Some simple modular Lie superalgebras

Transcription

1 Some simple modular Lie superalgebras Alberto Elduque Universidad de Zaragoza

2 Freudenthal-Tits Magic Square A supermagic rectangle A supermagic square Bouarroudj-Grozman-Leites Classification

3 Freudenthal-Tits Magic Square A supermagic rectangle A supermagic square Bouarroudj-Grozman-Leites Classification

4 Exceptional Lie algebras

5 Exceptional Lie algebras G 2, F 4, E 6, E 7, E 8

6 Exceptional Lie algebras G 2, F 4, E 6, E 7, E 8 G 2 = der O (Cartan 1914) F 4 = der H 3 (O) E 6 = str 0 H 3 (O) (Chevalley-Schafer 1950)

7 Tits construction (1966)

8 Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3,

9 Tits construction (1966) then C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, T (C, J) = der C (C 0 J 0 ) der J is a Lie algebra (char 2, 3) under a suitable Lie bracket: [a x, b y] = 1 3 tr(xy)d a,b+ ( [a, b] ( xy 1 3 tr(xy)1)) +2t(ab)d x,y.

10 Freudenthal-Tits Magic Square T (C, J) H 3 (k) H 3 (k k) H 3 (Mat 2 (k)) H 3 (C(k)) k A 1 A 2 C 3 F 4 k k A 2 A 2 A 2 A 5 E 6 Mat 2 (k) C 3 A 5 D 6 E 7 C(k) F 4 E 6 E 7 E 8

11 Tits construction rearranged

12 Tits construction rearranged J = H 3 (C ) k 3 ( 2 i=0ι i (C ) ), J 0 k 2 ( 2 i=0ι i (C ) ), der J tri(c ) ( 2 i=0ι i (C ) ),

13 Tits construction rearranged J = H 3 (C ) k 3 ( 2 i=0ι i (C ) ), J 0 k 2 ( 2 i=0ι i (C ) ), der J tri(c ) ( 2 i=0ι i (C ) ), T (C, J) = der C (C 0 J 0 ) der J der C (C 0 k 2 ) ( 2 i=0c 0 ι i (C ) ) ( tri(c ) ( 2 i=0ι i (C )) ) ( tri(c) tri(c ) ) ( 2 i=0ι i (C C ) )

14 Tits construction rearranged J = H 3 (C ) k 3 ( 2 i=0ι i (C ) ), J 0 k 2 ( 2 i=0ι i (C ) ), der J tri(c ) ( 2 i=0ι i (C ) ), T (C, J) = der C (C 0 J 0 ) der J der C (C 0 k 2 ) ( 2 i=0c 0 ι i (C ) ) ( tri(c ) ( 2 i=0ι i (C )) ) ( tri(c) tri(c ) ) ( 2 i=0ι i (C C ) ) tri(c) = {(d 0, d 1, d 2 ) so(c) 3 : d 0 (x y) = d 1 (x) y+x d 2 (y) x, y} (x y = xȳ) is the triality Lie algebra of C.

15 Tits construction rearranged

16 Tits construction rearranged The product in is given by: g(c, C ) = ( tri(c) tri(c ) ) ( 2 i=0ι i (C C ) ),

17 Tits construction rearranged The product in is given by: g(c, C ) = ( tri(c) tri(c ) ) ( 2 i=0ι i (C C ) ), tri(c) tri(c ) is a Lie subalgebra of g(c, C ), [(d 0, d 1, d 2 ), ι i (x x )] = ι i ( di (x) x ), [(d 0, d 1, d 2 ), ι i(x x )] = ι i ( x d i (x ) ), [ι i (x x ), ι i+1 (y y )] = ι i+2 ( ( xȳ) ( x ȳ ) ) (indices modulo 3), [ι i (x x ), ι i (y y )] = q (x, y )θ i (t x,y ) + q(x, y)θ i (t x,y ), where t x,y = ( q(x,.)y q(y,.)x, 1 2 q(x, y)1 R xr y, 1 2 q(x, y)1 L xl y ) (Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel)

