lie algebras from oriented partial linear spaces · introduction direct and indirect construction...
TRANSCRIPT
![Page 1: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/1.jpg)
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
![Page 2: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/2.jpg)
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
![Page 3: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/3.jpg)
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
![Page 4: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/4.jpg)
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
![Page 5: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/5.jpg)
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
![Page 6: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/6.jpg)
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
![Page 7: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/7.jpg)
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
![Page 8: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/8.jpg)
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
![Page 9: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/9.jpg)
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
![Page 10: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/10.jpg)
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
![Page 11: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/11.jpg)
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
![Page 12: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/12.jpg)
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
![Page 13: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/13.jpg)
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
![Page 14: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/14.jpg)
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
![Page 15: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/15.jpg)
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
![Page 16: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/16.jpg)
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
![Page 17: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/17.jpg)
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
![Page 18: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/18.jpg)
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
![Page 19: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/19.jpg)
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
![Page 20: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/20.jpg)
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
![Page 21: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/21.jpg)
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
![Page 22: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/22.jpg)
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
![Page 23: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/23.jpg)
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
![Page 24: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/24.jpg)
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
![Page 25: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/25.jpg)
IntroductionDirect Method
Indirect MethodSummary
Another Example
E.J. Postma Lie Algebras from Oriented Partial Linear Spaces
![Page 26: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/26.jpg)
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
![Page 27: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/27.jpg)
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
![Page 28: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/28.jpg)
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
![Page 29: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/29.jpg)
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
![Page 30: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/30.jpg)
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
![Page 31: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/31.jpg)
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
![Page 32: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/32.jpg)
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
![Page 33: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/33.jpg)
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
![Page 34: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/34.jpg)
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
![Page 35: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/35.jpg)
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
![Page 36: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/36.jpg)
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
![Page 37: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/37.jpg)
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
![Page 38: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/38.jpg)
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
![Page 39: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/39.jpg)
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
![Page 40: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/40.jpg)
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
![Page 41: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/41.jpg)
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
![Page 42: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/42.jpg)
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
![Page 43: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/43.jpg)
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
![Page 44: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/44.jpg)
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
![Page 45: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/45.jpg)
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
![Page 46: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/46.jpg)
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
![Page 47: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/47.jpg)
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
![Page 48: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/48.jpg)
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
![Page 49: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/49.jpg)
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
![Page 50: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/50.jpg)
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
![Page 51: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/51.jpg)
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
![Page 52: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/52.jpg)
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
![Page 53: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/53.jpg)
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
![Page 54: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/54.jpg)
IntroductionDirect Method
Indirect MethodSummary
Summary
(P, L, σ)
x
E.J. Postma Lie Algebras from Oriented Partial Linear Spaces
![Page 55: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/55.jpg)
IntroductionDirect Method
Indirect MethodSummary
Summary
(P, L, σ) L(P, L, σ)
x
E.J. Postma Lie Algebras from Oriented Partial Linear Spaces
![Page 56: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/56.jpg)
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
![Page 57: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/57.jpg)
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
![Page 58: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/58.jpg)
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
![Page 59: Lie Algebras from Oriented Partial Linear Spaces · Introduction Direct and indirect construction of Lie algebra from oriented partial linear space Kaplansky: new simple Lie algebras](https://reader030.vdocuments.net/reader030/viewer/2022041120/5f32e8c0ce0604000a3fd172/html5/thumbnails/59.jpg)
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