematicians and physicists: C. Briot, E. Ragoucy, J. Brundan, J. Brown, S. Goodwin, W. Wang, L. Zhao [2, 4, 15, 16]. Let g be a classical simple Lie superalgebra, i.e. g = g 0 g 1, g 0 is a reductive Lie algebra, and g has an invariant symmetric bilinear form. Let e g 0 be an even nilpotent element, and we fix sl(2) =< e, h, f >. The above definition of W χ makes sense, however Theorem of Kostant does not hold in this case since W χ must have a non-trivial odd part, and the center of U(g) is even. Kazhdan filtration on W χ can be defined exactly as in the Lie algebra case.

17 Proposition 6.1. Gr(W χ ) is supercommutative. 17 Remark 6.2. If dim(g 1 ) 1 is even, one can construct the similar map P : W χ S(g e ) by taking the monomials of the highest degree and the highest weight. If dim(g 1 ) 1 is odd, then there exists an odd element θ in g 1 l such that π(θ) W χ and π(θ) 2 = 1.

18 18 7. The case of g = sl(1 n) Form: (a b) = str(ab) e = , f = e is a regular nilpotent element. Elementary matrices: n (n 2) n 1 0 h = diag(0 n 1, n 3,..., 3 n, 1 n) E ij = h 0 µ 1 µ 2 µ 3 µ n ξ 1 h 1 e 1 ξ 2 f 1 h 2 e 2 ξ 3 f 2 h 3 e n 1 ξ n f n 1 h n h defines a Dynkin Z-grading of g whose degrees on the elementary matrices E ij are

19 0 1 n 3 n n 3 n 1 n n 2 n n 4 3 n 4 2n n 2 2n 2 0 dim(g e ) = (n 2) c = diag(n 1,, 1) is a central elements of g g e =< e, e 2,, e n 1, c ξ 1, µ n >, m = ( j 2 g j ) l, if n is odd, then l = 0, if n = 2k, then l =< ξ k+1 >. m is generated by f 1,..., f n ; µ 1,..., µ k 1, ξ k+1,..., ξ n, if n = 2k, and by f 1,..., f n ; µ 1,..., µ k, ξ k+2,..., ξ n, if n = 2k + 1. χ(f i ) = 1 χ(µ i ) = χ(ξ i ) = 0.

20 20 n is an ad(h)-invariant subspace of g: g = m n. Z-grading g = g 1 g 0 g 1 is consistent with Z 2 -grading. Fix an ad(h)-homogeneous bases: B(m 1 ) of m g 1, B(m 1 ) of m g 1, B(n 1 ) of n g 1, B(n 1 ) of n g 1. Set R 1 = ( x )( y ), R 2 = ( x B(m 1 ) y )( y B(n 1 ) y B(m 1 ) x B(n 1 ) x ). π(r 1 ) and π(r 2 ) are both Whittaker vectors, and hence π(r 1 ), π(r 2 ) W χ.

21 Remark 7.1. Casimir elements for gl(m n): 21 p(e ij ) = p(i) + p(j), { 0 if 1 i m p(i) = 1 if m + 1 i m + n Ω k = ( 1) p(i2)+ +p(ik) E i1 i 2 E i2 i 3... E ik i 1. i 1,i 2,...,i k Even generators of W χ : π(ω k ), k = 2,... n and π(c). The generators of W χ can be identified with elements of g e : ( 1) k+1 1 k π(ω k) π(c) π(r 1 ) π(r 2 ) P c, P ξ 1, P µ n. P e k 1, k = 2,..., n,

22 22 Theorem 7.2. ([12]). Let g be a Lie superalgebra of Type I and defect one (i.e. g = sl(1 n) or osp(2 2n 2)). Let n be the rank of g 0, c be a central element of g 0, and Ω 2,..., Ω n be the first n 1 Casimir elements in Z(g). Then W χ is a finite extension of C[π(c), π(ω 2 ),..., π(ω n )] with odd generators π(r 1 ), π(r 2 ) and defining relations π(r 1 ) 2 = π(r 2 ) 2 = 0, [π(c), π(r 1 )] = π(r 1 ), [π(c), π(r 2 )] = π(r 2 ), [π(ω i ), π(r 1 )] = [π(ω i ), π(r 2 )] = 0, i = 2,..., n, [π(r 1 ), π(r 2 )] = π( Ω), where Ω = is an element of Z(g). ad y( x) y B(g 1 ) x B(g 1 ) In this case W χ = U(g e ).