18 Freudenthal-Tits Magic Square (char 3)

19 Freudenthal-Tits Magic Square (char 3) This more symmetric construction is valid too in characteristic 3:

20 Freudenthal-Tits Magic Square (char 3) This more symmetric construction is valid too in characteristic 3: dim C g(c, C ) A 1 Ã 2 C 3 F 4 dim C 2 Ã 2 Ã 2 Ã 2 Ã 5 Ẽ 6 4 C 3 Ã 5 D 6 E 7 8 F 4 Ẽ 6 E 7 E 8

21 Freudenthal-Tits Magic Square (char 3) This more symmetric construction is valid too in characteristic 3: dim C g(c, C ) A 1 Ã 2 C 3 F 4 dim C 2 Ã 2 Ã 2 Ã 2 Ã 5 Ẽ 6 4 C 3 Ã 5 D 6 E 7 8 F 4 Ẽ 6 E 7 E 8 Ã 2 denotes a form of pgl 3, so [Ã2, Ã2] is a form of psl 3. Ã 5 denotes a form of pgl 6, so [Ã5, Ã5] is a form of psl 6. Ẽ 6 is not simple, but [Ẽ6, Ẽ6] is a codimension 1 simple ideal.

22 Freudenthal-Tits Magic Square A supermagic rectangle A supermagic square Bouarroudj-Grozman-Leites Classification

23 A look at the rows of Tits construction

24 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.)

25 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.) First row dim C = 1, T (C, J) = der J.

26 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.) First row dim C = 1, T (C, J) = der J. Second row dim C = 2, T (C, J) J 0 der J str 0 (J).

27 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.) First row dim C = 1, T (C, J) = der J. Second row dim C = 2, T (C, J) J 0 der J str 0 (J). Third row dim C = 4, so C = Q is a quaternion algebra and T (C, J) = ad Q0 (Q 0 J 0 ) der J (Q 0 J) der J. (This is Tits version (1962) of the Tits-Kantor-Koecher construction T KK(J))

28 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.) First row dim C = 1, T (C, J) = der J. Second row dim C = 2, T (C, J) J 0 der J str 0 (J). Third row dim C = 4, so C = Q is a quaternion algebra and T (C, J) = ad Q0 (Q 0 J 0 ) der J (Q 0 J) der J. (This is Tits version (1962) of the Tits-Kantor-Koecher construction T KK(J))

29 A look at the rows of Tits construction T (C, J) = der C (C 0 J 0 ) der J (C a composition algebra, J a suitable Jordan algebra.) First row dim C = 1, T (C, J) = der J. Second row dim C = 2, T (C, J) J 0 der J str 0 (J). Third row dim C = 4, so C = Q is a quaternion algebra and T (C, J) = ad Q0 (Q 0 J 0 ) der J (Q 0 J) der J. (This is Tits version (1962) of the Tits-Kantor-Koecher construction T KK(J)) Up to now, everything works for arbitrary Jordan algebras in characteristic 2, and even for Jordan superalgebras.

30 Fourth row

31 Fourth row Fourth row dim C = 8. If the characteristic is 2, 3, then der C = g 2 is simple of type G 2, C 0 is its smallest nontrivial irreducible module, and T (C, J) = g 2 (C 0 J 0 ) der J is a G 2 -graded Lie algebra. Essentially, all the G 2 -graded Lie algebras appear in this way [Benkart-Zelmanov 1996].

32 Fourth row Fourth row dim C = 8. If the characteristic is 2, 3, then der C = g 2 is simple of type G 2, C 0 is its smallest nontrivial irreducible module, and T (C, J) = g 2 (C 0 J 0 ) der J is a G 2 -graded Lie algebra. Essentially, all the G 2 -graded Lie algebras appear in this way [Benkart-Zelmanov 1996]. It makes sense to consider Jordan superalgebras, as long as their Grassmann envelopes satisfy the Cayley-Hamilton equation of degree 3.

