Construction of Chevalley Bases of Lie Algebras

Dan Roozemond Joint work with Arjeh M. Cohen

May 14th, 2009, Computeralgebratagung, Universität Kassel

http://www.win.tue.nl/~droozemo/ / department of mathematics and computer science 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?

/ department of mathematics and computer science What is a Lie Algebra? 3 of 30

n ‣ Vector space: F

/ department of mathematics and computer science What is a Lie Algebra? 3 of 30

n ‣ Vector space: F ‣ Multiplication [ , ]: L L L that is · · × "→ • Bilinear,

• Anti-symmetric,

• Satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z,[x, y]] = 0

/ department of mathematics and computer science What is a Lie Algebra? 3 of 30

n ‣ Vector space: F ‣ Multiplication [ , ]: L L L L that is · · × "→ • Bilinear, n • Anti-symmetric,F [ , ] • Satisfies Jacobi identity: · · [x, [y, z]] + [y, [z, x]] + [z,[x, y]] = 0

/ department of mathematics and computer science 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 1) E6, E7, E8 n ≥ Bn (n 2) F ≥ 4 Cn (n 3) G ≥ 2 D (n 4) n ≥

/ department of mathematics and computer science 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)

/ department of mathematics and computer science 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 • ...

/ department of mathematics and computer science Why Study Lie Algebras? 5 of 30

/ department of mathematics and computer science 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!

/ department of mathematics and computer science Root Systems 7 of 30

‣ A hexagon

/ department of mathematics and computer science Root Systems 7 of 30

‣ A hexagon

/ department of mathematics and computer science Root Systems 7 of 30

‣ A hexagon

/ department of mathematics and computer science Root Systems 7 of 30

Β Α#Β

‣ A hexagon $Α ‣ A root system of type A2 Α

$Α$Β $Β

/ department of mathematics and computer science Root Data 8 of 30

Definition (Root Datum)

R =(X, Φ,Y, Φ∨), , : X Y Z !· ·" × →

/ department of mathematics and computer science Root Data 8 of 30

Definition (Root Datum)

R =(X, Φ,Y, Φ∨), , : X Y Z !· ·" × →

‣ X, Y: dual free Z -modules, ‣ put in duality by , , !· ·" ‣ Φ X : roots, ⊆ ‣ Φ ∨ Y : coroots. ⊆

/ department of mathematics and computer science Root Data 8 of 30

Definition (Root Datum)

R =(X, Φ,Y, Φ∨), , : X Y Z !· ·" × →

‣ X, Y: dual free Z -modules, ‣ put in duality by , , !· ·" ‣ Φ X : roots, ⊆ ‣ Φ ∨ Y : coroots. ⊆ Several Root Data: One Root System “adjoint” ... { “simply connected”

/ department of mathematics and computer science Root Data 9 of 30

Definition (Root Datum)

R =(X, Φ,Y, Φ∨), , : X Y Z !· ·" × →

Several Root Data: One Root System “adjoint” ... { “simply connected”

Irreducible Root Data: An· , Bn· , Cn· , Dn· , E6· , E7· , E8· , F4· , G2· .

/ department of mathematics and computer science Root Data 10 of 30

Β Α#Β

‣ A hexagon $Α ‣ A root system of type A2 Α

$Α$Β $Β

/ department of mathematics and computer science Root Data 10 of 30

xΒ Β Α#x ΑΒ#Β

‣ A hexagon $xΑ ‣ A root system of type A2 $Α H

Αx Α ‣ A Lie algebra of type A2

$Αx$$ΒΑ$Β $xΒ$Β

/ department of mathematics and computer science Chevalley Basis 11 of 30

Definition (Chevalley Basis)

Formal basis: L = Fhi Fxα ⊕ i=1,...,n α Φ ! !∈ Bilinear anti-symmetric multiplication satisfies ( i, j 1 , . . . , n ; α , β Φ ) : ∈ { } ∈ [hi,hj]=0,

[xα,hi]= α,fi xα, ! n " [x α,xα]= ei, α∨ hi, − i=1! " Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ " 0 otherwise, and the Jacobi identity.

/ department of mathematics and computer science Chevalley Basis 11 of 30

Definition (Chevalley Basis)

Formal basis: L = Fhi Fxα ⊕ i=1,...,n α Φ ! !∈ Bilinear anti-symmetric multiplication satisfies ( i, j 1 , . . . , n ; α , β Φ ) : ∈ { } ∈ [hi,hj]=0, x x Β Α#Β [xα,hi]= α,fi xα, ! n " [x α,xα]= ei, α∨ hi, − i=1! " Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ x $Α H " 0 otherwise,

x Α and the Jacobi identity.

x$Α$Β x $Β

/ department of mathematics and computer science 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.