23 Remark 7.3. The finite W -algebras for g = gl(m n) were described as certain truncations of a shifted version of the super-yangian Y (gl(1 1)) by J. Brown, J. Brundan and S. Goodwin in 2012 [4]. 23 The connection between the finite W -algebra for gl(m n) and Y (gl(1 1)) was foreseen by C. Briot and E. Ragoucy in 2003 [2].

24 24 8. The case of g = Q(n) Let otr ( A B B A ) ( A B Q(n) := { B A = trb. ) A, B are n n matrices} Remark 8.1. Q(n) has one-dimensional center < z >, z = 1 2n. Let SQ(n) = {X Q(n) otrx = 0}. The Lie superalgebra SQ(n)/ < z > is simple. Let e i,j and f i,j be standard bases in A and B respectively: e i,j = ( ) Eij 0, f 0 E i,j = ij ( 0 Eij E ij 0 ). Q(n) admits an odd nondegenerate g-invariant super symmetric bilinear form (x y) := otr(xy) for x, y g.

25 Let sl(2) =< e, h, f >, where n 1 e = e i,i+1, h = diag(n 1, n 3,..., 3 n, 1 n), f = i=1 n 1 i=1 i(n i)e i+1,i. 25 e is a regular nilpotent element. h defines an even Dynkin Z-grading of g whose degrees on the elementary matrices are 0 2 2n n n n 4 2 2n 0 2 2n n n n n 4 2 2n 0 2 2n 0

26 26 Replace e = n 1 i=1 e i,i+1 by E = n 1 i=1 f i,i+1 E is odd. Let χ g be defined by χ(x) = (x E). g E = {z, e, e 2,..., e n 1 H 0, H 1,..., H n 1 }, dim(g E ) = (n n), where H 0 = n i=1 ( 1)i+1 f i,i, H 1 = n 1 i=1 ( 1)i f i,i+1,..., H n 1 = ( 1) n+1 f 1,n H 0 = m = n j=2 g 2 2j and it is generated by e i+1,i and f i+1,i where i = 1,..., n 1, χ(e i+1,i ) = 1, χ(f i+1,i ) = 0.

27 27 A. Sergeev defined by induction the elements e (m) i,j and f (m) i,j belonging to U(Q(n)) [14]: e (m) i,j f (m) i,j = n k=1 e i,ke (m 1) k,j = n k=1 e i,kf (m 1) k,j + ( 1) m+1 f i,k f (m 1) k,j, + ( 1) m+1 f i,k e (m 1) k,j. Then [e i,j, e (m) k,l ] = δ j,ke (m) i,l δ i,l e (m) k,j, [f i,j, e (m) k,l ] = ( 1)m+1 δ j,k f (m) i,l δ i,l f (m) k,j, [e i,j, f (m) k,l ] = δ j,k f (m) i,l δ i,l f (m) kj, [f i,j, f (m) k,l ] = ( 1) m+1 δ j,k e (m) i,l + δ i,l e (m) k,j. Proposition 8.2. A. Sergeev [14] The elements n i=1 e(2m+1) i,i generate Z(Q(n)).

28 28 Proposition 8.3. π(e (m) n,1 (m) ) and π(f ) are Whittaker vectors. n,1 Theorem 8.4. W χ has n even generators π(e (n+k 1) n,1 ) and n odd generators π(f (n+k 1) k = 1,..., n. The generators of W χ can be identified with elements of g E : π(e (n) n,1 ) P z, π(e (n+1) n,1 ) P e, π(f (n) n,1 ) P H 0, π(f (n+1) n,1 ) P H 1, n,1 ), Corollary 8.5. W χ = U(Q(n)) adm.

29 Let p := j 0 g j. Let f =< e i,i, f i,i i = 1,..., n >, and let ϑ : U(p) U(f) be the Harish-Chandra homomorphism. 29 Proposition 8.6. The Harish-Chandra homomorphism is injective. Denote x i = e i,i, ξ i = ( 1) i+1 f i,i. Theorem 8.7. Under the Harish-Chandra homomorphism: ϑ(π(e (n+k 1) n,1 )) = [ i 1 i 2... i k (x i1 + ( 1) k+1 ξ i1 )... (x ik 1 ξ ik 1 )(x ik + ξ ik )] even, ϑ(π(f (n+k 1) n,1 )) = [ i 1 i 2... i k (x i1 + ( 1) k+1 ξ i1 )... (x ik 1 ξ ik 1 )(x ik + ξ ik )] odd.