33 Fourth row Fourth row dim C = 8. If the characteristic is 2, 3, then der C = g 2 is simple of type G 2, C 0 is its smallest nontrivial irreducible module, and T (C, J) = g 2 (C 0 J 0 ) der J is a G 2 -graded Lie algebra. Essentially, all the G 2 -graded Lie algebras appear in this way [Benkart-Zelmanov 1996]. It makes sense to consider Jordan superalgebras, as long as their Grassmann envelopes satisfy the Cayley-Hamilton equation of degree 3. T (C, J(V )) G(3) (dim V = 2, V = V 1 ).

34 Fourth row Fourth row dim C = 8. If the characteristic is 2, 3, then der C = g 2 is simple of type G 2, C 0 is its smallest nontrivial irreducible module, and T (C, J) = g 2 (C 0 J 0 ) der J is a G 2 -graded Lie algebra. Essentially, all the G 2 -graded Lie algebras appear in this way [Benkart-Zelmanov 1996]. It makes sense to consider Jordan superalgebras, as long as their Grassmann envelopes satisfy the Cayley-Hamilton equation of degree 3. T (C, J(V )) G(3) (dim V = 2, V = V 1 ). T (C, D 2 ) F (4).

35 Fourth row Fourth row dim C = 8. If the characteristic is 2, 3, then der C = g 2 is simple of type G 2, C 0 is its smallest nontrivial irreducible module, and T (C, J) = g 2 (C 0 J 0 ) der J is a G 2 -graded Lie algebra. Essentially, all the G 2 -graded Lie algebras appear in this way [Benkart-Zelmanov 1996]. It makes sense to consider Jordan superalgebras, as long as their Grassmann envelopes satisfy the Cayley-Hamilton equation of degree 3. T (C, J(V )) G(3) (dim V = 2, V = V 1 ). T (C, D 2 ) F (4). T (C, K 10 ) in characteristic 5!! This is a new simple modular Lie superalgebra, whose even part is so 11 and odd part its spin module.

36 A supermagic rectangle T (C, J) H 3 (k) H 3 (k k) H 3 (Mat 2 (k)) H 3 (C(k)) J(V ) D t K 10 k A 1 A 2 C 3 F 4 A 1 B(0, 1) B(0, 1) B(0, 1) k k A 2 A 2 A 2 A 5 E 6 B(0, 1) A(1, 0) C(3) Mat 2 (k) C 3 A 5 D 6 E 7 B(1, 1) D(2, 1; t) F (4) C(k) F 4 E 6 E 7 E 8 G(3) F (4) T (C(k), K 10 ) (t = 2) (char 5)

37 A supermagic rectangle: the new columns T (C, J) J(V ) D t K 10 k A 1 B(0, 1) B(0, 1) B(0, 1) k k B(0, 1) A(1, 0) C(3) Mat 2 (k) B(1, 1) D(2, 1; t) F (4) C(k) G(3) F (4) T (C(k), K 10 ) (t = 2) (char 5)

38 Fourth row, characteristic 3

39 Fourth row, characteristic 3 If the characteristic is 3 and dim C = 8, then der C is no longer simple, but contains the simple ideal ad C 0 (a form of psl 3 ). It makes sense to consider: T (C, J) = ad C 0 (C 0 J 0 ) der J (C 0 J) der J.

40 Fourth row, characteristic 3 If the characteristic is 3 and dim C = 8, then der C is no longer simple, but contains the simple ideal ad C 0 (a form of psl 3 ). It makes sense to consider: T (C, J) = ad C 0 (C 0 J 0 ) der J (C 0 J) der J. T (C, J) becomes a Lie algebra if and only if J is a commutative and alternative algebra (these conditions imply the Jordan identity).

41 Fourth row, characteristic 3 If the characteristic is 3 and dim C = 8, then der C is no longer simple, but contains the simple ideal ad C 0 (a form of psl 3 ). It makes sense to consider: T (C, J) = ad C 0 (C 0 J 0 ) der J (C 0 J) der J. T (C, J) becomes a Lie algebra if and only if J is a commutative and alternative algebra (these conditions imply the Jordan identity). The simple commutative alternative algebras are just the fields, so nothing interesting appears here.

