From Fischer spaces to (Lie) algebras Max Horn joint work with H. - PowerPoint PPT Presentation

From Fischer spaces to (Lie) algebras Max Horn joint work with H. Cuypers, J. in ’t panhuis, S. Shpectorov Technische Universit¨ at Braunschweig Buildings 2010

Overview From Fischer spaces to (Lie) algebras Max Horn 3-transposition groups and Fischer spaces 1 3-transposition groups and Fischer spaces Algebras from Fischer spaces 2 Algebras from Fischer spaces Vanishing sets Vanishing sets 3 Lie algebras Some computations Lie algebras 4 Some computations 5

Overview From Fischer spaces to (Lie) algebras Max Horn 3-transposition groups and Fischer spaces 1 3-transposition groups and Fischer spaces Algebras from Fischer spaces 2 Algebras from Fischer spaces Vanishing sets Vanishing sets 3 Lie algebras Some computations Lie algebras 4 Some computations 5

3-transposition groups From Fischer A class of 3-transpositions in a group G is a conjugacy class D spaces to (Lie) algebras of G such that Max Horn 1 the elements of D are involutions and 3-transposition 2 for all d , e ∈ D the order of de is equal to 1, 2 or 3. groups and Fischer spaces G is called 3-transposition group if G = � D � . Algebras from Fischer spaces Examples Vanishing sets Lie algebras Transpositions in G = Sym( n ); D = (12) G Some Transvections in G = U( n , 2); D = d G where computations 0 0 ... 0 1   0 1 ... 0 0 . . ...  (in GAP’s version of this group) d = . .  . . 0 0 ... 1 0 1 0 ... 0 0 Fi 22 , Fi 23 , Fi 24 (note: the simple group is Fi ′ 24 )

3-transposition groups From Fischer A class of 3-transpositions in a group G is a conjugacy class D spaces to (Lie) algebras of G such that Max Horn 1 the elements of D are involutions and 3-transposition 2 for all d , e ∈ D the order of de is equal to 1, 2 or 3. groups and Fischer spaces G is called 3-transposition group if G = � D � . Algebras from Fischer spaces Examples Vanishing sets Lie algebras Transpositions in G = Sym( n ); D = (12) G Some Transvections in G = U( n , 2); D = d G where computations 0 0 ... 0 1   0 1 ... 0 0 . . ...  (in GAP’s version of this group) d = . .  . . 0 0 ... 1 0 1 0 ... 0 0 Fi 22 , Fi 23 , Fi 24 (note: the simple group is Fi ′ 24 )

3-transposition groups From Fischer A class of 3-transpositions in a group G is a conjugacy class D spaces to (Lie) algebras of G such that Max Horn 1 the elements of D are involutions and 3-transposition 2 for all d , e ∈ D the order of de is equal to 1, 2 or 3. groups and Fischer spaces G is called 3-transposition group if G = � D � . Algebras from Fischer spaces Examples Vanishing sets Lie algebras Transpositions in G = Sym( n ); D = (12) G Some Transvections in G = U( n , 2); D = d G where computations 0 0 ... 0 1   0 1 ... 0 0 . . ...  (in GAP’s version of this group) d = . .  . . 0 0 ... 1 0 1 0 ... 0 0 Fi 22 , Fi 23 , Fi 24 (note: the simple group is Fi ′ 24 )

3-transposition groups From Fischer A class of 3-transpositions in a group G is a conjugacy class D spaces to (Lie) algebras of G such that Max Horn 1 the elements of D are involutions and 3-transposition 2 for all d , e ∈ D the order of de is equal to 1, 2 or 3. groups and Fischer spaces G is called 3-transposition group if G = � D � . Algebras from Fischer spaces Examples Vanishing sets Lie algebras Transpositions in G = Sym( n ); D = (12) G Some Transvections in G = U( n , 2); D = d G where computations 0 0 ... 0 1   0 1 ... 0 0 . . ...  (in GAP’s version of this group) d = . .  . . 0 0 ... 1 0 1 0 ... 0 0 Fi 22 , Fi 23 , Fi 24 (note: the simple group is Fi ′ 24 )

Classification of 3-transpositions groups From Fischer spaces to (Lie) algebras Max Horn Fischer (around 1970) classified finite 3-transposition 3-transposition groups with no non-trivial normal solvable subgroups. groups and Fischer spaces � classification of finite simple groups Algebras from Fischer spaces Cuypers and Hall (90s) classified all (possibly infinite) Vanishing sets 3-transposition groups with trivial center, using geometric Lie algebras methods (Fischer spaces). Some computations Cuypers and Hall: If center is non-trivial, then G / Z ( G ) is 3-transposition group with trivial center.

