/department of mathematics and computer science

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1 Construction of Chevalley Bases of Lie Algebras Dan Roozemond Joint work with Arjeh M. Cohen May 14th, 2009, Computeralgebratagung, Universität Kassel

2 Outline 2 of 30 What is a Lie algebra? What is a Chevalley basis? How to compute Chevalley bases? Does it work? What next? Any questions?

3 What is a Lie Algebra? 3 of 30 Vector space: F n

4 What is a Lie Algebra? 3 of 30 Vector space: Multiplication Bilinear, F n [, ] :L L L that is Anti-symmetric, Satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z,[x, y]] = 0

5 What is a Lie Algebra? 3 of 30 Vector space: Multiplication F n L [, ] :L L L that is Bilinear, Anti-symmetric, F n [, ] Satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z,[x, y]] = 0

6 Simple Lie algebras 4 of 30 Classification (Killing, Cartan) If char(f) = 0 or big enough then the only simple Lie algebras are: A n (n 1) B n (n 2) C n (n 3) D n (n 4) E 6, E 7, E 8 F 4 G 2

7 Why Study Lie Algebras? 5 of 30

8 Why Study Lie Algebras? 5 of 30 Study groups by their Lie algebras: Simple algebraic group G <-> Unique Lie algebra L Many properties carry over to L Easier to calculate in L G Aut(L), often even G = Aut(L)

9 Why Study Lie Algebras? 5 of 30 Study groups by their Lie algebras: Simple algebraic group G <-> Unique Lie algebra L Many properties carry over to L Easier to calculate in L G Aut(L), often even G = Aut(L) Opportunities for: Recognition Conjugation...

10 Why Study Lie Algebras? 5 of 30 Study groups by their Lie algebras: Simple algebraic group G <-> Unique Lie algebra L Many properties carry over to L Easier to calculate in L G Aut(L), often even G = Aut(L) Opportunities for: Recognition Conjugation... Because there are problems to be solved!... and a thesis to be written...

11 Chevalley Bases 6 of 30 x Β x Α Β x Α H x Α x Α Β x Β Many Lie algebras have a Chevalley basis!

12 Root Systems 7 of 30 A hexagon

13 Root Systems 7 of 30 A hexagon

14 Root Systems 7 of 30 Β A hexagon Α

15 Root Systems 7 of 30 Β Α Β A hexagon A root system of type A2 Α Α Α Β Β

16 Root Data 8 of 30 Definition (Root Datum) R =(X, Φ, Y, Φ ),, : X Y Z

17 Root Data 8 of 30 Definition (Root Datum) R =(X, Φ, Y, Φ ),, : X Y Z X, Y: dual free -modules, put in duality by,, : roots, Φ X Z : coroots. Φ Y

18 Root Data 8 of 30 Definition (Root Datum) R =(X, Φ, Y, Φ ),, : X Y Z X, Y: dual free -modules, put in duality by,, : roots, Φ X Z : coroots. Φ Y One Root System { Several Root Data: adjoint... simply connected

19 Root Data 9 of 30 Definition (Root Datum) R =(X, Φ, Y, Φ ),, : X Y Z One Root System { Several Root Data: adjoint... simply connected Irreducible Root Data: A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2.

20 Root Data 10 of 30 Β Α Β A hexagon A root system of type A2 Α Α Α Β Β

21 Root Data 10 of 30 xβ Β x Α Β Α Β A hexagon A root system of type A2 A Lie algebra of type A2 x Α Α H Αx Α x Α Β Α Β x Β Β

22 Chevalley Basis 11 of 30 Definition (Chevalley Basis) Formal basis: L = Fh i Fx α i=1,...,n α Φ Bilinear anti-symmetric multiplication satisfies ( i, j {1,..., n}; α, β Φ) : [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity.

23 Chevalley Basis 11 of 30 Definition (Chevalley Basis) Formal basis: L = Fh i Fx α i=1,...,n α Φ Bilinear anti-symmetric multiplication satisfies ( i, j {1,..., n}; α, β Φ) : x Α x Β H x Α Β x Α [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity. x Α Β x Β

24 Why? 12 of 30 Why Chevalley bases? Because transformation between two Chevalley bases is an automorphism of L, So we can test isomorphism between two Lie algebras (and find isomorphisms!) by computing Chevalley bases.