42 Fourth superrow, characteristic 3

43 Fourth superrow, characteristic 3 But there are simple commutative alternative superalgebras [Shestakov 1997] (characteristic 3):

44 Fourth superrow, characteristic 3 But there are simple commutative alternative superalgebras [Shestakov 1997] (characteristic 3): (i) fields,

45 Fourth superrow, characteristic 3 But there are simple commutative alternative superalgebras [Shestakov 1997] (characteristic 3): (i) fields, (ii) J(V ), the Jordan superalgebra of a superform on a two dimensional odd space V,

46 Fourth superrow, characteristic 3 But there are simple commutative alternative superalgebras [Shestakov 1997] (characteristic 3): (i) fields, (ii) J(V ), the Jordan superalgebra of a superform on a two dimensional odd space V, (iii) B = B(Γ, D, 0) = Γ Γu, where Γ is a commutative associative algebra, D der Γ such that Γ is D-simple, a(bu) = (ab)u = (au)b, (au)(bu) = ad(b) D(a)b, a, b Γ.

47 Fourth superrow, characteristic 3 Example (Divided powers) Γ = O(1; n) = span { t (r) : 0 r 3 n 1 }, t (r) t (s) = ( ) r+s r t (r+s), D(t (r) ) = t (r 1).

48 Bouarroudj-Leites superalgebras

49 Bouarroudj-Leites superalgebras Over an algebraically closed field of characteristic 3:

50 Bouarroudj-Leites superalgebras Over an algebraically closed field of characteristic 3: T ( C(k), J(V ) ) is a simple Lie superalgebra specific of characteristic 3 of (super)dimension 10 14,

51 Bouarroudj-Leites superalgebras Over an algebraically closed field of characteristic 3: T ( C(k), J(V ) ) is a simple Lie superalgebra specific of characteristic 3 of (super)dimension 10 14, T ( C(k), O(1, n) O(1, n)u ) = Bj(1; n 7) is a simple Lie superalgebra of (super)dimension n n.

52 Bouarroudj-Leites superalgebras Over an algebraically closed field of characteristic 3: T ( C(k), J(V ) ) is a simple Lie superalgebra specific of characteristic 3 of (super)dimension 10 14, T ( C(k), O(1, n) O(1, n)u ) = Bj(1; n 7) is a simple Lie superalgebra of (super)dimension n n.

53 Bouarroudj-Leites superalgebras Over an algebraically closed field of characteristic 3: T ( C(k), J(V ) ) is a simple Lie superalgebra specific of characteristic 3 of (super)dimension 10 14, T ( C(k), O(1, n) O(1, n)u ) = Bj(1; n 7) is a simple Lie superalgebra of (super)dimension n n. Both simple Lie superalgebras have been considered in a completely different way by Bouarroudj and Leites (2006).

54 Freudenthal-Tits Magic Square A supermagic rectangle A supermagic square Bouarroudj-Grozman-Leites Classification

55 Composition superalgebras

56 Composition superalgebras Definition A superalgebra C = C 0 C 1, endowed with a regular quadratic superform q = (q 0, b), called the norm, is said to be a composition superalgebra in case q 0 (x 0 y 0 ) = q 0 (x 0 )q 0 (y 0 ), b(x 0 y, x 0 z) = q 0 (x 0 )b(y, z) = b(yx 0, zx 0 ), b(xy, zt) + ( 1) x y + x z + y z b(zy, xt) = ( 1) y z b(x, z)b(y, t), The unital composition superalgebras are termed Hurwitz superalgebras.

57 Composition superalgebras: examples (Shestakov)

58 Composition superalgebras: examples (Shestakov) B(1, 2) = k1 V, char k = 3, V a two dim l vector space with a nonzero alternating bilinear form.., with 1x = x1 = x, uv = u v 1, q 0 (1) = 1, b(u, v) = u v, is a Hurwitz superalgebra. (As a superalgebra, this is just our previous J(V ).)

59 Composition superalgebras: examples (Shestakov)

60 Composition superalgebras: examples (Shestakov) B(4, 2) = End k (V ) V, k and V as before, End k (V ) is equipped with the symplectic involution f f, ( f (u) v = u f (v) ), the multiplication is given by: the usual multiplication (composition of maps) in End k (V ), v f = f (v) = f v for any f End k (V ) and v V, u v =. u v ( w w u v ) End k (V ) for any u, v V, and with quadratic superform q 0 (f ) = det f, b(u, v) = u v, is a Hurwitz superalgebra.

