# The “wrong” definitions of “Algebra” and “Lie Algebra”

I am reading Robert Mann's "An Introduction to Particle Physics and the Standard Model" and a review from Amazon (with 12 upvotes) concerns me:

...The definition of algebra on page 54 is wrong. The definition of Lie algebra also on page 54 is wrong. I don't mean typo, I mean wrong...

The definition of Algebra in the book is:

An algebra is a vector space $$V = \{v_I\}$$ with a binary (combining) operation $$\circ$$ that obeys this rule: $$v^I \circ v^J = \sum_{k} C^{IJ}_kv^k = C^{IJ}_kv^k$$, where $$C^{IJ}_k$$ is known as structure constant.

Whilst the definition of Lie Algebra is:

A Lie algebra is an algebra whose combining operation is the commutator: $$\circ = [,]$$ where $$[T^a,T^b] = if^{ab}_cT^c$$ and it is the Taylor-series expansion of a Lie Group. The group associativity implies the Jacobi identity: $$[[T^a,T^b],T^c] + [[T^b,T^c],T^a] + [[T^c,T^a],T^b] = 0$$.

As a novice in the subject, I can't tell what is wrong with these definitions. Can anyone clarify what's wrong with them and give the correct definitions?

• They look about right to me (an algebra can be abstracted further to modules, I believe). Not everyone on the internet is correct. Maybe that includes me. In any case, Lie algebras at a minimum are common enough there's a torrent of sources stating the definition from various points of view (equivalent, but the words aren't always the same) you can compare against. – Richard Myers Mar 16 at 4:24
• I think the definition of a Lie algebra needs some additional constraints on the structure constants, but I'm not sure without checking the details. Other than that I think these definitions are fine, although I can see how a mathematically inclined reader might find them imprecise. – d_b Mar 16 at 4:30
• You can have mathematical fun with the differences between associative and non-associative algebras, algebras over rings and algebras over fields, unitary and non-unitary algebras (or should that be "unital" not "unitary?), etc. Or you can just get on and do some physics. From the title, I guess this book isn't meant for mathematicians to want to quibble over details. – alephzero Mar 16 at 5:17
• You could compare these definitions with those of Wikipedia for algebra and Lie algebra. Mathematicians don’t like thinking of Lie algebras in terms of commutators. $[a,b]$ in a Lie algebra is supposed to be just a “Lie bracket”, a bilinear product producing another element of the algebra; there is no assumption that $ab$ or $ba$ or $ab-ba$ is defined. – G. Smith Mar 16 at 5:23
• I wasn't sure which definition of "algebra" should I look for until @G.Smith gave the relevant Wikipedia page. It's good to know that the definitions above are acceptable from the physics perspective. Appreciate the answers. – cZe99 Mar 16 at 9:16

2. The book never states that the binary (multiplication/bracket) operator for the Lie algebra will always be anticommutative, $$[x,y]=-[y,x]$$ (note this is anticommutative where $$A$$ times $$B$$ is $$[A,B]$$ rather than $$AB$$). Since we are going to use the commutator $$[x,y]=-[y,x]$$ is obvious.
There is a technical third point that the lie algebra binary operator isn't stated to satisfy $$[x,x] = 0$$. Even if we correct point 2, if the field that the algebra is defined over has field characteristic 2 then we find that $$2a = 0 \quad \forall a$$. This means that while normally $$[x,y] = -[y,x] \Rightarrow [x,x] = 0$$, we instead find that $$[x,x] = -[x,x]$$ but that is true for any element of your lie algebra, not just for zero and so we could have $$[x,x] \neq 0$$. We tend to work over the real or complex numbers, so the field is of characteristic 0 and this point again isn't an issue.