Geometries from structurablealgebras and inner ideals

Jeroen MeulewaeterNovember 5, 2019Malaga University

Content

1 Structurable algebras• Associated Lie algebra• Inner ideals• Structurable division algebras

2 Construction of some low-rank geometries• Moufang sets• Moufang hexagons• Moufang triangles

3 Extremal geometries• Original definition and results• Extended definition

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Credits

◮ Most work is joint with Tom De Medts (Ghent University),my supervisor.

◮ Work on extremal geometries is joint with Hans Cuypers(Eindhoven University of Technology).

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Overview/motivation

◮ Structurable algebras are a class of non-associative,non-commutative algebras.

◮ Boelaert, De Medts, Stavrova, ’19: Connection betweenstructurable division algebras and so-called Moufang sets(”geometries of rank 1”).

◮ Cohen, Ivanyos, ’06: 1-dimensional inner ideals in a Liealgebra are related to point-line geometries (”extremalgeometries”).

◮ Tits-Kantor-Koecher-procedure, ’78: We can associate aLie algebra to a structurable algebra.

◮ Question: Constructions for higher rank geometries?◮ Question: Relation with extremal geometries?◮ Ingredient: Inner ideals!

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Structurable algebras: what are these?1

AssumptionAll algebras are finite-dimensional and defined over a field k ofcharacteristic different from 2 and 3! Moreover, we do notassume them to be associative nor commutative.

Definition.Consider an algebra A with involution. (I.e., xy = y .x)Set

Vx ,y z = (xy)z + (zy)x − (zx)y .

Then we call A structurable if

[Vx ,y , Va,b] = VVx,y a,b − Va,Vy,x b.

Hence 〈Vx ,y | x , y ∈ A〉 ≤ End(A) is a Lie subalgebra of gl(A).•••• •••••••••••••••• •••••••••••••••••• •••••••• 5

Jordan algebras1

◮ Assume that the involution is the identity.◮ Then xy = xy = y · x = yx . I.e., A is commutative.◮ Moreover, one can show that A is a Jordan algebra. (So

satisfying (xy)x2 = x(yx2))◮ Example: A associative, then A+ = (A, ◦) Jordan, with

a ◦ b = (ab + ba)/2.◮ Example: Jordan algebra associated to a quadratic form.◮ Exceptional example: 27-dimensional Albert algebra,

constructed from octonions.

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Classification1

Theorem.If A is a central simple structurable algebra, it is isomorphic to(at least) one of the following:◮ Central simple associative algebra with involution.◮ Central simple Jordan algebra.◮ Hermitian type.◮ Central simple structurable algebra of skew-dimension one.◮ Forms of tensor product of two composition algebras.◮ Smirnov algebra.

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Classification1

The skew-dimension is dim(S), with

S := {a ∈ A | a = −a}.

Classification due to:

◮ Characteristic 0: Allison, ’79.◮ Characteristic ∕= 2, 3, 5: Smirnov, ’91.◮ Characteristic = 5: Boelaert, De Medts, Stavrova, ’19,

together with separate article of Stavrova, ’19.

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Tensor product of composition algebras1

Definition.If Ci is a composition algebra over k with involution σi , fori = 1, 2, then the k-algebra C1 ⊗k C2, together with theinvolution

− = σ := σ1 ⊗ σ2

is a structurable algebra.

◮ One notes S = {s1 ⊗ 1 + 1 ⊗ s2 | si ∈ Si}, with Si the setof skew elements w.r.t. the involution σi . In particular, it isof dimension 7, 8, 10 or 14 if C1 is octonion.

◮ There only exist forms if dim(C1) = dim(C2).

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Skew-dimension one1

Let J be a Jordan algebra, T : J × J → k be a symmetricbilinear form, × : J × J → J be a symmetric bilinear map, andN : J → k be a cubic form such that one of the following holds:

◮ J is a cubic Jordan algebra with a non-degenerateadmissible form N, with basepoint 1, trace form T , andFreudenthal cross product ×.

◮ J is a Jordan algebra of a non-degenerate quadratic form qwith basepoint 1, and T is the linearization of q. In thiscase, N and × are the zero maps.

◮ J = 0, and the maps N, T and × are the zero maps. Inthis case, J is not unital.

