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| {{DISPLAYTITLE:E<sub>''n''</sub> (Lie algebra)}}
| | The author is known by the title of Numbers Lint. Managing people is what I do and the wage has been truly satisfying. California is our birth location. Playing baseball is the pastime he will never quit doing.<br><br>Have a look at my web blog :: [http://www.webmdbook.com/index.php?do=/profile-11685/info/ http://www.webmdbook.com] |
| {| align=right class=wikitable
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| |+ Dynkin diagrams
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| |-
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| !colspan=2|Finite
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| |-
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| |'''E<sub>3</sub>'''='''A<sub>2</sub>A<sub>1</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|2|node_n3}}
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| |-
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| |'''E<sub>4</sub>'''='''A<sub>4</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch}}
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| |-
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| |'''E<sub>5</sub>'''='''D<sub>5</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4}}
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| |-
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| |'''E<sub>6</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5}}
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| |-
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| |'''E<sub>7</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6}}
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| |-
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| |'''E<sub>8</sub>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7}}
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| |-
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| !colspan=2|Affine (Extended)
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| |-
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| |'''E<sub>9</sub>''' or '''E<sub>8</sub><sup>(1)</sup>''' or '''E<sub>8</sub><sup>+</sup>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8}}
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| |-
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| !colspan=2|Hyperbolic (Over-extended)
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| |-
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| |'''E<sub>10</sub>''' or '''E<sub>8</sub><sup>(1)^</sup>''' or '''E<sub>8</sub><sup>++</sup>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9}}
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| |-
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| !colspan=2|Lorentzian (Very-extended)
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| |-
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| |'''E<sub>11</sub>''' or '''E<sub>8</sub><sup>+++</sup>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9|3|nodeg_n10}}
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| |-
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| !colspan=2|Kac–Moody
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| |-
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| |'''E<sub>12</sub>''' or '''E<sub>8</sub><sup>++++</sup>'''
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| |{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9|3|nodeg_n10|3|nodeg_n11}}
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| |-
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| |colspan=2|...
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| |}
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| In [[mathematics]], especially in [[Lie algebra|Lie]] theory, '''E<sub>''n''</sub>''' is the [[Kac–Moody algebra]] whose [[Dynkin diagram]] is a bifurcating graph with three branches of length 1,2, and ''k'', with ''k''=''n-4''.
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| In some older books and papers, ''E''<sub>2</sub> and ''E''<sub>4</sub> are used as names for [[G2 (mathematics)|''G''<sub>2</sub>]] and [[F4 (mathematics)|''F''<sub>4</sub>]].
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| ==Finite dimensional Lie algebras==
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| The E<sub>n</sub> group is similar to the A<sub>n</sub> group, except the nth node is connected to the 3rd node. So the [[Cartan matrix]] appears similar, -1 above and below the diagonal, except for the last row and column, have -1 in the third row and column. The determinant of the Cartan matrix for E<sub>n</sub> is 9-''n''.
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| *'''E<sub>3</sub>''' is another name for the Lie algebra ''A''<sub>1</sub>''A''<sub>2</sub> of dimension 11, with Cartan determinant 6.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 \\
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| -1 & 2 & 0 \\
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| 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''E<sub>4</sub>''' is another name for the Lie algebra ''A''<sub>4</sub> of dimension 24, with Cartan determinant 5.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 \\
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| -1 & 2 & -1& 0 \\
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| 0 & -1 & 2 & -1 \\
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| 0 & 0 & -1 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''E<sub>5</sub>''' is another name for the Lie algebra ''D''<sub>5</sub> of dimension 45, with Cartan determinant 4.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 \\
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| 0 & -1 & 2 & -1 & -1 \\
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| 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''[[E6 (mathematics)|E<sub>6</sub>]]''' is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & -1 \\
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| 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''[[E7 (mathematics)|E<sub>7</sub>]]''' is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & 0 & -1 \\
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| 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''[[E8 (mathematics)|E<sub>8</sub>]]''' is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\
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| 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| ==Infinite dimensional Lie algebras==
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| *'''E<sub>9</sub>''' is another name for the infinite dimensional [[affine Lie algebra]] <math>{\tilde{E}}_8</math> (also as '''E<sub>8</sub><sup>+</sup>''' or E<sub>8</sub><sup>(1)</sup> as a (one-node) '''extended''' E<sub>8</sub>) (or [[E8 lattice]]) corresponding to the Lie algebra of type [[E8 (mathematics)|E<sub>8</sub>]]. '''E<sub>9</sub>''' has a Cartan matrix with determinant 0.