61 Composition superalgebras: classification

62 Composition superalgebras: classification Theorem (E.-Okubo 2002) Any unital composition superalgebra is either: a Hurwitz algebra, a Z 2 -graded Hurwitz algebra in characteristic 2, isomorphic to either B(1, 2) or B(4, 2) in characteristic 3.

63 Composition superalgebras: classification Theorem (E.-Okubo 2002) Any unital composition superalgebra is either: a Hurwitz algebra, a Z 2 -graded Hurwitz algebra in characteristic 2, isomorphic to either B(1, 2) or B(4, 2) in characteristic 3. Hurwitz superalgebras can be plugged into the symmetric construction of Freudenthal-Tits Magic Square g(c, C ).

64 Supermagic Square (char 3, Cunha-E. 2007) g(c, C ) k k k Mat 2 (k) C(k) B(1, 2) B(4, 2) k A 1 à 2 C 3 F k k à 2 à 2 à 5 Ẽ Mat 2 (k) D 6 E C(k) E B(1, 2) B(4, 2) 78 64

65 Supermagic Square (char 3, Cunha-E. 2007) g(c, C ) k k k Mat 2 (k) C(k) B(1, 2) B(4, 2) k A 1 à 2 C 3 F k k à 2 à 2 à 5 Ẽ Mat 2 (k) D 6 E C(k) E B(1, 2) B(4, 2) Notation: g(n, m) will denote the superalgebra g(c, C ), with dim C = n, dim C = m.

66 Lie superalgebras in the Supermagic Square B(1, 2) B(4, 2) k psl 2,2 sp 6 (14) k k ( ) ( ) sl2 pgl 3 (2) psl3 pgl 6 (20) Mat 2 (k) ( ) ( ) sl2 sp 6 (2) (13) so 12 spin 12 C(k) ( ) ( ) sl2 f 4 (2) (25) e 7 (56) B(1, 2) so 7 2spin 7 sp 8 (40) B(4, 2) sp 8 (40) so 13 spin 13

67 Lie superalgebras in the Supermagic Square

68 Lie superalgebras in the Supermagic Square All these Lie superalgebras are simple, with the exception of g(2, 3) and g(2, 6), both of which contain a codimension one simple ideal.

69 Lie superalgebras in the Supermagic Square All these Lie superalgebras are simple, with the exception of g(2, 3) and g(2, 6), both of which contain a codimension one simple ideal. Only g(1, 3) psl 2,2 has a counterpart in Kac s classification in characteristic 0. The other Lie superalgebras in the Supermagic Square, or their derived algebras, are new simple Lie superalgebras, specific of characteristic 3.

70 Lie superalgebras in the Supermagic Square All these Lie superalgebras are simple, with the exception of g(2, 3) and g(2, 6), both of which contain a codimension one simple ideal. Only g(1, 3) psl 2,2 has a counterpart in Kac s classification in characteristic 0. The other Lie superalgebras in the Supermagic Square, or their derived algebras, are new simple Lie superalgebras, specific of characteristic 3. The simple Lie superalgebra g(2, 3) = [g(2, 3), g(2, 3)] is isomorphic to our previous T (C(k), J(V )).

71 The Supermagic Square and Jordan algebras

72 The Supermagic Square and Jordan algebras k k k Mat 2 (k) C(k) B(1, 2) psl 2,2 ( sl2 pgl 3 ) ( (2) psl3 ) ( sl2 sp 6 ) ( (2) (13) ) ( sl2 f 4 ) ( (2) (25) ) B(4, 2) sp 6 (14) pgl 6 (20) so 12 spin 12 e 7 (56)

73 The Supermagic Square and Jordan algebras k k k Mat 2 (k) C(k) B(1, 2) psl 2,2 ( sl2 pgl 3 ) ( (2) psl3 ) ( sl2 sp 6 ) ( (2) (13) ) ( sl2 f 4 ) ( (2) (25) ) B(4, 2) sp 6 (14) pgl 6 (20) so 12 spin 12 e 7 (56) g(3, r) = ( sl 2 der J ) ( (2) Ĵ ), r = 1, 2, 4, 8, Ĵ = J 0 /k1, J = H 3 (C).