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Skew-dimension one1

LetA =

!"k1 j1j2 k2

#

| k1, k2 ∈ k, j1, j2 ∈ J$

,

and define the multiplication and the involution by"

k1 j1j2 k2

# "k ′

1 j ′1

j ′2 k ′

2

#

="

k1k ′1 + T (j1, j ′

2) k1j ′1 + k ′

2j1 + j2 × j ′2

k ′1j2 + k2j ′

2 + j1 × j ′1 k2k ′

2 + T (j2, j ′1)

#

,

"k1 j1j2 k2

#

="

k2 j1j2 k1

#

,

for all k1, k2, k ′1, k ′

2 ∈ k and j1, j2, j ′1, j ′

2 ∈ J .

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Skew-dimension one1

◮ This is called a matrix structurable algebra, denoted byM(J , 1).

◮ Similar construction for M(J , η), η ∈ k×.◮ Assume A skew-dimension one.◮ Consider 0 ∕= s0 ∈ S, then s2

0 = µ · 1, for some µ ∈ k×.◮ Allison-Faulkner ’84: A ∼= M(J , 1) ⇐⇒ µ is a square.◮ In particular: There exists a quadratic field extension m/k

such that A ⊗m ∼= M(J , 1).◮ Example cubic Jordan algebra: Any Albert algebra.◮ Other example: Hermitian (3 × 3)-matrices over a

composition algebra. These have dimension 6, 9, 15 or 27.

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TKK Lie algebras1

◮ Recall that Instrl(A) := 〈Vx ,y | x , y ∈ A〉 ≤ End(A) is a Liesubalgebra of gl(A).

◮ If A is a structurable algebra, then

K (A) = S− ⊕ A− ⊕ Instrl(A) ⊕ A+ ⊕ S+

is a 5-graded Lie algebra. Denote its i-th component by Li .◮ So [Li , Lj ] ≤ Li+j .◮ A+ and A− are two copies of A.◮ E.g.: [x+, y−] = Vx ,y .◮ E.g.: [s+, t−] = LsLt .◮ E.g.: [Vx ,y , z+] = (Vx ,y z)+.

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Exceptional Lie algebras1

Definition.A Lie algebra L is of (absolute) type Xn if L ⊗ k is of type Xn.

Some examples:

Example.We get the Freudenthal-Tits-magic square for the tensorproduct of two composition algebras:

dim(Ci) 1 2 4 81 A1 A2 C3 F42 A2 A2 × A2 A5 E64 C3 A5 D6 E78 F4 E6 E7 E8

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Exceptional Lie algebras1

Example.If J is a cubic Jordan algebra, we get (at least if char(k) = 0):

dim(J) dim(M(J , 1)) type of K (M(J , 1))6 14 F49 20 E615 32 E727 56 E8

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Inner ideals1

Definition.An inner ideal of a Lie algebra L is a subspace I such that[I, [I, L]] ≤ I. If I is 1-dimensional, any non-zero element of I iscalled extremal.

Any ideal is an inner ideal, 0 and L as well.

ExampleFor L = sl2 we have a basis {e, f , h} with [e, f ] = h, [h, e] = 2eand [h, f ] = −2f . Then e is extremal by[e, [e, h]] = 0 = [e, [e, e]] and [e, [e, f ]] = [e, h] = −2e.

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Canonical inner ideals1

◮ [Li , [Li , Lj ]] ≤ L2i+j .◮ S+ will always be an inner ideal!◮ If S = 0, then A+ will always be an inner ideal.◮ If A is central simple, any proper inner ideal I ≤ K (A)

satisfies [I, I] = 0.◮ A Lie algebra automorphism maps inner ideals on to inner

ideals.

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Some automorphisms1

◮ For any a ∈ A and s ∈ S we get ad(a+ + s+)5 = 0.◮ If A is central simple

e+(a, s) := exp(ad(a+ + s+)) =4%

i=0

1i! ad(a+ + s+)i

is a Lie algebra automorphism. (Quite delicate incharacteristic 5!)

◮ Set E+(A) = {e+(a, s) | s ∈ S, a ∈ A} and similarlyE−(A).

◮ E (A) is the subgroup of Aut(L) generated by E+(A) andE−(A).

