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gptkbp:instanceOf
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gptkb:root
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gptkbp:appearsIn
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classification of semisimple Lie algebras
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gptkbp:associatedWith
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Lie algebra sl_n
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gptkbp:Cartan_matrix
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tridiagonal with 2 on diagonal, -1 on off-diagonal
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gptkbp:dimensions
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n-1
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gptkbp:dual_root_system
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gptkb:A_{n-1}_root_system
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gptkbp:Dynkin_diagram
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A_{n-1}
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gptkbp:firstAppearance
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19th century mathematics
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gptkbp:fundamental_weights
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sum_{i=1}^k e_i - k/n sum_{i=1}^n e_i
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gptkbp:highest_root
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e_1 - e_n
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https://www.w3.org/2000/01/rdf-schema#label
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A {n-1} root system
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gptkbp:isSimplyLaced
|
true
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gptkbp:namedFor
|
gptkb:Wilhelm_Killing
gptkb:Élie_Cartan
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gptkbp:notation
|
A_{n-1}
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gptkbp:number_of_roots
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n(n-1)
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gptkbp:rank
|
n-1
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gptkbp:relatedTo
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permutation group
special linear group SL_n
type A Lie algebras
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gptkbp:root_lattice
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vectors in R^n with integer coordinates summing to zero
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gptkbp:roots
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e_i - e_j, i ≠ j
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gptkbp:simple_roots
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e_i - e_{i+1}
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gptkbp:type
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classical root system
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gptkbp:usedIn
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gptkb:geometry
physics
representation theory
combinatorics
algebraic groups
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gptkbp:weight_lattice
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vectors in R^n with rational coordinates summing to zero
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gptkbp:Weyl_group
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gptkb:symmetric_group_S_n
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gptkbp:bfsParent
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gptkb:A_{n-1}_root_lattice
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gptkbp:bfsLayer
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7
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