30 30 Theorem 8.8. π(e (n+1) n,1 ) = π( 1 2 n e 2 i,i + i=1 n 1 i=1 e i,i+1 + i<j ( 1) i j f i,i f j,j z2 z). One can define odd generators Φ 0,..., Φ n 1 of W χ as follows: Φ 0 = π(f (n) n,1 ) = π(h 0), Φ 1 = [π(e (n+1) n,1 ), Φ 0 ],... Φ n 1 = [π(e (n+1) n,1 ), Φ n 2 ]. Then [Φ m, Φ p ] = 0, if m + p is odd, [Φ m, Φ p ] Z(Q(n)), if m + p is even.

31 Lemma 8.9. For odd m and p we have 31 [π(e (n+m) n,1 ), π(e (n+p) n,1 )] = 0. We set z i = π(e (n+i) n,1 ) for odd i, z i = [Φ 0, Φ i ] for even i. Theorem 8.10 (P-S). Elements z 0,..., z n 1 are algebraically independent in W χ. Together with Φ 0,..., Φ n 1 they form a complete set of generators in W χ. Conjecture In the case when dim(g 1 ) 1 is even, it is possible to find a set of generators of W χ such that even generators commute, and the commutators of odd generators are in Z(g).

32 32 9. Yangian of Q(n) For a finite-dimensional semi-simple Lie algebra g, the Yangian of g is an infinite-dimensional Hopf algebra Y (g). It is a deformation of the universal enveloping algebra of the Lie algebra of polynomial currents of g. Super-Yangian Y (Q(n)) was studied by M. Nazarov and A. Sergeev [11]. Y (Q(n)) is the associative unital superalgebra over C with the countable set of generators T (m) i,j, where m = 1, 2,... and i, j = ±1, ±2,..., ±n. The Z 2 -grading of the algebra Y (Q(n)): p(t (m) i,j ) = p(i) + p(j), where p(i) = 0 if i > 0 and p(i) = 1 if i < 0.

33 Formal series in Y (Q(n))[[u 1 ]]: 33 The relations in Y (Q(n))[[u 1, v 1 ]]: T i,j (u) = δ ij 1 + T (1) i,j u 1 + T (2) i,j u We also have the relations (u 2 v 2 )[T i,j (u), T k,l (v)] ( 1) p(i)p(k)+p(i)p(l)+p(k)p(l) = (u + v)(t k,j (u)t i,l (v) T k,j (v)t i,l (u)) (u v)(t k,j (u)t i,l (v) T k, j (v)t i, l (u)) ( 1) p(k)+p(l). (1) T i,j ( u) = T i, j (u). (2)

34 34 Relations (1) and (2) are equivalent to the following defining relations: ([T (m+1) i,j T (m) k,j T (r 1) i,l, T (r 1) k,l ] [T (m 1) i,j, T (r+1) k,l ]) ( 1) p(i)p(k)+p(i)p(l)+p(k)p(l) = + T (m 1) k,j T (r) i,l +( 1) p(k)+p(l) ( T (m) k,j T (r 1) i,l T (r 1) k,j T (m) i,l T (r) k,j T (m 1) i,l + T (m 1) k,j T (r) i,l + T (r 1) k, j T (m) i, l T (r) k, j T (m 1) i, l ), (1 ) T (m) i, j = ( 1)m T (m) i,j, (2 ) where m, r = 1,... and T (0) ij = δ ij. Theorem 9.1 (P-S). There exists a surjective homomorphism: defined as follows: φ : Y (Q(1)) W χ φ(t (k) 1,1 ) = ( 1)k π(e (n+k 1) n,1 ), φ(t (k) 1,1 ) = ( 1)k π(f (n+k 1) n,1 ), for k = 1, 2,...

35 10. The case of g = Γ(σ 1, σ 2, σ 3 ) σ 1, σ 2, σ 3 are complex numbers, σ 1 + σ 2 + σ 3 = Γ(σ 1, σ 2, σ 3 ) = Γ 0 Γ 1, Γ 0 = sl(2) 1 sl(2) 2 sl(2) 3, Γ 1 = V 1 V 2 V 3. sl(2) i =< X i, H i, Y i >, V i =< e i, f i >, i = 1, 2, 3. P i : V i V i sl(2) i is sl(2) i -invariant bilinear mapping: P i (e i, e i ) = 2X i, P i (f i, f i ) = 2Y i, P i (e i, f i ) = P i (f i, e i ) = H i. ψ i is a non-degenerate skew-symmetric form on V i : ψ i (e i, f i ) = 1.