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Structurable division algebras1

We define the operator Ux by Ux (y) = Vx ,y x .

Definition.An element u ∈ A is called conjugate invertible if there existsu ∈ A such that Vu,u = Id. (This implies Uu invertible.)The structurable algebra A is called division if every non-zeroelement of A is conjugate invertible.

Some examples:

◮ Associative division algebra with involution. Then u = u−1.◮ Albert division algebra.◮ Jordan algebra associated with anisotropic quadratic form.

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Some more exceptional division examples1

◮ Octonion division algebra.◮ Consider a quartic field extension whose splitting field has

Galois group Alt(4) or Sym(4). Then one can apply ageneralized Cayley-Dickson process to this algebra to obtaina skew-dimension one structurable division algebra (ofdimension 8).

◮ Exceptional cases: quite hard to determine explicitly!

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Inner ideals of K (A)2

Theorem (De Medts - M.).Let J be a central simple Jordan division algebra.Then any non-trivial proper inner ideal of K (J) distinct from J−equals e−(j)(J+) for a unique j ∈ J .

Note: sl2 = K (k).

Theorem (De Medts - M.).Let A be a central simple structurable division algebra withS ∕= 0.Then any non-trivial proper inner ideal of K (A) distinct fromS− equals e−(a, s)(S+) for unique a ∈ A and s ∈ S.

In particular: all inner ideals have the same dimension.

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Associated geometry2

DefinitionLet X be a set and {Ux | x ∈ X} a collection of subgroups ofSym(X ). The data (X , {Ux }x∈X ) is a Moufang set if:◮ For each x ∈ X , Ux fixes x and acts sharply transitively on

X\{x}.◮ For each g ∈ G := 〈Ux | x ∈ X 〉 ≤ Sym(X ) and each

y ∈ X we have g−1Uy g = Uy .g .

Corollary of previous result: If A is central simple division, theset of non-trivial proper inner ideals of K (A) is a Moufang set.

Note: US− = E−(A).

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Skew-dimension one structurable algebras2

◮ Recall the skew-dimension one structurable algebraM(J) := M(J , 1) with as elements

"a ij b

#

,

with a, b ∈ k and i , j ∈ J .◮ Assumption: J is a cubic Jordan division algebra.

(N(j) = 0 implies j = 0)◮ Example of J : an Albert division algebra.◮ Other example: k with cubic form N(a) = a3.

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Reduction2

Lemma (De Medts - M.).Any non-trivial inner ideal of K (M(J)) is the image of an innerideal containing S+ under an element of E (A).

◮ S+ is 1-dimensional.◮ What are the inner ideals containing S+?◮ The inner ideals of a structurable algebra A are the

subspaces I satisfying UI(A) ≤ I.◮ In this case all non-trivial proper inner ideals of M(J) are

1-dimensional and form a Moufang set.

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A tale of two types of inner ideals2

Lemma (De Medts - M.).The only non-trivial inner ideals properly containing S+ areS+ ⊕ 〈a+〉, with 〈a〉 an inner ideal.

◮ Each non-trivial proper inner ideal is 1 or 2-dimensional.◮ If its dimension is 2, all subspaces are inner ideals.◮ In particular: the Moufang set in M(J) embeds in the

lattice of inner ideals of K (M(J)).◮ What geometry does this lattice of inner ideals form?◮ First, we have to introduce some incidence-geometry

concepts!

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Point-line geometries2

Definition.A point-line geometry is a tuple (P, L), with P a set, called theset of points, and L a subset of 2P , called the set of lines.

We say that a point p ∈ P is on a line l ∈ L if p ∈ l .

Definition.We say that (P, L) is a partial linear space if for any twodistinct p1, p2 ∈ P there exists at most one line containing bothp1 and p2 and similarly, for any two distinct lines, there is atmost one point on both lines.

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Ordinary polygons2

Example.An ordinary n-gon.Explicitly: P = {1, . . . , n}, L = {{1, 2}, {2, 3}, . . . , {n, 1}}.

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Generalized polygons2Let n ≥ 3. A generalized n-gon is a point-line geometry suchthat

◮ It is a partial linear space.◮ It does not have ordinary m-gons as subgeometries, for all

2 < m < n.◮ It does have an ordinary n-gon as subgeometry.◮ For any two points p1, p2, there exists an ordinary n-gon as

subgeometry, containing both points.◮ Similarly, for any two lines and any point and line, there

exists an ordinary n-gon as subgeometry, containing boththese elements.

