Quantum groups and non-commutative geometry Hopf algebra

all examples above either commutative (i.e. multiplication commutative) or co-commutative (i.e. Δ = t ∘ Δ twist map t: h ⊗ h → h ⊗ h defined t(x ⊗ y) = y ⊗ x). other interesting hopf algebras deformations or quantizations of example 3 neither commutative nor co-commutative. these hopf algebras called quantum groups, term far loosely defined. important in noncommutative geometry, idea being following: standard algebraic group described standard hopf algebra of regular functions; can think of deformed version of hopf algebra describing non-standard or quantized algebraic group (which not algebraic group @ all). while there not seem direct way define or manipulate these non-standard objects, 1 can still work hopf algebras, , indeed 1 identifies them hopf algebras. hence name quantum group .