25 Why? 13 of 30 L F n [, ]

26 Why? 13 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β

27 Why? 13 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β L F n [, ]

28 Why? 13 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β F n [, ] x Β x Α Β L x Α H x Α x Α Β x Β

29 Why? 13 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β equal F n [, ] x Β x Α Β L x Α H x Α x Α Β x Β

30 Why? 13 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β isomorphic! equal F n [, ] x Β x Α Β L x Α H x Α x Α Β x Β

31 Why? 14 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β

32 Why? 14 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β L F n [, ]

33 Why? 14 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β x 3Α 2Β L x Β x Α Β x 2Α Β x 3Α Β F n [, ] x Α H x Α x 3Α Β x 2Α Β x Α Β x Β x 3Α 2Β

34 Why? 14 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β not equal x 3Α 2Β L x Β x Α Β x 2Α Β x 3Α Β F n [, ] x Α H x Α x 3Α Β x 2Α Β x Α Β x Β x 3Α 2Β

35 Why? 14 of 30 L x Β x Α Β x Α H F n [, ] x Α x Α Β x Β non-isomorphic! not equal x 3Α 2Β L x Β x Α Β x 2Α Β x 3Α Β F n [, ] x Α H x Α x 3Α Β x 2Α Β x Α Β x Β x 3Α 2Β

36 Outline 15 of 30 What is a Lie algebra? What is a Chevalley basis? How to compute Chevalley bases? Does it work? What next? Any questions?

37 The Mission 16 of 30 Given a Lie algebra (on a computer), Want to know which Lie algebra it is, So want to compute a Chevalley basis for it. Root datum, field Group of Lie type Matrix Lie algebra L Chevalley Basis x Β x Α Β F n [, ] x Algorithm Α x Α H x Α Β x Β...

38 The Mission 17 of 30 Root datum, field Group of Lie type Matrix Lie algebra L Chevalley Basis x Β x Α Β F n [, ] x Algorithm Α x Α H x Α Β x Β... Assume splitting Cartan subalgebra H is given (Cohen/Murray, indep. Ryba); Assume root datum R is given

39 The Mission 18 of 30 Root datum, field Group of Lie type Matrix Lie algebra L Chevalley Basis x Β x Α Β F n [, ] x Algorithm Α x Α H x Α Β x Β... Char. 0, p 5: De Graaf, Murray; implemented in GAP, MAGMA

40 The Mission 18 of 30 Root datum, field Group of Lie type Matrix Lie algebra L Chevalley Basis x Β x Α Β F n [, ] x Algorithm Α x Α H x Α Β x Β... Char. 0, p 5: De Graaf, Murray; implemented in GAP, MAGMA Char. 2,3: R., 2009, Implemented in MAGMA

41 The Problems 19 of 30 Normally: Diagonalise L using action of H on L (gives set of ), Use Cartan integers α, β to identify the, Solve easy linear equations. x α x α x Α x Β H x Α Β x Α [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, x Α Β x Β and the Jacobi identity.

42 The Problems 19 of 30 Normally: Diagonalise L using action of H on L (gives set of ), Use Cartan integers α, β to identify the, x α x α Solve easy linear equations. x Α x Β H x Α Β x Α [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, x Α Β x Β and the Jacobi identity.

43 The Problems 19 of 30 Normally: Diagonalise L using action of H on L (gives set of ), Use Cartan integers α, β to identify the, Solve easy linear equations. x α x α x Α x Β H x Α Β x Α [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, x Α Β x Β and the Jacobi identity.

44 Diagonalising (A1, char. 2) 20 of 30 x Α H x Α [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity.

45 Diagonalising (A1, char. 2) 20 of 30 x Α H x Α A Ad 1 : X = Y = Z Φ = {α =1, α = 1}, Φ = {α =2, α = 2}, [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity.

46 Diagonalising (A1, char. 2) 20 of 30 x Α H x Α A Ad 1 : X = Y = Z Φ = {α =1, α = 1}, Φ = {α =2, α = 2}, [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity. L = Fh Fx α Fx α

47 Diagonalising (A1, char. 2) 20 of 30 x Α H x Α A Ad 1 : X = Y = Z Φ = {α =1, α = 1}, Φ = {α =2, α = 2}, [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity. L = Fh Fx α Fx α x α x α h x α 0 e 1, α h α,f 1 x α x α 0 α,f 1 x α h 0

48 Diagonalising (A1, char. 2) 20 of 30 x Α H x Α A Ad 1 : X = Y = Z Φ = {α =1, α = 1}, Φ = {α =2, α = 2}, [h i,h j ] = 0, [x α,h i ] = α,f i x α, [x α,x α ] = n { i=1 e i, α h i, N α,β x α+β if α + β Φ, [x α,x β ] = 0 otherwise, and the Jacobi identity. L = Fh Fx α Fx α x α x α h x α 0 e 1, α h α,f 1 x α x α 0 α,f 1 x α h 0 x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0