We call a point-line geometry a generalized polygon if it is ageneralized n-gon, for some n ≥ 3.•••• •••••••••••••••• •••••••••••••••••• •••••••• 28

Generalized triangles2

Recall the (axiomatic) definition of a projective plane:

Definition.A point-line geometry is a projective plane if◮ Any two points are on a unique line.◮ Any two lines intersect in a unique point.◮ There are at least two lines.

Then one easily sees that a generalized 3-gon is precisely thesame as a projective plane!

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Moufang polygons2

◮ Classifying projective planes is quite a tedious task! (Evenif the point set is finite.)

◮ Classification is completed if one assumes thickness andsome (transitivity) conditions on the automorphism groupof the generalized polygon.

◮ This condition is called the Moufang condition, and thegeneralized polygons satisfying it the Moufang polygons.

◮ Implies n = 3, 4, 6 or 8.◮ Classification due to Jacques Tits and Richard Weiss (’02).◮ Now: describe some embeddings into Lie algebras, using

inner ideals.

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Moufang hexagon2

◮ A generalized hexagon is a point-line geometry which doesnot contain triangles, quadrangles or pentagons and suchthat any two distinct points, any two distinct lines and anypoint and line lie in a hexagon.

◮ In particular: two points are at distance at most 3.

Theorem (De Medts - M.).The point-line geometry with as points the 1-dimensional innerideals of K (M(J)), as lines all proper inner ideals of dimension> 1 and inclusion as incidence is a Moufang hexagon.

◮ S+ and S− are at distance 3.

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A hexagon2

The generic hexagon in K (M(J)) is:〈( 1 0

0 0 )−〉〈( 1 00 0 )+〉

S+

〈( 0 00 1 )+〉 〈( 0 0

0 1 )−〉

S−

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Some remarks2

◮ In Groups with Steinberg Relations and Coordinatization ofPolygonal Geometries (’77) Faulkner constructs a Liealgebra starting from a cubic Jordan division algebra.He then constructs a generalized hexagon with as pointsand lines some specific inner ideals.

◮ Similar ideas can also be used to determine inner ideals ofsome other TKK Lie algebras that will correspond toprojective planes. We will describe this now.

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Alternative algebras2

◮ Let F be an alternative algebra (i.e., satisfying x2y = x(xy)and yx2 = (yx)x).

◮ Consider A := F ⊕ F , with multiplication(x , y)(a, b) = (xa, by) and involution (x , y) 0→ (y , x).

◮ Then A is a structurable algebra.◮ S = {(f , −f ) | f ∈ F} has the same dimension as F .◮ So, in general not a skew-dimension one structurable

algebra.◮ Assume from now on that F is division.

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Again a reduction2

Lemma (De Medts - M.).Any non-trivial inner ideal of K (A) is the image of an inner idealcontaining S+ under an element of E (A).

◮ What are the inner ideals containing S+?◮ There are precisely two of these: S+ ⊕ (F ⊕ 0)+ and

S+ ⊕ (0 ⊕ F )+.◮ Hence all non-trivial proper inner ideals have dimension

dim(F ) or 2 dim(F ).

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Constructing a geometry2

◮ P = {I | I is a minimal non-trivial inner ideal}◮ L = {I | I is a proper non-trivial non-minimal inner ideal}.◮ Then (P, L) is a point-line geometry with containment as

incidence.◮ The situation is more involved, since the minimal inner

ideals are not 1-dimensional.

Lemma (De Medts-M.).Let I, J ∈ P be two distinct points at distance d in (P, L). Then

d = 1 ⇐⇒ [I, J ] = 0,

d = 2 ⇐⇒ [I, J ] ∕= 0 and [I, [I, J ]] = 0,

d = 3 ⇐⇒ [I, [I, J ]] ∕= 0.

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Thin generalized hexagon2

We call a geometry thin, if every point is on precisely two lines.

Theorem (De Medts-M.).The point-line geometry (P, L) is a thin generalized hexagon. Itis the so-called double of a Moufang triangle.

