In early schooling, concepts are taught such as multiplication and division.  One first is taught a few examples, such as what 2 + 4 is.  At some point they realize 2 + 4 = 4 + 2 and 3 + 6 = 6 + 3 and that this sort of symmetry works for any numbers.  In general a + b = b + a for any a and b.  This is called the commutative law.  Notice that it also holds for multiplication as ab=ba for any real numbers a and b.  Another important law is the associative law: (a + b) + c = a + (b + c).  This one holds for multiplication as well, since (ab)c=a(bc) for any real numbers a, b, and c.
            What is addition really?  We all have an instinctive feel for the concept.  Basically addition takes two numbers, and gives back a third.  Thus it is really a function, a sort of machine.  We put in any two elements, which we take from the set (or collection) of numbers and we get a third.  This third output number satisfies certain laws like the commutative and associative law above.
            Addition and multiplication satisfy many similar rules.  One can step aside from these two examples of objects satisfying these rules and study all such objects which satisfy them.  Taking the commutative law, associative law, and two other laws, we get a particular structure we call an abelian group.  We can start to look for all the different examples of abelian groups instead of just individual examples.  We can do the same for different algebraic structures as well.
            The laws of commutativity seem somewhat intuitive, as one is used to doing addition and subtraction from grade school.  However one could look at a different set of laws and ask what structures fulfill those laws as well.  A Lie Algebra is a special example of a set with a function that satisfies different laws.  These laws may look strange, but do arise naturally in certain situations.  First of all, the set upon which these operations take place is called a vector space and has certain properties of its own.  Upon this set there is an operation which takes two elements and spits out a third.  Instead of writing a + b or ab for the output we write [ab].  Here [ab] stands for the output of a and b.  A Lie Algebra, L, satisfies a rule for being bilinear (which deals with the vector space) as well as the following rules:
1) [aa]=0 for any a
2) [a[bc]]+[b[ca]]+[c[ab]]=0 for any a, b, and c

 
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