Lie algebras as non-unital, non-associative algebras #
It is useful to have a special typeclass for Lie algebras because:
- it enables us to use the traditional notation
⁅x, y⁆for the Lie multiplication,
- associative algebras carry a natural Lie algebra structure via the ring commutator and so we need
them to carry both
- more generally, Poisson algebras (not yet defined) need both typeclasses.
However there are times when it is convenient to be able to regard a Lie algebra as a general algebra and we provide some basic definitions for doing so here.
Main definitions #
CommutatorRingturns a Lie ring into a
NonUnitalNonAssocSemiringby turning its
⁅, ⁆) into a
lie algebra, non-unital, non-associative