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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | )abbrev category LIECAT LieAlgebra
++ Author: Michel Petitot (petitot@lifl.fr).
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Description:
++ The category of Lie Algebras.
++ It is used by the domains of non-commutative algebra,
++ LiePolynomial and XPBWPolynomial.
LieAlgebra(R) : Category == SIG where
R : CommutativeRing
SIG ==> Module(R) with
construct : ($,$) -> $
++ \axiom{construct(x,y)} returns the Lie bracket of \axiom{x}
++ and \axiom{y}.
NullSquare
++ \axiom{NullSquare} means that \axiom{[x,x] = 0} holds.
JacobiIdentity
++ \axiom{JacobiIdentity} means that
++ \axiom{[x,[y,z]]+[y,[z,x]]+[z,[x,y]] = 0} holds.
if R has Field then
"/" : ($,R) -> $
++ \axiom{x/r} returns the division of \axiom{x} by \axiom{r}.
add
if R has Field then x / r == inv(r)$R * x
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