/ department of mathematics and computer science Why? 13 of 30

L n [ , ] F · ·

/ department of mathematics and computer science Why? 13 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

/ department of mathematics and computer science Why? 13 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · ·

x$Α$Β x $Β

· ]

· ,

[

L n F

/ department of mathematics and computer science Why? 13 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · ·

x$Α$Β x $Β

· x x ] Β Α#Β

· ,

[ x $Α H

x Α

L n F

x$Α$Β x $Β

/ department of mathematics and computer science Why? 13 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

equal

· x x ] Β Α#Β

· ,

[ x $Α H

x Α

L n F

x$Α$Β x $Β

/ department of mathematics and computer science Why? 13 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

isomorphic! equal

· x x ] Β Α#Β

· ,

[ x $Α H

x Α

L n F

x$Α$Β x $Β

/ department of mathematics and computer science Why? 14 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

/ department of mathematics and computer science Why? 14 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

L n [ , ] F · ·

/ department of mathematics and computer science Why? 14 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

x3Α#2Β

x Β x Α#Β x2Α#Β L x3Α#Β H n [ , ] x $Α x Α F · · x$3Α$Β x$2Α$Β x$Α$Β x $Β

x$3Α$2Β

/ department of mathematics and computer science Why? 14 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

not equal

x3Α#2Β

x Β x Α#Β x2Α#Β L x3Α#Β H n [ , ] x $Α x Α F · · x$3Α$Β x$2Α$Β x$Α$Β x $Β

x$3Α$2Β

/ department of mathematics and computer science Why? 14 of 30

L x Β x Α#Β

x $Α H n [ , ] x Α F · · x$Α$Β x $Β

non-isomorphic! not equal

x3Α#2Β

x Β x Α#Β x2Α#Β L x3Α#Β H n [ , ] x $Α x Α F · · x$3Α$Β x$2Α$Β x$Α$Β x $Β

x$3Α$2Β

/ department of mathematics and computer science 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?

/ department of mathematics and computer science 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

x Β x Α#Β Group of Lie type L “Chevalley Basis x $Α H n x Α Matrix Lie F [ , ] Algorithm” algebra · · x$Α$Β x $Β

....

/ department of mathematics and computer science The Mission 17 of 30

Root datum, field

x Β x Α#Β Group of Lie type L “Chevalley Basis x $Α H n x Α Matrix Lie F [ , ] Algorithm” algebra · · x$Α$Β x $Β

....

‣ Assume splitting Cartan subalgebra H is given (Cohen/Murray, indep. Ryba);

‣ Assume root datum R is given

/ department of mathematics and computer science The Mission 18 of 30

Root datum, field

x Β x Α#Β Group of Lie type L “Chevalley Basis x $Α H n x Α Matrix Lie F [ , ] Algorithm” algebra · · x$Α$Β x $Β

....

‣ Char. 0, p ≥ 5: De Graaf, Murray; implemented in GAP, MAGMA

/ department of mathematics and computer science The Mission 18 of 30

Root datum, field

x Β x Α#Β Group of Lie type L “Chevalley Basis x $Α H n x Α Matrix Lie F [ , ] Algorithm” algebra · · x$Α$Β x $Β

....

‣ Char. 0, p ≥ 5: De Graaf, Murray; implemented in GAP, MAGMA

‣ Char. 2,3: R., 2009, Implemented in MAGMA

/ department of mathematics and computer science The Problems 19 of 30 Normally:

‣ Diagonalise L using action of H on L (gives set of x α ),

‣ Use Cartan integers α , β to “identify” the x , ! " α ‣ Solve easy linear equations.

x Β x Α#Β [hi,hj]=0,

[xα,hi]= α,fi xα, ! n " x $Α [x α,xα]= ei, α∨ hi, H − i=1! " x Α Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ " 0 otherwise, and the Jacobi identity. x$Α$Β x $Β

/ department of mathematics and computer science The Problems 19 of 30 Normally:

‣ Diagonalise L using action of H on L (gives set of x α ),

‣ Use Cartan integers α , β to “identify” the x , ! " α ✓ ‣ Solve easy linear equations.

x Β x Α#Β [hi,hj]=0,

[xα,hi]= α,fi xα, ! n " x $Α [x α,xα]= ei, α∨ hi, H − i=1! " x Α Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ " 0 otherwise, and the Jacobi identity. x$Α$Β x $Β

/ department of mathematics and computer science The Problems 19 of 30 Normally:

✗ ‣ Diagonalise L using action of H on L (gives set of x α ),

✗ ‣ Use Cartan integers α , β to “identify” the x α , ! " ✓ ‣ Solve easy linear equations.

x Β x Α#Β [hi,hj]=0,

[xα,hi]= α,fi xα, ! n " x $Α [x α,xα]= ei, α∨ hi, H − i=1! " x Α Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ " 0 otherwise, and the Jacobi identity. x$Α$Β x $Β

/ department of mathematics and computer science Diagonalising (A1, char. 2) 20 of 30

x !Α H x Α

[hi,hj]=0,

[xα,hi]= α,fi xα, ! n " [x α,xα]= ei, α∨ hi, − i=1! " Nα,βxα+β if α + β Φ, [xα,xβ]=! ∈ " 0 otherwise, and the Jacobi identity.

