01462nas a2200145   4500008004100000245005600041210005600097260005100153520096300204100002501167700002401192700002701216700002201243856005101265       2014                        en  d00aQuantum gauge symmetries in noncommutative geometry0 aQuantum gauge symmetries in noncommutative geometry  bEuropean Mathematical Society Publishing House3 aWe discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group).1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDas, Biswarup, Krishna1 aDabrowski, Ludwik  uhttp://urania.sissa.it/xmlui/handle/1963/3489700860nas a2200133   4500008004100000245008400041210006900125260001300194520041200207100002500619700002400644700002200668856003600690       2011                        en  d00aQuantum Isometries of the finite noncommutative geometry of the Standard  Model0 aQuantum Isometries of the finite noncommutative geometry of the   bSpringer3 aWe compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDabrowski, Ludwik  uhttp://hdl.handle.net/1963/4906