Oct 7, 2011 - Why study Lie algebras? ⢠Study groups by their Lie algebras: ⣠Simple algebraic group G unique Lie algebra L. ⣠Many properties carry over ...
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3 Algorithms for Lie Algebras of Algebraic Groups
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Outline
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Recognising Lie algebras 0 11
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Applying new algorithms 01
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“Special” cases
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2 What is a Lie algebra?
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What is a Lie Algebra? Definition (Lie algebra L)
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• • With a multiplication ‣ Bilinear, ‣ Anti-symmetric, and 4 ‣ Satisfies the Jacobi identity 3 0 00 0 0 10 01
Vector space
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that is
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What is a Lie Algebra? Definition (Lie algebra L)
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• • With a multiplication ‣ Bilinear, ‣ Anti-symmetric, and 4 ‣ Satisfies the Jacobi identity 3 0 00 0 0 10 01
Vector space
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that is
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1 ⇠ =
L
0L
⇠ 6=
L 3 /23
Simple Lie algebras
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Classification (Killing, Cartan) algebraically closed and the only simple Lie algebras are: 0 00 0 0 10 01
For
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Why study Lie algebras?
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Many properties carry over to L 0 00
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Easier to calculate in L 0 11
Opportunities for:
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‣ ‣ ‣
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G ≤ Aut(L), often even G = Aut(L) 0 11 1 0
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unique Lie algebra L
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‣ ‣ ‣ ‣
2 Simple algebraic group G 0 00 0 0 10 01
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Study groups by their Lie algebras:
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Recognition,
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3 Solving conjugation problems, ... 1 0 11 01
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Because there are problems to be solved!
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... and there was a thesis to be written... 5 /23
• Example
Matrix Lie algebra: elements are 2x2 matrices of trace 0, called
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Basis:
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2
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Multiplication:
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Definition (Lie algebra L)
✔ • Vector space ✔ • With a multiplication 01
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111
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3
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4 that is
‣✔ Bilinear, and ‣✔ Anti-symmetric, 1 ✔ ‣ Satisfies the Jacobi identity 0 0 11 01
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6 /23
Example
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Basis: Multiplication:
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Matrix Lie algebra: elements are 2x2 matrices of trace 0, called
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The same Lie algebra 01
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1 0
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[·, ·] E F H
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E 0 H 2E
F H 0 2F
H 2E 2F 0 7 /23
Outline
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Recognising Lie algebras 0 11
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111
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Applying new algorithms 01
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1 0
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“Special” cases
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2 What is a Lie algebra?
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The multiplication and the field
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The multiplication and the field
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field 00
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multiplication [ , ]
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The multiplication and the field
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multiplication [ , ]
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The multiplication and the field
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Example, revisited
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[·, ·] E F H
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AAd 1 (F2 ) !
[·, ·] E F H
E 0 0 E
F 0 0 F
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L7
H 0 0 0
⇠ 6=
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01
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1 00 Ad 0 0 10 A1 (Q) 1 1 11 10
[·, ·] E F H
!
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⇠ = 11 01
2E 2F 0
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E 0 H 2E
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[·, ·] E F H
2H
SC A1 (F2 )00000000 0
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sl2 (Q) =
SC A1 (Q)
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L H E F 0 10/23
Outline
• •
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Recognising Lie algebras 0 11
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Applying new algorithms 01
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“Special” cases
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2 What is a Lie algebra?
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Chevalley Bases
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Many Lie algebras have a Chevalley basis! 12/23
Why Chevalley bases? 00
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2 A Chevalley basis for L certifiably states whether 6 L is A , B C D E , E , E , F , G (or a 5 combination thereof). 4 And we can test isomorphism between two Lie 3 isomorphisms!) by computing algebras (and find Chevalley 1 bases. n,
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Why Chevalley bases?
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isomorphic!
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equal
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H xa
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Why Chevalley bases?
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nonisomorphic!
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x a+b
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x -a-b
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x -b
not equal
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x 3a+2b xb
x a+b
x 2a+b x 3a+b H
x -a x-3a-b
xa
x-2a-b
x -a-b
x -b
x-3a-2b
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Computing Chevalley bases Existing algorithms: Often (especially over Q ) 0 00 0 0 10 01
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G2
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4 But sometimes (especially over F ) 3 1 ?? 0 1 00
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New algorithms
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In almost all cases, including characteristic 2: 00
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Algorithms
2 Carefully study the “special cases”,
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Several algorithms per type, 0 11
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Quite “high-level”. 01
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Algorithms
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Magma 2010
Schoonschip 1960’s 19/23
Outline
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Recognising Lie algebras 0 11
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111
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Applying new algorithms 01
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1 0
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“Special” cases
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2 What is a Lie algebra?
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Straightforward applications
2 Recognition of Lie algebras, 0
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Isomorphism testing and construction for Lie algebras, 0 00
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4 Indirectly: matrix group recognition. 3 1 111
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Distance-Transitive Graphs
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One of open cases: H =
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in G = E7(2).
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Use the new algorithms to construct in Magma,
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and search for different H-orbits on Lie(G),
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proving non-existence.
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Long-term effort (Cohen, Liebeck, Saxl) to “classify the graphs on which an almost simple group of exceptional Lie type acts distance-transitively”. 0 00
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Outline
• •
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“Special” cases
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Recognising Lie algebras 0 11
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Applying new algorithms 01
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3 • Thank you! 1 0 0 11 01
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2 What is a Lie algebra?
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