/ department of mathematics and computer science Diagonalising (A1, char. 2) 20 of 30

x !Α H x Α

[hi,hj]=0,

Ad [xα,hi]= α,fi xα, A1 : X = Y = Z ! n " [x α,xα]= ei, α∨ hi, Φ = α =1, α = 1 , − i=1! " Nα,βxα+β if α + β Φ, { − − } [xα,xβ]=! ∈ 0 otherwise, Φ∨ = α∨ =2, α∨ = 2 , " { − − } and the Jacobi identity.

/ department of mathematics and computer science Diagonalising (A1, char. 2) 20 of 30

x !Α H x Α

[hi,hj]=0,

L = Fh Fxα Fx α ⊕ ⊕ −

/ department of mathematics and computer science Diagonalising (A1, char. 2) 20 of 30

x !Α H x Α

[hi,hj]=0,

L = Fh Fxα Fx α ⊕ ⊕ −

xα x α h − x 0 e , α∨ h α,f x α ! 1 " ! 1" α x α 0 α,f1 x α −h !− 0 " −

/ department of mathematics and computer science Diagonalising (A1, char. 2) 20 of 30

x !Α H x Α

[hi,hj]=0,

L = Fh Fxα Fx α ⊕ ⊕ −

xα x α h xα x α h − − x 0 e , α∨ h α,f x xα 0 2hxα α ! 1 " ! 1" α − x α 0 α,f1 x α x α 2h 0 x α − !− " − − − − h 0 h xα x α 0 − −

/ department of mathematics and computer science Diagonalising (A1, char. 2) 21 of 30

xα x α h − x 0 2hx α − α x α 2h 0 x α − − − h xα x α 0 − −

/ department of mathematics and computer science Diagonalising (A1, char. 2) 21 of 30

xα x α h − x 0 2hx α − α x α 2h 0 x α − − − h xα x α 0 − −

Basis transformation.... x = xα x α − − y =2xα + x α −

/ department of mathematics and computer science Diagonalising (A1, char. 2) 21 of 30

xα x α h − x 0 2hx α − α x α 2h 0 x α − − − h xα x α 0 − −

Basis transformation.... x = xα x α − − y =2xα + x α −

x y h 1 2 x 0 6h 3 x + 3 y − −4 1 y 6h 0 3 x + 3 y h 1 x 2 y 4 x 1 y 0 3 − 3 − 3 − 3

/ department of mathematics and computer science Diagonalising (A1, char. 2) 21 of 30

xα x α h − x 0 2hx α − α x α 2h 0 x α Algorithm: − − − h xα x α 0 ‣ Diagonalize L wrt H − − ‣ Find 1-dim eigenspaces: Basis transformation.... S1,S 1,S0 − x = xα x α − − ‣ Take y =2xα + x α − x + y S1 1 ∈ x y S 1 x y h − 2 ∈ − h S0 1 2 ∈ x 0 6h 3 x + 3 y − −4 1 ‣ Done! y 6h 0 3 x + 3 y h 1 x 2 y 4 x 1 y 0 3 − 3 − 3 − 3

/ department of mathematics and computer science Diagonalising (A1, char. 2) 21 of 30

/ department of mathematics and computer science Diagonalising (G2, char. 3) 22 of 30

x3Α#2Β

x Β x Α#Β x2Α#Β x3Α#Β

H x $Α x Α

x$3Α$Β x$2Α$Β x$Α$Β x $Β

x$3Α$2Β

char. not 3

/ department of mathematics and computer science Diagonalising (G2, char. 3) 22 of 30

x3Α#2Β x3Α#2Β

x Β x Α#Β x2Α#Β x Β x Α#Β x2Α#Β x3Α#Β x3Α#Β

H H x $Α x Α x $Α 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

/ department of mathematics and computer science Diagonalising (overview) 23 of 30 COMPUTING CHEVALLEY BASES IN SMALL CHARACTERISTICS 7

