If the symmetry corresponding to the conserved currents via Noether's theorem preserves the given Lagrangian only up to a divergence term, then the current algebra is a central extension of the Lie algebra of the underlying symmetries. This effect makes current algebras tend to be subtle and of particular mathematical interest.
Particularly famous is the case of the WZW sigma model field theory for a string propagating on a Lie group . In this case one chiral half of the algebra of currents is the corresponding affine Lie algebra. In parts of the (mathematical) literature it is this special case which meant by default by the term current algebra.
With local prequantum field theory in dimension formulated as de Donder-Weyl field theory, every local Lagrangian is incarnated as a principal d-form connection on some space , such that its curvature form determines by its kernel the tangents to the solutions of the equations of motion. (see cftcht, sections 3.1, 3.2, 3.4).
What is traditionally considered in the literature is the special case of this where is the (dualized, first) jet bundle of a field bundle over a smooth manifold (see at multisymplectic geometry) and the underlying principal infinity-bundle of is trivial, so that is incarnated as a globally defined differential d-form. Moreover, this is traditionally expressed as the product of a smooth function times a prescribed volume form. This function then is what in much of the traditional literure is referred to as the Lagrangian of the theory.
together with an equivalence
Under Lie differentiation this gives the Poisson bracket Lie n-algebra . In its dg-Lie algebra version spelled out here and restricted to the special case of globally defined Lagrangian form, this has, in degree 0, precisely the Lie bracket of conserved currents as known from traditional literature, for instance (AGIT89, equations (13), (14)). Details are in cftcht, 3.3.
But the contains in addition the equivalence between two central terms, coming from higher gauge transformations. Taking this into account, the extension theorem for says that its truncation to a Lie algebra is an extension by . This has been informally argued for instance in AGIT 89, p. 8.
Steven Weinberg, Current Algebra and Gauge Theories. I Phys. Rev. D 8, 605–625 (1973)
Herbert Pietschmann, On the Early History of Current Algebra (arXiv:1101.2748)
General discussion and application to the Green-Schwarz super p-brane sigma models is in
Discussion from an nPOV is in