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Moufang quadrangles?2

◮ For some classical class of Moufang quadrangles we alsohave a description in terms of inner ideals. (Related toquadratic forms.)

◮ There are also Moufang quadrangles of type E6, E7 and E8.◮ It seems that one obtains this as inner ideal geometry if you

consider a suitable structurable algebra.◮ Hopefully, we can attack higher rank geometries with this

approach as well.◮ This will be related to extremal geometries.

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Extremal geometriesFrom now on: joint with Hans Cuypers (TU Eindhoven)

3

◮ Recall: 0 ∕= x ∈ L is extremal if [x , [x , L]] ≤ 〈x〉.◮ E = set of extremal elements.◮ x is a zero divisor if [x , [x , L]] = 0.◮ If L does not contain a non-zero zero divisor, it is called

non-degenerate.◮ Assume that L is a simple non-degenerate Lie algebra

generated by its extremal elements.◮ P = {〈x〉 | x ∈ E}.◮ L = {〈x , y〉 | [x , y ] = 0 and λx + µy ∈ E , ∀λ, µ ∈

k, (λ, µ) ∕= (0, 0)}.◮ Γ := (P, L) is a point-line geometry, called the extremal

geometry.◮ Introduced by Arjeh Cohen and Gabor Ivanyos. (’06)

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Results Cohen and Ivanyos3Theorem.If L ∕= ∅, then Γ is isomorphic to a root shadow space of typeAn,{1,n} (n ≥ 2), BCn,2 (n ≥ 3), Dn,2 (n ≥ 4), E6,2, E7,1, E8,8,F4,1 or G2,2.

◮ Precise definition does not matter. What is important: theyare classified and more or less known.

◮ Example: An,{1,n} is associated with a projective space.◮ Example: G2,2 are precisely these generalized hexagons.◮ The extremal geometry in a split Lie algebra of type Xn has

the same type.◮ Moreover, their approach works in any characteristic.◮ However, there is one root shadow space missing, namely

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Inner line ideal geometry3

◮ In order to fix this, we extended the line set.◮ We call I an inner line ideal if it is a proper inner ideal

containing two linearly independent extremal elements andis minimal with these properties.

◮ Set L′ := {I | I is an inner line ideal}.◮ We call Γ′ = (P, L′) the inner line ideal geometry of L.

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Result3

Theorem (Cuypers-M.).Suppose L is a simple non-degenerate Lie algebra generated byextremal elements over a field of characteristic not 2. Then wehave one of the following:(a) The extremal geometry of L contains lines and coincides

with the inner line ideal geometry.(b) The extremal geometry of L contains no lines, but L

contains two commuting, but linearly independent extremalelements; the inner line ideal geometry is a non-degeneratepolar space of rank at least 2.

(c) The Lie algebra L does not contain commuting, but linearlyindependent extremal elements, and the inner line idealgeometry has no lines.

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Connection with structurable algebras?3

Relying on results of Stavrova:

Theorem (Cuypers - M.).If L is a non-degenerate non-symplectic simple Lie algebragenerated by its extremal elements, then L = K (A) for someskew-dimension one structurable algebra A.

◮ Moreover, the inner line ideal geometry coincides with theextremal geometry if and only if A is isomorphic to amatrix structurable algebra.

◮ If the inner line ideal geometry has no lines, it is comingfrom a skew-dimension one structurable division algebra.Hence, it is a Moufang set!

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Some remarks3

◮ In particular one sees that one does not recover all Moufangsets, using the extremal geometry/inner line ideal geometryapproach.

◮ A geometry of type A2,{1,2} is precisely the double of aprojective plane.

◮ So we only get the double of the projective plane over theground field as extremal geometry.

◮ For other projective planes, the minimal inner ideals need tohave dimension strictly bigger than 1.

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Some more questions3

◮ What happens in characteristic 2 (and 3)? Extremalelements in characteristic 2 are well-defined, what is “good”definition for inner ideals of Lie algebras?

◮ Inner ideals of a Jordan algebra correspond to inner idealsof its Lie algebra, what happens when structurable algebrais not Jordan?

◮ Good definition for structurable algebras in characteristic 2and/or 3?

◮ Explicit construction for forms of matrix structurablealgebras corresponding to quadrangles of type E6, E7, E8?

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End3

Thanks for your attention!

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