Compact quantum group

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In mathematics, compact quantum groups are generalisations of compact groups, where the commutative ${\displaystyle \mathrm {C} ^{*}}$-algebra of continuous complex-valued functions on a compact group is generalised to an abstract structure on a not-necessarily commutative unital ${\displaystyle \mathrm {C} ^{*}}$-algebra, which plays the role of the "algebra of continuous complex-valued functions on the compact quantum group".^[1]

The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a commutative C*-algebra. On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.

S. L. Woronowicz^[2] introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.