Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Hilbert's fourteenth problem
gave the following counterexample to Hilbert's problem. The field "k" is a field containing 48 elements "a", ...,"a", for "i"=1, 2, 3 that are algebraically independent over the prime field. The ring "R" is the polynomial ring "k"["x"...,"x", "t"...,"t"] in 32 variables. The vector space "V" is a 13-dimensional vector space over "k" consisting of all vectors ("b"...,"b") in "k" orthogonal to each of the three vectors ("a", ...,"a") for "i"=1, 2, 3. The vector space "V" is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on "R" by fixing all elements "t" and taking "x" to "x" + "b""t". Then the ring of elements of "R" invariant under the action of the group "V" is not a finitely generated "k"-algebra.