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 / 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 ‣ 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... / 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 Β ‣ A hexagon Α / 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 ...