What Are the Differences Between Group and Algebra Terminology?

[bronxman]

(Again, I am sorry for the simplicity of these questions. I am a mechanical engineer learning this stuff late in life.)

I have read the following terms or phrases:
group
algebra
group algebra
the algebra of a group
an algebra group
an algebraic group
a group of algebras

So... could someone please explain this to me?
I understand the stand-alone definitions. But this mixing of terms completely baffles me.
Could some one please take a moment to give a coherent and stand-alone definition of each of these things... You can go easy on the math -- I am slowly getting it (I think): I have an understanding of groups, rings, fields, etc. I am as interested in the syntax of how these words are used.

Are they nouns or adjectives or adverbs? Yes, that is the level of my confusion.

Is algebra a "discipline of math" or " a "thing" or a "class of things."

Here is an example: I have read: Lie algebras are closely related to Lie Groups

(Again, I am sorry: I KNOW these are stupid questions.)

I can say a Macintosh and a Golden Delicious are closely related because they are both apples.
I can say a computer and a desk are closely related because they are both things.
But I cannot say that GREEN and LOVE are related: they are different things.
So what is it in the definition of groups and algebras that can be related?

 

[Stephen Tashi]

The first thing to understand is that mathematical definitions are not created by defining individual words. For example, in calculus, you studied a definition for "The limit of f(x) as x approaches 'a' = L". That definition (the "epsilon-delta definition") does not contain a discussion of the meaning "approaches". In fact, that definition does not contain a discussion of the meaning of the isolated word "limit", even though people refer to the definition as "the definition of limit". The definition gives the meaning of a statement (i.e. a complete sentence). The sentence incorporates the words "approaches" and "limit", but the definition does not give a statement for what the individual words mean.

For example, what people call "the definition of an group algebra" is not actually a definition of the isolated words "group" and "algebra". Technically, it is a definition of the complete sentence such as " G is a group algebra". The appearance of the individual words "group" and "algebra" can be explained, but the explanation is essentially a cultural and historical explanation.

There are systems of terminology that have rigorous rules, such as schemes for writing the names of chemical compounds. In those systems, one may interpret the meaning of a "phrase" (such as C_2\ H_{12}\ O_{11} ) by looking at the individual pieces that make up the term. Much mathematical terminology is not that systematic. (For example, there is the famous saying that "A random variable is not random and it is not a variable".) When it comes to "groups" and "algebras", interpreting the individual words in a phrase like "group algebra" can give you a hint about what the definition says, but it isn't sufficient to let you deduce the exact definition. Once you have read the actual definition of "group algebra", the individual words might aid you in remembering it. You can say to yourself "Aha! It involves an algebra constructed by using a group."

Your questions can be answered in the sense that we can explain "What is the motivation for the teminology "Lie Algebra"?". But it isn't possible to answer the question: "How can I deduce the precise definition of 'Lie Algebra' if I know already know the meanings of
'Lie Group' and 'Algebra' ?" The mathematical terminology isn't systematic enough to let you do that.