lie algebras from oriented partial linear spaces · introduction direct and indirect construction...

IntroductionDirect Method

Indirect MethodSummary

Lie Algebras from Oriented Partial LinearSpaces

E.J. Postmaepostma@win.tue.nl

Technische Universiteit Eindhoven

DIAMANT/EIDMA symposium 2005

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Introduction

Direct and indirect construction of Lie algebra fromoriented partial linear space

Kaplansky: new simple Lie algebras from symplectic space

Hall: similar construction with partial linear space

Cuypers earlier studied the case char F = 2

Joint work with Cohen and Cuypers

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Introduction

Direct and indirect construction of Lie algebra fromoriented partial linear space

Kaplansky: new simple Lie algebras from symplectic space

Hall: similar construction with partial linear space

Cuypers earlier studied the case char F = 2

Joint work with Cohen and Cuypers

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Introduction

Direct and indirect construction of Lie algebra fromoriented partial linear space

Kaplansky: new simple Lie algebras from symplectic space

Hall: similar construction with partial linear space

Cuypers earlier studied the case char F = 2

Joint work with Cohen and Cuypers

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Introduction

Direct and indirect construction of Lie algebra fromoriented partial linear space

Kaplansky: new simple Lie algebras from symplectic space

Hall: similar construction with partial linear space

Cuypers earlier studied the case char F = 2

Joint work with Cohen and Cuypers

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Introduction

Direct and indirect construction of Lie algebra fromoriented partial linear space

Kaplansky: new simple Lie algebras from symplectic space

Hall: similar construction with partial linear space

Cuypers earlier studied the case char F = 2

Joint work with Cohen and Cuypers

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Definitions

Lie algebra: an algebra with

[x , x ] = 0 (Implies [x , y ] = −[y , x ])

[x , [y , z]] + [y , [z, x ]] + [z, [x , y ]] = 0.

Partial linear space: (P, L) with a pair of points connected by atmost one line. We consider only spaces with lines of length 3.

Oriented partial linear space: (P, L, σ : L → Sym(P)) with σ(`)equal to (p, q, r) or (p, r , q).

qp r qp r

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

The Kaplansky Algebra

Let L(P, L, σ) = FP with multiplication

[p, q] =

0 if p = q or p 6∼ q (“p ⊥ q”),

r if ` = {p, q, r} ∈ L and pσ(`) = q,

−r if ` = {p, q, r} ∈ L and qσ(`) = p.

Example:

ji k

Subalgebra of the quaternions isomorphic to sl2.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Geometrical Classification

If L(P, L, σ) is a Lie algebra, then its planes are all isomorphicto the dual affine plane of order 2 or the Fano plane. The Fanoplane can only occur if char F = 3.

Dual affine plane of order 2 Fano plane

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Geometrical Classification

If L(P, L, σ) is a Lie algebra, then its planes are all isomorphicto the dual affine plane of order 2 or the Fano plane. The Fanoplane can only occur if char F = 3.

Dual affine plane of order 2 Fano plane

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Geometrical Classification

If L(P, L, σ) is a Lie algebra, then its planes are all isomorphicto the dual affine plane of order 2 or the Fano plane. The Fanoplane can only occur if char F = 3.

Dual affine plane of order 2 Fano plane

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Geometrical Classification

If L(P, L, σ) is a Lie algebra, then its planes are all isomorphicto the dual affine plane of order 2 or the Fano plane. The Fanoplane can only occur if char F = 3.

Dual affine plane of order 2 Fano plane

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Another Example

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

New Start

Assumptions:

Only dual affine planes of order 2

No two points have the same set of neighbours, orequivalently, L(P, L, σ) is simple

Let V = F2P/〈p + q + r | {p, q, r} ∈ L〉. Then (P, L) embedsinto PG(V ).

Quadratic form Q on F2P determined by:

Q(p) = 1; Q(p + q) =

{0, if p ⊥ q;

1, if p ∼ q.

Q is nondegenerate on V . So dim V = 2n.E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

O−2 (4) with form Q(x , y , z, w) = x2 + xy + y2 + zw :

00110100 0101 01101000 1001 10101100 1101 1110

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

O−2 (4) with form Q(x , y , z, w) = x2 + xy + y2 + zw :

00110100 0101 01101000 1001 10101100 1101 1110

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

O−2 (4) with form Q(x , y , z, w) = x2 + xy + y2 + zw :

0011

0101

1001

11001010

1110 1000

0100

0110

1101

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Extraspecial Group

G = G(P, L, σ) = 21+2n, then |G| = 21+2n

Z (G) = 〈z〉 of order 2G/Z (G) = F2n

2 = V ; choose a preimage v for every v ∈ V

v2 =

{1, if Q(v) = 0,

z, if Q(v) = 1;and in the second case,

v−1 = zv .

uv u−1v−1 =

{1, if B(u, v) = 0,

z, if B(u, v) = 1.If the preimages are chosen right, then for lines {p, q, r}:

pqr =

{z, if pσ(`) = q,

1, if qσ(`) = p.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

0011

0101

1001

11001010

1110 1000

0100

0110

1101 O−2 (4), so

21+4− = Q8 ◦ D8.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

0011

0101

1001

11001010

1110 1000

0100

0110

1101 O−2 (4), so

21+4− = Q8 ◦ D8.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

Back to the Second Example

1001

11001010

1110 1000

1101

0100

0101

0011

0110

O−2 (4), so

21+4− = Q8 ◦ D8.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Plesken Lie Algebra

Let G be a group. Consider the Lie algebra of G’s group algebraLie(F[G]): formal basis G, multiplication [g, h] = gh − hg.

Let g = g − g−1, then

L(G) = 〈g | g ∈ G〉

is a sub-Lie algebra of Lie(F[G]) linearly spanned by elementsg.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

The Plesken Lie Algebra

Let G be a group. Consider the Lie algebra of G’s group algebraLie(F[G]): formal basis G, multiplication [g, h] = gh − hg.

Let g = g − g−1, then

L(G) = 〈g | g ∈ G〉

is a sub-Lie algebra of Lie(F[G]) linearly spanned by elementsg.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Orthogonal spaceExtraspecial GroupPlesken Lie AlgebraEmbedding

An Embedding

Let ϕ : L(P, L, σ) → L(G(P, L, σ)) be a linear map determinedby ϕ(p) = 1

4ˆp = 1

4(1 − z)p. Then

[ϕ(p), ϕ(q)] = [14(1 − z)p,

14(1 − z)q] =

116

(1−z)2(pq−qp) =

0 = ϕ([p, q]), if p ⊥ q,1

16(1 − z)3r = ϕ([p, q]), if pσ(`) = q,

− 116(1 − z)3r = ϕ([p, q]), if qσ(`) = p.

(1 − z)3 = 4(1 − z), pq =

{r , if pσ(`) = q,

zr , if qσ(`) = p.

So ϕ respects multiplication. Since L(P, L, σ) is simple, ϕcannot have a kernel, so it is a Lie algebra embedding.

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ)

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ) L(P, L, σ)

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ) L(P, L, σ)

Orthogonal spaceV

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ) L(P, L, σ)

Orthogonal spaceV

G(P, L, σ)

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ) L(P, L, σ)

Orthogonal spaceV

G(P, L, σ)

L(G(P, L, σ))

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces

IntroductionDirect Method

Indirect MethodSummary

Summary

(P, L, σ) L(P, L, σ)

Orthogonal spaceV

G(P, L, σ)

L(G(P, L, σ))

x

E.J. Postma Lie Algebras from Oriented Partial Linear Spaces