Classification of 3-transpositions groups From Fischer spaces to (Lie) algebras Max Horn Fischer (around 1970) classified finite 3-transposition 3-transposition groups with no non-trivial normal solvable subgroups. groups and Fischer spaces � classification of finite simple groups Algebras from Fischer spaces Cuypers and Hall (90s) classified all (possibly infinite) Vanishing sets 3-transposition groups with trivial center, using geometric Lie algebras methods (Fischer spaces). Some computations Cuypers and Hall: If center is non-trivial, then G / Z ( G ) is 3-transposition group with trivial center.

Classification of 3-transpositions groups From Fischer spaces to (Lie) algebras Max Horn Fischer (around 1970) classified finite 3-transposition 3-transposition groups with no non-trivial normal solvable subgroups. groups and Fischer spaces � classification of finite simple groups Algebras from Fischer spaces Cuypers and Hall (90s) classified all (possibly infinite) Vanishing sets 3-transposition groups with trivial center, using geometric Lie algebras methods (Fischer spaces). Some computations Cuypers and Hall: If center is non-trivial, then G / Z ( G ) is 3-transposition group with trivial center.

Fischer spaces From Fischer spaces to (Lie) algebras Max Horn Throughout the rest of this talk, let D be a class of 3-transposition 3-transpositions generating a 3-transposition group G , and groups and Fischer spaces Z ( G ) = 1. Algebras from Fischer spaces ⇒ d � = d e = e d � = e Vanishing sets o ( de ) = 3 ⇐ ⇒ de � = ed ⇐ Lie algebras Some The Fischer space Π( D ) is the partial linear space with D computations as point set, and the triples { d , e , d e } as lines (when o ( de ) = 3).

Fischer spaces From Fischer spaces to (Lie) algebras Max Horn Throughout the rest of this talk, let D be a class of 3-transposition 3-transpositions generating a 3-transposition group G , and groups and Fischer spaces Z ( G ) = 1. Algebras from Fischer spaces ⇒ d � = d e = e d � = e Vanishing sets o ( de ) = 3 ⇐ ⇒ de � = ed ⇐ Lie algebras Some The Fischer space Π( D ) is the partial linear space with D computations as point set, and the triples { d , e , d e } as lines (when o ( de ) = 3).

Fischer spaces From Fischer spaces to (Lie) algebras Max Horn Throughout the rest of this talk, let D be a class of 3-transposition 3-transpositions generating a 3-transposition group G , and groups and Fischer spaces Z ( G ) = 1. Algebras from Fischer spaces ⇒ d � = d e = e d � = e Vanishing sets o ( de ) = 3 ⇐ ⇒ de � = ed ⇐ Lie algebras Some The Fischer space Π( D ) is the partial linear space with D computations as point set, and the triples { d , e , d e } as lines (when o ( de ) = 3).

Characterizing Fischer spaces From Fischer spaces to (Lie) algebras Max Horn Proposition (Buekenhout) 3-transposition A partial linear space is a Fischer space if and only if every pair groups and Fischer spaces of intersecting lines generates a subspace isomorphic to the Algebras from dual of an affine plane of order 2, or an affine plane of order 3. Fischer spaces Vanishing sets Lie algebras Some computations ( F 2 F 2 2 ) dual � 3 �

Overview From Fischer spaces to (Lie) algebras Max Horn 3-transposition groups and Fischer spaces 1 3-transposition groups and Fischer spaces Algebras from Fischer spaces 2 Algebras from Fischer spaces Vanishing sets Vanishing sets 3 Lie algebras Some computations Lie algebras 4 Some computations 5

Algebras from Fischer spaces From Fischer spaces to (Lie) algebras Denote by F 2 D the F 2 vector space with basis D . Max Horn Vectors are finite subsets of D ; sum of two sets is their 3-transposition groups and symmetric difference. Fischer spaces Algebras from Fischer spaces Define the 3-transposition algebra A ( D ) with underlying Vanishing sets vector space F 2 D ; multiplication is linear expansion of Lie algebras multiplication defined on d , e ∈ D by Some computations d + e + e d = { d , e , e d } � if o ( de ) = 3 d ∗ e := 0 otherwise. A ( D ) is a non-associative commutative algebra.

Algebras from Fischer spaces From Fischer spaces to (Lie) algebras Denote by F 2 D the F 2 vector space with basis D . Max Horn Vectors are finite subsets of D ; sum of two sets is their 3-transposition groups and symmetric difference. Fischer spaces Algebras from Fischer spaces Define the 3-transposition algebra A ( D ) with underlying Vanishing sets vector space F 2 D ; multiplication is linear expansion of Lie algebras multiplication defined on d , e ∈ D by Some computations d + e + e d = { d , e , e d } � if o ( de ) = 3 d ∗ e := 0 otherwise. A ( D ) is a non-associative commutative algebra.