equation 14.3 eq:lie-generators

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Supports

• depends_on proposition 14.3 A one-parameter subgroup is an exponential ¶
• step_from equation 14.4 eq:lie-gengrlie ¶
  • step_from equation 14.5 eq:lie-structconst ¶
    • depends_on definition 14.14 Invariant symmetric tensor ¶
      • depends_on lemma A.350 Invariant polynomials on $\mathfrak{g}$ ¶
        • depends_on theorem A.351 Chevalley restriction theorem, quoted ¶
      • depends_on theorem 14.16 Invariant tensors give Casimir operators ¶
        • depends_on corollary A.328 The Casimir elements are not accidentally zero ¶
        • depends_on corollary 14.17 The quadratic Casimir ¶
        • depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
    • depends_on definition 14.10 Killing form ¶
      • depends_on corollary A.343 The Killing form of a simple ideal ¶
        • depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
      • depends_on corollary 14.12 Total antisymmetry of the structure constants ¶
        • depends_on corollary 14.17 The quadratic Casimir ¶ ↺
      • depends_on definition A.424 Loop algebra and residue ¶
        • depends_on lemma A.425 The residue of a derivative vanishes ¶
        • depends_on theorem A.426 The Kac–Moody cocycle ¶
      • depends_on definition A.400 Invariant form, dual basis, Casimir ¶
        • depends_on lemma A.401 The invariance identity ¶
      • depends_on lemma A.350 Invariant polynomials on $\mathfrak{g}$ ¶ ↺
      • depends_on lemma 14.11 Invariance of the Killing form ¶
        • depends_on corollary 14.12 Total antisymmetry of the structure constants ¶ ↺
        • depends_on definition A.400 Invariant form, dual basis, Casimir ¶ ↺
        • depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
        • depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
        • depends_on theorem A.426 The Kac–Moody cocycle ¶ ↺
      • depends_on proposition A.334 Nondegeneracy implies no abelian ideal ¶
      • depends_on proposition 14.84 Properties of a contraction ¶
      • depends_on theorem A.330 Cartan's criterion for semisimplicity ¶ ↺
      • depends_on theorem 14.13 Cartan's criterion ¶
        • depends_on corollary 14.17 The quadratic Casimir ¶ ↺
        • depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶ ↺
        • depends_on proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶
        • depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
    • depends_on definition 14.5 Universal enveloping algebra ¶
      • depends_on definition A.320 Filtration by degree ¶
        • depends_on definition A.325 Symmetric algebra and symbol ¶
        • depends_on lemma A.321 The ordered monomials span ¶
      • depends_on definition A.326 Symmetrization ¶
        • depends_on proposition A.327 Symmetrization is an isomorphism of $\mathfrak{g}$-modules ¶
      • depends_on definition 14.8 Casimir element ¶
        • depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
        • depends_on lemma A.347 Central is the same as invariant ¶
        • depends_on lemma 14.87 A rescaled Casimir stays central ¶
        • depends_on proposition 14.60 The quadratic invariant ¶
        • depends_on theorem A.346 Racah ¶
        • depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶
        • depends_on theorem 14.16 Invariant tensors give Casimir operators ¶ ↺
        • depends_on theorem 14.21 Racah ¶
      • depends_on lemma A.347 Central is the same as invariant ¶ ↺
      • depends_on theorem A.319 Poincaré–Birkhoff–Witt ¶
        • depends_on corollary A.328 The Casimir elements are not accidentally zero ¶ ↺
        • depends_on definition A.325 Symmetric algebra and symbol ¶ ↺
        • depends_on proposition A.327 Symmetrization is an isomorphism of $\mathfrak{g}$-modules ¶ ↺
      • depends_on theorem 14.16 Invariant tensors give Casimir operators ¶ ↺
      • depends_on theorem 14.7 Poincaré–Birkhoff–Witt ¶
        • depends_on lemma 14.87 A rescaled Casimir stays central ¶ ↺
    • depends_on lemma 14.102 Two identities for a $\mathfrak{g}$-valued $1$-form ¶
      • depends_on theorem 14.107 Bianchi identity ¶
    • depends_on theorem 14.110 thm:lie-invpoly ¶
      • depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
      • depends_on theorem 14.111 Chern–Weil ¶
        • depends_on definition 14.115 Characteristic forms ¶
        • depends_on definition 14.112 Chern–Simons form ¶
        • depends_on proposition 14.114 Chern–Weil forms descend to the base ¶
        • depends_on proposition 106.81 The topological charge is an integer ¶