49 Diagonalising (A1, char. 2) x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0 21 of 30

50 Diagonalising (A1, char. 2) 21 of 30 x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0 Basis transformation... x = x α x α y = 2x α + x α

51 Diagonalising (A1, char. 2) x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0 21 of 30 Basis transformation... x = x α x α y = 2x α + x α x y h x 0 6h 1 3 x y 4 y 6h 0 3 x y h 1 3 x 2 3 y 4 3 x 1 3 y 0

52 Diagonalising (A1, char. 2) x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0 Basis transformation... x = x α x α y = 2x α + x α x y h x 0 6h 1 3 x y 4 y 6h 0 3 x y h 1 3 x 2 3 y 4 3 x 1 3 y 0 Algorithm: Diagonalize L wrt H Find 1-dim eigenspaces: Take Done! S 1,S 1,S 0 x + y S 1 x 1 2 y S 1 h S 0 21 of 30

53 Diagonalising (A1, char. 2) x α x α h x α 0 2h x α x α 2h 0 x α h x α x α 0 Basis transformation... x = x α x α y = 2x α + x α x y h x 0 6h 1 3 x y 4 y 6h 0 3 x y Algorithm: Diagonalize L wrt H Find 1-dim eigenspaces: Take Done! S 1,S 1,S 0 x + y S 1 x 1 2 y S 1 h S 0 h 1 3 x 2 3 y 4 3 x 1 3 y 0 But if char. is of 30

54 x Β x Α Β x 2Α Β x 3Α Β Diagonalising (G2, char. 3) 22 of 30 x 3Α 2Β x Α H x Α x 3Α Β x 2Α Β x Α Β x Β x 3Α 2Β char. not 3

55 Diagonalising (G2, char. 3) 22 of 30 x 3Α 2Β x 3Α 2Β x Β x Α Β x 2Α Β x 3Α Β x Β x Α Β x 2Α Β x 3Α Β x Α H x Α x Α H x Α x 3Α Β x 2Α Β x Α Β x Β x 3Α Β x 2Α Β x Α Β x Β x 3Α 2Β x 3Α 2Β char. not 3 char. 3

56 Diagonalising (overview) 23 of 30 R(p) Mults Soln R(p) Mults Soln A 2 sc (3) 3 2 [Der] G 2 (3) 1 6, 3 2 [C] A sc,(2) 3 (2) 4 3 [Der] B ad 2 (2) 2 2, 4 [C] B n ad (2) (n 3) 2 n, 4 (n 2) [C] B 2 sc (2) 4, 4 [B 2 sc ] B 3 sc (2) 6 3 [Der] B 4 sc (2) 2 4, 8 3 [Der] B n sc (2) (n 5) 2 n, 4 (n 2) [C] C n ad (2) (n 3) 2n, 2 n(n 1) [C] C sc n (2) (n 3) 2n, 4 (n 2) [B sc 2 ] D (1),(n 1),(n) 4 (2) 4 6 [Der] D sc 4 (2) 8 3 [Der] D (1) n (2) (n 5) 4 2) (n [Der] D n sc (2) (n 5) 4 (n 2) [Der] F 4 (2) 2 12, 8 3 [C] G 2 (2) 4 3 [Der] all remaining(2) 2 Φ+ [A 2 ] Table 1. Multidimensional root spaces

57 Diagonalising (overview) 23 of 30 R(p) Mults Soln R(p) Mults Soln A 2 sc (3) 3 2 [Der] G 2 (3) 1 6, 3 2 [C] A sc,(2) 3 (2) 4 3 [Der] B ad 2 (2) 2 2, 4 [C] B n ad (2) (n 3) 2 n, 4 (n 2) [C] B 2 sc (2) 4, 4 [B 2 sc ] B 3 sc (2) 6 3 [Der] B 4 sc (2) 2 4, 8 3 [Der] B n sc (2) (n 5) 2 n, 4 (n 2) [C] C n ad (2) (n 3) 2n, 2 n(n 1) [C] C sc n (2) (n 3) 2n, 4 (n 2) [B sc 2 ] D (1),(n 1),(n) 4 (2) 4 6 [Der] D sc 4 (2) 8 3 [Der] D (1) n (2) (n 5) 4 2) (n [Der] D n sc (2) (n 5) 4 (n 2) [Der] F 4 (2) 2 12, 8 3 [C] G 2 (2) 4 3 [Der] all remaining(2) 2 Φ+ [A 2 ] Table 1. Multidimensional root spaces

