  We develop a new approach to RG flows and show that one-loop flows in
multiscalar theories can be described by commutative but non-associative
Euclidean algebras. As an example related to $D$-brane field theories and
tensor models, we study the algebra of a theory with $M$ $SU(N)$ adjoint
scalars and its large $N$ limits. A block diagonal decomposition of the RG flow
into independent sets of couplings is achieved by a complete decomposition of
the algebra into a sum of simple ideals, permitted by its Euclidean metric. The
algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are
used to further characterise the RG flows.
  We classify and describe all large N limits of algebras of multiadjoint
scalar models: the standard `t Hooft matrix theory limit, a `multi-matrix'
limit, each with one free parameter, and an intermediate case with extra
symmetry and no free parameter of the algebra, but an emergent free parameter
from a line of one-loop fixed points. The algebra identifies these limits
without diagrammatic or combinatorial analysis.

High Energy Physics - Theory;High Energy Physics - Phenomenology