# How about the Lie algebra over commutative ring?

It is just like the linear algebra over commutative ring (maybe advanced linear algebra), that is a nature extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic.

I just know Weibel’s book “Homological Algebra” discuss something, but that is just a form, do not pay more attention to the Lie algebra and show some difference between field coefficient and ring coefficient.

Does anybody know something else？

• Meanwhile I am missing a text systematically treating Hopf algebras over rings. For some reason, everybody seems to enjoy throwing around properties of vector spaces (bases, flatness etc.) without any need. – darij grinberg Dec 25 '11 at 6:34
• Why all the close votes? It is a reasonable reference request. – darij grinberg Dec 25 '11 at 8:59
• If you are interested in PBW for Lie algebras over rings, here is a nice (and wellwritten) paper: P.J. Higgins, Baer Invariants and the Birkhoff-Witt theorem, J. of Alg. 11, 469-482, (1969). (I can send you the paper; google my name for my email address.) – darij grinberg Dec 25 '11 at 8:59
• So many answers!I really do not know which one is the best. – Strongart Dec 28 '11 at 10:52

Lie algebras over rings (Lie rings) are important in group theory. For instance, to every group $G$ one can associate a Lie ring

$$L(G)=\bigoplus _{i=1}^\infty \gamma _i(G)/\gamma _{i+1}(G),$$

where $\gamma _i(G)$ is the $i$-th term of the lower central series of $G$. The addition is defined by the additive structure of $\gamma _i{G}/\gamma _{i+1}(G)$, and the Lie product is defined on homogeneous elements by $[x\gamma _{i+1}(G),y\gamma _{j+1}(G)]=[x,y]\gamma _{i+j+1}(G)$, and then extended to L(G) by linearity.

There are several other ways of constructing Lie rings associated to groups, and there are numerous applications of these. One of the most notorious ones is the solution of the Restricted Burnside Problem by Zelmanov, see the book M. R. Vaughan-Lee, "The Restricted Burnside Problem". There's other books related to these rings, for example, Kostrikin, "Around Burnside", Huppert, Blackburn, "Finite groups II", Dixon, du Sautoy, Mann, Segal, "Analytic pro-$p$ groups".

• Thank you,but I do not know much about group theory.For the group algebra kG,we can induce a Lie structure by [a,b]=ab-ba naturelly,that is why I care about the Lie algebra over commutative ring. – Strongart Dec 28 '11 at 10:59
• I have always seen the lower central series $\gamma$ indexed from $\gamma_1=G$. I always found this peculiar as the upper central series is indexed from $0$. Perhaps there is good reason to. – Ali Caglayan Feb 7 '17 at 2:53

Integral Lie algebras are also important in homotopy theory, if $X$ is a simply connected space and $\pi_*(X)=\bigoplus_{n\geq 1}\pi_n(X)$ are its homotopy groups, $\pi_*(X)[-1]$ is a graded Lie algebra. The Lie bracket is the Whitehead product. This Lie algebras satisfy the feature that $[x,x]\neq 0$ in general since over the integers this is not equivalent to antisymmetry.

There is a joke definition of a Lie algebra, due to my adviser John Moore, that is relevant. His definition of a Lie algebra over a commutative ring $R$ is that it is a module $L$ with a bracket operation such that there exists an associative $R$-algebra $A$ and a monomorphism $L \to A$ of $R$-modules that takes the bracket operation to the commutator in $A$. The point is to try to build in the PBW and dodge the question of which identities characterize Lie algebras. It is equivalent to the usual definition when $R$ is a field, as one sees by proving PBW using only the standard identities, but not so over a general commutative ring.

Even over a field (char $\neq 2$ for simplicity) there is an interesting contrast with the definition of a Jordan algebra. There the analogue of the commutator is $1/2 (ab + ba)$. One writes down the identities this satisfies and defines a Jordan algebra to be a vector space that satisfies the identities. But Jordan algebras do not generally embed in associative algebras (those that do are called special).

• Are you sure you want $A$ to be a commutative $R$-algebra? – Qiaochu Yuan Dec 26 '11 at 5:19
• He perhaps means it to be `associative'. – Shripad Dec 26 '11 at 17:34
• Just thank you to Qiaochu Yuan and Shripad. Answer has been edited. – Peter May Dec 27 '11 at 0:17

Some basic notions and results about Lie algebras are given in the generality you want in Chapter 1 of Bourbaki's "Lie Groups and Lie Algebras", and also in the book by J.-P. Serre "Lie Algebras and Lie Groups".

Actually, it is possible to go even further and define Lie algebras over noncommutative rings, see the paper Lie algebras and Lie groups over noncommutative rings by Berenstein and Retakh.

1. J.F. Hurley in a series of papers studied Lie algebras obtained by taking the multiplication table (with integer coefficients, due to Chevalley) of simple Lie algebras of classical or exceptional type and considering them over a commutative ring. The results describe center, ideal structure, etc. of such algebras in terms of the underlying ring. See, for example: Ideals in Chevalley algebras, Trans. Amer. Math. Soc. 137 (1969), 245-258; Composition series in Chevalley algebras Pacific J. Math. 32 (1970), 429-434; Centers of Chevalley algebras, J. Math. Soc. Japan 34 (1982), No.2, 219-222. In the joint paper with J. Morita (Affine Chevalley algebras, J. Algebra 72 (1981), N2, 359-373) he does something similar for some Kac--Moody algebras.

2. Some questions in free Lie algebras were considered over commutative rings, for example: D.Z. Djokovic, On some inner derivations of free Lie algebras over commutative rings, J. Algebra 119 (1988), 233-245, where centralizers of a member of a free generating set are studied. The latter reference is more or less random, probably more can be found in some books (Reutenauer?).

There are more instances of considering Lie algebras over commutative rings (for example, plenty of papers about automorphisms of some triangular or close to them algebras), but, unlike in the case of Lie algebras over fields, all these are some isolated examples, rather than a coherent theory. The book(s) of Bourbaki recommended by Anatoly Kochubei are, probably, interesting in that regard. Bourbaki tend to state things in the utmost generality, and it is educational to see how quickly they have to give up considering Lie algebras over rings and have to "throw around properties of vector spaces" (quoting Darij Grinberg).

Perhaps the question could be augmented slightly by asking what is the reason for the absence of such a theory for Lie algebras (as opposed, for example, for associative algebras). Perhaps this is related somehow to the fact that classifying some natural (e.g., simple) classes of associative algebras is easier then that of Lie algebras (e.g., root space decomposition technique for Lie algebras which works over algebraically closed fields vs. "idempotent" technique for associative algebras which works over arbitrary fields and even over rings), but I venture into a sheer speculation here.

One interesting fact about Lie algebras over commutative rings is that the PBW theorem can fail. I highly recommend the paper A remark on the Birkhoff-Witt theorem by P.M. Cohn. The fact that PBW can fail is probably addressed already in darij's comment to your original question, but I wanted to cite this paper separately because it is short (7 pages) and very well-written. In it, Cohn proves PBW for Lie algebras (over arbitrary commutative rings) whose additive groups are torsion-free, and also gives counterexamples where the additive group of the Lie algebra is a $p$-group.