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We investigate the geometric phases and the Bargmann invariants associatedwith a multi-level quantum systems. In particular, we show that a full set of`gauge-invariant' objects for an $n$-level system consists of $n$ geometricphases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmanninvariants. In the process of establishing this result we develop a canonicalform for U(n) matrices which is useful in its own right. We show that therecently discovered `off-diagonal' geometric phases [N. Manini and F.Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in termsof the basic building blocks developed in this work. This result liberates theoff-diagonal phases from the assumption of adiabaticity used in arriving atthem.Comment: 13 pages, latex, no figure

Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: A unitary-group approach