Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction...
Transcript of Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction...
IntroductionDirect Method
Indirect MethodSummary
Lie Algebras from Oriented Partial LinearSpaces
E.J. [email protected]
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