36 36 [Γ 0, Γ 1] is the natural representation. [Γ 1, Γ 1] is given by [x 1 x 2 x 3, y 1 y 2 y 3 ] = σ 1 ψ 2 (x 2, y 2 )ψ 3 (x 3, y 3 )P 1 (x 1, y 1 )+ σ 2 ψ 1 (x 1, y 1 )ψ 3 (x 3, y 3 )P 2 (x 2, y 2 ) + σ 3 ψ 1 (x 1, y 1 )ψ 2 (x 2, y 2 )P 3 (x 3, y 3 ). Remark The superalgebra Γ(σ 1, σ 2, σ 3 ) is simple σ i 0 for i = 1, 2, 3. Γ(σ 1, σ 2, σ 3 ) = Γ(σ 1, σ 2, σ 3) the sets {σ i } and {σ i} are obtained from each other by a permutation and multiplication of all elements of one set by a nonzero complex number. Γ(σ 1, σ 2, σ 3 ) is a one-parameter family of deformations of osp(4 2). Γ(1, 1 α, α) = D(2, 1; α), where α 0, 1.

37 Consider the form on g is determined by (X i, Y i ) = 1 σ i. 37 sl(2) =< e, h, f >, where e = X 1 + X 2 + X 3, h = H 1 + H 2 + H 3, f = Y 1 + Y 2 + Y 3. e is a regular nilpotent element, h defines a Dynkin Z-grading of g: g = 3 j= 3g j, where g 3 =< e 1 e 2 e 3 >, g 2 =< X 1, X 2, X 3 >, g 1 =< e 1 e 2 f 3, e 1 f 2 e 3, f 1 e 2 e 3 >, g 0 =< H 1, H 2, H 3 >, g 1 =< e 1 f 2 f 3, f 1 e 2 f 3, e 1 e 2 f 3 >, g 2 =< Y 1, Y 2, Y 3 >, g 3 =< f 1 f 2 f 3 >. dim(g e ) = (3 3). (g e ) 0 =< X 1, X 2, X 3 >, (g e ) 1 =< e 1 f 2 e 3 e 1 e 2 f 3, f 1 e 2 e 3 e 1 e 2 f 3, e 1 e 2 e 3 >.

38 38 m = g 3 g 2 l, l =< e 1 f 2 f 3 >. m is generated by Y 1, Y 2, Y 3 and e 1 f 2 f 3. χ(y i ) = 1 σ i, χ(e 1 f 2 f 3 ) = 0. θ = f 1 e 2 f 3 f 1 f 2 e 3 g 1 l, π(θ) W χ, π(θ) 2 = 2.

39 Even generators of W χ : 39 C 1 = π(2x 1 + σ 1 ( H2 1 2 H 1 )), C 2 = π(2x 2 + σ 2 H (f 1 e 2 f 3 )(e 1 e 2 f 3 )), C 3 = π(2x 3 + σ 3 H (f 1 f 2 e 3 )(e 1 f 2 e 3 )). Odd generators of W χ : R 1 = π(2(e 1 f 2 e 3 e 1 e 2 f 3 ) + σ 1 H 1 (f 1 e 2 f 3 f 1 f 2 e 3 )), R 2 = π(2(f 1 e 2 e 3 e 1 e 2 f 3 ) + (σ 1 H 1 σ 3 H 3 )(f 1 e 2 f 3 ) σ 2 H 2 (f 1 f 2 e 3 )), R 3 = π(4(e 1 e 2 e 3 ) σ 1 H 1 R 2 4σ 1 (f 1 e 2 f 3 )X 1 2(σ 1 H 1 (e 1 e 2 f 3 ) + σ 2 H 2 (e 1 f 2 e 3 ) + σ 3 H 3 (e 1 e 2 f 3 ))), and π(θ). where Ω is the Casimir element of g. π(ω) = C 1 + C 2 + C R 1π(θ),

40 40 Theorem 10.2 (P-S). W χ is generated by even elements π(ω), C 1 and C 2, and odd element π(θ). The relations are [C 1, C 2 ] = 0, [π(θ), C i ] = R i σ i 2 π(θ), i = 1, 2, [C 2, R 1 ] = σ 2 2 R 1 + R 3, [C 1, R 2 ] = σ 1 2 R 2 + R 3, [R i, R i ] = 8σ i C i 2σ i R i π(θ), i = 1, 2, [R 1, R 2 ] = 4(σ 1 C 2 + σ 2 C 1 + σ 3 π(ω)) + (σ 1 R 2 + σ 2 R 1 )π(θ), [R i, π(θ)] = 2σ i, i = 1, 2, [π(θ), π(θ)] = 4, [π(ω), π(θ)] = 0, [π(ω), C i ] = 0, i = 1, 2, [π(ω), R i ] = 0, i = 1, 2, 3.

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