definition 14.72 Central extension

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Supports

• 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 proposition A.428 Uniqueness in degree zero ¶
  • depends_on theorem A.426 The Kac–Moody cocycle ¶ ↺
• depends_on definition A.430 The hypotheses ¶
  • depends_on lemma A.431 Positivity of the double commutator ¶
    • depends_on theorem A.432 Schwinger's theorem ¶
  • depends_on theorem A.432 Schwinger's theorem ¶ ↺
• depends_on definition A.421 Witt algebra ¶
  • depends_on theorem A.422 The Virasoro extension ¶
• depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶
  • depends_on definition 14.74 Coboundary and triviality ¶
    • depends_on definition 14.75 The classifying group ¶
      • depends_on definition A.397 Cochains and the differential ¶
        • depends_on lemma A.398 The differential squares to zero in the degrees used ¶
        • depends_on theorem A.403 Vanishing when the Casimir is invertible ¶
      • depends_on example 14.80 The Galilei algebra ¶
      • depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶
        • depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
        • depends_on proposition A.408 The two groups vanish for the trivial module ¶
        • depends_on theorem A.414 $H^{2}$ of the Galilei algebra ¶
        • depends_on theorem A.422 The Virasoro extension ¶ ↺
      • depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
        • depends_on proposition A.428 Uniqueness in degree zero ¶ ↺
        • depends_on proposition 14.84 Properties of a contraction ¶
      • depends_on theorem A.414 $H^{2}$ of the Galilei algebra ¶ ↺
      • depends_on theorem A.422 The Virasoro extension ¶ ↺
    • depends_on example 14.79 The Heisenberg algebra ¶
    • depends_on proposition A.419 The coboundaries ¶
    • depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶ ↺
  • depends_on definition 14.75 The classifying group ¶ ↺
  • depends_on example 14.79 The Heisenberg algebra ¶ ↺
  • depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶ ↺
  • depends_on proposition 25.14 The Galilei cocycle is not a coboundary ¶
    • depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶
      • depends_on example 15.64 The whole chapter at $D=4$ ¶
      • depends_on remark 15.59 The central charge is what makes the form work ¶
    • depends_on example 14.80 The Galilei algebra ¶ ↺
  • depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶
    • depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶ ↺
    • depends_on corollary 15.35 Where the central charges are ¶
      • depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶ ↺
      • depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
• depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶ ↺
• depends_on proposition 25.14 The Galilei cocycle is not a coboundary ¶ ↺
• depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶ ↺