74 The Supermagic Square and Jordan algebras k k k Mat 2 (k) C(k) B(1, 2) psl 2,2 ( sl2 pgl 3 ) ( (2) psl3 ) ( sl2 sp 6 ) ( (2) (13) ) ( sl2 f 4 ) ( (2) (25) ) B(4, 2) sp 6 (14) pgl 6 (20) so 12 spin 12 e 7 (56) g(3, r) = ( sl 2 der J ) ( (2) Ĵ ), r = 1, 2, 4, 8, Ĵ = J 0 /k1, J = H 3 (C). g(6, r) = (der T ) T, ( ) k J r = 1, 2, 4, 8, T =, J = H J k 3 (C).

75 Freudenthal-Tits Magic Square A supermagic rectangle A supermagic square Bouarroudj-Grozman-Leites Classification

76 Simple modular Lie superalgebras with a Cartan matrix

77 Simple modular Lie superalgebras with a Cartan matrix The finite dimensional modular Lie superalgebras with indecomposable symmetrizable Cartan matrices (or contragredient Lie superalgebras) over algebraically closed fields have been classified by Bouarroudj, Grozman and Leites (2009), under some extra technical hypotheses.

78 Simple modular Lie superalgebras with a Cartan matrix The finite dimensional modular Lie superalgebras with indecomposable symmetrizable Cartan matrices (or contragredient Lie superalgebras) over algebraically closed fields have been classified by Bouarroudj, Grozman and Leites (2009), under some extra technical hypotheses. For characteristic p 3, apart from the Lie superalgebras obtained as the analogues of the Lie superalgebras in the classification in characteristic 0, by reducing the Cartan matrices modulo p, there are only the following exceptions:

79 Simple modular Lie superalgebras with a Cartan matrix

80 Simple modular Lie superalgebras with a Cartan matrix 1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5). (Dimensions and )

81 Simple modular Lie superalgebras with a Cartan matrix 1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5). (Dimensions and ) 2. The family of exceptions given by the Lie superalgebras in the Supermagic Square in characteristic 3.

82 Simple modular Lie superalgebras with a Cartan matrix 1. Two exceptions in characteristic 5: br(2; 5) and el(5; 5). (Dimensions and ) 2. The family of exceptions given by the Lie superalgebras in the Supermagic Square in characteristic Another two exceptions in characteristic 3, similar to the ones in characteristic 5: br(2; 3) and el(5; 3). (Dimensions 10 8 and )

83 Moreover, The superalgebra br(2; 3) appeared first related to an eight dimensional symplectic triple system (E. 2006).

84 Moreover, The superalgebra br(2; 3) appeared first related to an eight dimensional symplectic triple system (E. 2006). The superalgebra el(5; 5) is the Lie superalgebra T (C(k), K 10 ) considered previously.

85 Moreover, The superalgebra br(2; 3) appeared first related to an eight dimensional symplectic triple system (E. 2006). The superalgebra el(5; 5) is the Lie superalgebra T (C(k), K 10 ) considered previously. The superalgebra el(5; 3) lives (as a natural maximal subalgebra) in the Lie superalgebra g(3, 8) of the Supermagic Square.

86 Moreover, The superalgebra br(2; 3) appeared first related to an eight dimensional symplectic triple system (E. 2006). The superalgebra el(5; 5) is the Lie superalgebra T (C(k), K 10 ) considered previously. The superalgebra el(5; 3) lives (as a natural maximal subalgebra) in the Lie superalgebra g(3, 8) of the Supermagic Square.

87 Moreover, The superalgebra br(2; 3) appeared first related to an eight dimensional symplectic triple system (E. 2006). The superalgebra el(5; 5) is the Lie superalgebra T (C(k), K 10 ) considered previously. The superalgebra el(5; 3) lives (as a natural maximal subalgebra) in the Lie superalgebra g(3, 8) of the Supermagic Square. That s all. Thanks