Gvalued

G-valued is an adjective used in mathematics and theoretical physics to denote that a given object takes its values in a Lie algebra g associated with a Lie group G. This convention appears in differential geometry, gauge theory, and representation theory. When fields or forms are defined on a manifold, a g-valued object assigns to each point an element of g rather than a real number or a vector space.

In notation, g typically denotes Lie(G). A g-valued differential form on a manifold M is a form whose values lie in g, for example a g-valued 1-form A or a g-valued 2-form F. A g-valued function f on M assigns to each x in M an element f(x) in g. Operations such as the Lie bracket [ , ] remain inside g, allowing natural algebraic structures to be defined on these objects.

In gauge theory, the connection form is g-valued, and its curvature is a g-valued 2-form. The theory uses invariant inner products on g, such as the Killing form, to construct actions and trace expressions. The designation "g-valued" signals that the field content carries Lie-algebraic, noncommutative structure, with commutator terms central to dynamics and symmetry.

Notes: The term is context-dependent and describes the codomain or target of a field, form, or map, rather than naming a distinct object. The standard notation for the Lie algebra is fraktur g or g with context clarifying the corresponding Lie group G. See also Lie algebra, Lie group, connection, curvature, gauge theory.