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| *:<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & -1 \\
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| 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''E<sub>10</sub>''' (or '''E<sub>8</sub><sup>++</sup>''' or '''E<sub>8</sub><sup>(1)^</sup>''' as a (two-node) '''over-extended''' E<sub>8</sub>) is an infinite dimensional [[Kac–Moody algebra]] whose root lattice is the even Lorentzian [[unimodular lattice]] II<sub>9,1</sub> of dimension 10. Some of its root multiplicities have been calculated; for small roots the multiplicities seem to be well behaved, but for larger roots the observed patterns break down. '''E<sub>10</sub>''' has a Cartan matrix with determinant -1:
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| **<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 \\
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| 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| *'''E<sub>11</sub>''' (or '''E<sub>8</sub><sup>+++</sup>''' as a (three-node) '''very-extended''' E<sub>8</sub>) is a [[Lorentzian algebra]], containining one time-like imaginary dimension, that has been conjectured to generate the symmetry "group" of [[M-theory]].
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| *'''E<sub>''n''</sub>''' for ''n''≥12 is an infinite dimensional [[Kac–Moody algebra]] that has not been studied much.
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| ==Root lattice==
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| The root lattice of '''E'''<sub>''n''</sub> has determinant 9−''n'', and can be constructed as the
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| lattice of vectors in the [[unimodular lattice|unimodular Lorentzian lattice]] '''Z'''<sub>''n'',1</sub> that are orthogonal to the vector (1,1,1,1,....,1|3) of norm ''n''×1<sup>2</sup> − 3<sup>2</sup> = ''n'' − 9.
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| ==E7½==
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| {{main|E7½}}
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| Landsberg and Manivel extended the definition of E<sub>''n''</sub> for integer ''n'' to include the case ''n'' = 7½. They did this in order to fill the "hole" in dimension formulae for representations of the E<sub>''n''</sub> series which was observed by Cvitanovic, Deligne, Cohen and de Man. E<sub>7½</sub> has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional [[Heisenberg algebra]] as its [[Nilradical of a Lie algebra|nilradical]].
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| == See also ==
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| * [[Semiregular k 21 polytope|k<sub>21</sub>]], [[Uniform 2 k1 polytope|2<sub>k1</sub>]], [[Uniform 1 k2 polytope|1<sub>k2</sub>]] polytopes based on E<sub>n</sub> Lie algebras.
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| ==References==
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| *{{Cite book | last1=Kac | first1=Victor G | last2=Moody | first2=R. V. | last3=Wakimoto | first3=M. | title=Differential geometrical methods in theoretical physics (Como, 1987) | publisher=Kluwer Acad. Publ. | location=Dordrecht | series=NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. | mr=981374 | year=1988 | volume=250 | chapter=On E<sub>10</sub> | pages=109–128 | postscript=<!--None-->}}
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| == Further reading ==
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| *{{cite journal |title=E<sub>11</sub> and M Theory |year=2001 |version= |last1=West | first1=P. |doi=10.1088/0264-9381/18/21/305 |journal=Classical and Quantum Gravity |volume=18 |issue=21 |pages=4443–4460 |arxiv=hep-th/0104081}} Class.Quant.Grav. 18 (2001) 4443-4460
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| *{{cite arxiv|title=E<sub>10</sub> for beginners|year=1994|version=|eprint=hep-th/9411188|last1=Gebert | first1=R. W.|last2=Nicolai | first2=H.|class=hep-th}} Guersey Memorial Conference Proceedings '94
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| * Landsberg, J. M. Manivel, L. [http://arxiv.org/abs/math.RT/0402157'' The sextonions and E<sub>7½</sub>'']. Adv. Math. 201 (2006), no. 1, 143-179.
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| * ''Connections between Kac-Moody algebras and M-theory'', Paul P. Cook, 2006 [http://arxiv.org/abs/0711.3498]
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| * ''A class of Lorentzian Kac-Moody algebras'', Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002[http://arxiv.org/abs/hep-th/0205068]
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| [[Category:Lie groups]]
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