58 Diagonalising (overview) 23 of 30 R(p) Mults Soln R(p) Mults Soln A 2 sc (3) 3 2 [Der] G 2 (3) 1 6, 3 2 [C] A sc,(2) 3 (2) 4 3 [Der] B ad 2 (2) 2 2, 4 [C] B n ad (2) (n 3) 2 n, 4 (n 2) [C] B 2 sc (2) 4, 4 [B 2 sc ] B 3 sc (2) 6 3 [Der] B 4 sc (2) 2 4, 8 3 [Der] B n sc (2) (n 5) 2 n, 4 (n 2) [C] C n ad (2) (n 3) 2n, 2 n(n 1) [C] C sc n (2) (n 3) 2n, 4 (n 2) [B sc 2 ] D (1),(n 1),(n) 4 (2) 4 6 [Der] D sc 4 (2) 8 3 [Der] D (1) n (2) (n 5) 4 2) (n [Der] D n sc (2) (n 5) 4 (n 2) [Der] F 4 (2) 2 12, 8 3 [C] G 2 (2) 4 3 [Der] all remaining(2) 2 Φ+ [A 2 ] Table 1. Multidimensional root spaces

59 Diagonalising (overview) 23 of 30 R(p) Mults Soln R(p) Mults Soln A 2 sc (3) 3 2 [Der] G 2 (3) 1 6, 3 2 [C] A sc,(2) 3 (2) 4 3 [Der] B ad 2 (2) 2 2, 4 [C] B n ad (2) (n 3) 2 n, 4 (n 2) [C] B 2 sc (2) 4, 4 [B 2 sc ] B 3 sc (2) 6 3 [Der] B 4 sc (2) 2 4, 8 3 [Der] B n sc (2) (n 5) 2 n, 4 (n 2) [C] C n ad (2) (n 3) 2n, 2 n(n 1) [C] C sc n (2) (n 3) 2n, 4 (n 2) [B sc 2 ] D (1),(n 1),(n) 4 (2) 4 6 [Der] D sc 4 (2) 8 3 [Der] D (1) n (2) (n 5) 4 2) (n [Der] D n sc (2) (n 5) 4 (n 2) [Der] F 4 (2) 2 12, 8 3 [C] G 2 (2) 4 3 [Der] all remaining(2) 2 Φ+ [A 2 ] Table 1. Multidimensional root spaces

60 Diagonalising (B3, char. 2) 24 of 30

61 Diagonalising (B3, char. 2) 24 of 30

62 Diagonalising (B3, char. 2) 24 of 30

63 Diagonalising (A2, char. 3) 25 of 30 x Β x Α Β Type Eigenspaces Composition x Α H x Α Ad (2,) SC (2,) x Α Β x Β

64 Diagonalising (A2, char. 3) 25 of 30 x Β x Α Β Type Eigenspaces Composition x Α H x Α Ad (2,) SC (2,) x Α Β x Β Observations: There is only one 7, Der(L SC ) = L Ad.

65 Outline 26 of 30 What is a Lie algebra? What is a Chevalley basis? How to compute Chevalley bases? Does it work? What next? Any questions?

66 A tiny demo: A3 / sl4 27 of 30

67 A tiny demo: A3 / sl4 27 of 30

68 A graph 28 of 30 Runtime (s) O(n 10 ) Dn 1000 Cn Bn O(n 6 ) An n

69 Conclusion 29 of 30 Main challenges for computing Chevalley bases in small characteristic: Multidimensional eigenspaces,

70 Conclusion 29 of 30 Main challenges for computing Chevalley bases in small characteristic: Multidimensional eigenspaces, Broken root chains;

71 Conclusion 29 of 30 Main challenges for computing Chevalley bases in small characteristic: Multidimensional eigenspaces, Broken root chains; Found solutions for all cases, and implemented these in MAGMA;

72 Conclusion 29 of 30 Main challenges for computing Chevalley bases in small characteristic: Multidimensional eigenspaces, Broken root chains; Found solutions for all cases, and implemented these in MAGMA; To do: Compute split Cartan subalgebras in small characteristic;

73 Conclusion 29 of 30 Main challenges for computing Chevalley bases in small characteristic: Multidimensional eigenspaces, Broken root chains; Found solutions for all cases, and implemented these in MAGMA; To do: Compute split Cartan subalgebras in small characteristic; Bigger picture: Recognition of groups or Lie algebras, Finding conjugators for Lie group elements, Finding automorphisms of Lie algebras,...

74 Outline 30 of 30 What is a Lie algebra? What is a Chevalley basis? How to compute Chevalley bases? Does it work? What next? Any questions?