R(p) Mults Soln R(p) Mults Soln sc 2 ad n(n 1) A2 (3) 3 [Der] Cn (2) (n 3) 2n, 2 − [C] ≥ n G (3) 16, 32 [C] C sc(2) (n 3) 2n, 4(2) [B sc] 2 n ≥ 2 sc,(2) 3 (1),(n 1),(n) 6 A3 (2) 4 [Der] D4 − (2) 4 [Der] ad 2 sc 3 B2 (2) 2 , 4 [C] D4 (2) 8 [Der] ad n (n) (1) (n) Bn (2) (n 3) 2 , 4 2 [C] Dn (2) (n 5) 4 2 [Der] ≥ ≥ n sc sc sc ( ) B2 (2) 4, 4 [B2 ] D (2) (n 5) 4 2 [Der] n ≥ sc 3 12 3 B3 (2) 6 [Der] F4(2) 2 , 8 [C] sc 4 3 3 B4 (2) 2 , 8 [Der] G2(2) 4 [Der] n + B sc(2) (n 5) 2n, 4(2) [C] all remaining(2) 2 Φ [A ] n ≥ | | 2 Table 1. Multidimensional root spaces

trivial,/ department meaning of mathematics that and computer= scienceFx x E H is the required result. The remaining cases are identified byX Proposition{ | 3,∈ and\ the} algorithms for these cases are indicated sc by [A2], [C], [Der], [B2 ] in Table 1 and explained in Section 3. In IdentifyRoots we compute Cartan integers and use these to make the identification ι between the root system Φ of R and the Chevalley frame computed previously. This identification is again made on a case-by-case basis dependingX on the root datum R. See Section 4 for details. The algorithm ends with ScaleToBasis where the vectors Xα (α Φ) be- longing to members of the Chevalley frame are picked in such a∈ way that 0 X X =(Xα)α Φ is part of a Chevalley basis with respect to H and R, and a suitable 0 ∈ basis H = h1, . . . , hn of H is computed, so that they satisfy the Chevalley basis multiplication{ rules.} This step involves the solving of several systems of linear 8 equations, similar to the procedure explained in [6], which takes time O∼(n log(q)). Finally, in Section 5, we finish the proof of Theorem 1 and discuss some further problems for which our algorithm may be of use.

2. Multidimensional root spaces In this section we prove Proposition 3, but first we explain the notation in Table 1. As already mentioned, the first column contains the root datum R specified by means of the Dynkin type with a superscript for the isogeny type, as well as (between parentheses) the characteristic p. A root datum of type A3 can have any of three isogeny types: adjoint, simply connected, or an intermediate one, corresponding to the subgroup of order 1, 4, and 2 of its fundamental group Z/4Z, (2) respectively. We denote the intermediate type by A3 . For computations we fix root and coroot matrices for each isomorphism class of root data, as indicated at the end of Section 1.2. For A3, for example, the Cartan matrix is 2 10 − C = 12 1 . −0 12−  −   Diagonalising (overview) 23 of 30 COMPUTING CHEVALLEY BASES IN SMALL CHARACTERISTICS 7

/ department of mathematics and computer science Diagonalising (B3, char. 2) 24 of 30

/ department of mathematics and computer science Diagonalising (A2, char. 3) 25 of 30

x Β x Α#Β

Type Eigenspaces Composition x $Α H 1 Ad (2,) 16 x Α 7 7 SC (2,) 32 1 x$Α$Β x $Β

/ department of mathematics and computer science Diagonalising (A2, char. 3) 25 of 30

x Β x Α#Β

Type Eigenspaces Composition x $Α H 1 Ad (2,) 16 x Α 7 7 SC (2,) 32 1 x$Α$Β x $Β

Observations: ‣ There is only one “7”,

‣ Der(LSC) = LAd.

/ department of mathematics and computer science 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?

/ department of mathematics and computer science A tiny demo: A3 / sl4 27 of 30

/ department of mathematics and computer science A graph 28 of 30

100000 O(n10)

10000 Runtime (s) Dn

Cn 1000 Bn

100 O(n6) 10 An

0.1

0.01 3 4 5 6 7 8 9

/ department of mathematics and computer science

x Rank An Bn Cn Dn O(n^6) O(n^10) 3 10.8 0.7 0.4 0.1 0.3 0.0243 0.59049 4 19.2 0.1 1.9 1.1 3.2 0.1365 10.48576

x Rank An Bn Cn Dn O(n^6) O(n^10) 5 30 0.2 4.8 10 22 0.5208 97.65625 6 43.2 0.6 20 40 121 1.5552 604.66176 7 58.8 1.5 54 172 545 3.9216 2824.7525 8 76.8 3.6 172 693 1994 8.7381 10737.418 9 97.2 7.9 493 2212 6396 17.715 34867.844 Conclusion 29 of 30

‣ Main challenges for computing Chevalley bases in small characteristic: • Multidimensional eigenspaces,

/ department of mathematics and computer science Conclusion 29 of 30

‣ Main challenges for computing Chevalley bases in small characteristic: • Multidimensional eigenspaces, • Broken root chains;

/ department of mathematics and computer science 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, • ...

/ department of mathematics and computer science 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?

/ department of mathematics and computer science