{"id":25,"date":"2015-10-06T07:03:45","date_gmt":"2015-10-06T14:03:45","guid":{"rendered":"http:\/\/alexmennen.com\/?p=25"},"modified":"2022-01-28T17:08:26","modified_gmt":"2022-01-29T01:08:26","slug":"nonabelian-modules","status":"publish","type":"post","link":"http:\/\/alexmennen.com\/index.php\/2015\/10\/06\/nonabelian-modules\/","title":{"rendered":"Nonabelian modules"},"content":{"rendered":"\n

<\/p>\n\n\n\n

This is a rough overview of my thoughts on a thing I’ve been thinking about, and as such is incomplete and may contain errors. Proofs have been omitted when writing them out would be at all tedious.<\/p>\n\n\n\n

Edit: It has been pointed out to me<\/a> that near-ring modules have already been defined, and the objects I describe in this post are just near-ring modules where the near-ring happens to be a ring.<\/p>\n\n\n\n

Introduction<\/h3>\n\n\n\n

As you all know (those of you who have the background for this post, anyway), an \"R\"-module is an abelian group \"M\" (written additively) together with a multiplication map \"R\times M\rightarrow M\" such that for all \"\alpha,\beta\in R\" and \"x,y\in M\", \"\alpha\cdot\left(x+y\right)=\alpha\cdot x+\alpha\cdot y\", \"\left(\alpha+\beta\right)\cdot x=\alpha\cdot x+\beta\cdot x\", \"\left(\alpha\beta\right)\cdot x=\alpha\cdot\left(\beta\cdot x\right)\", and \"1\cdot x=x\".<\/p>\n\n\n\n

What if we don’t want to restrict attention to abelian groups? One\u00a0could attempt to define a nonabelian module using the same axioms,\u00a0but without the restriction that the group be abelian. As it is customary\u00a0to write groups multiplicatively if they are not assumed to be abelian,\u00a0we will do that, and the map \"R\times M\rightarrow M\" will be written\u00a0as exponentiation (since exponents are written on the right, I’ll\u00a0follow the definition of right-modules, rather than left-modules).\u00a0The axioms become: for all \"\alpha,\beta\in R\" and \"x,y\in M\", \"\left(xy\right)^{\alpha}=x^{\alpha}y^{\alpha}\",\u00a0\"x^{\alpha+\beta}=x^{\alpha}x^{\beta}\", \"x^{\left(\alpha\beta\right)}=\left(x^{\alpha}\right)^{\beta}\",\u00a0and \"x^{1}=x\".<\/p>\n\n\n\n

What has changed? Absolutely nothing, as it turns out. The first axiom\u00a0says again that \"M\" is abelian, because \"yx=x^{-1}\left(xy\right)^{2}y^{-1}=x^{-1}\left(x^{2}y^{2}\right)y^{-1}=xy\".\u00a0We’ll have to get rid of that axiom. Our new definition, which it\u00a0seems to me captures the essence of a module except for abelianness:<\/p>\n\n\n\n

A nonabelian \"R\"-module is a group \"M\" (written multiplicatively)\u00a0together with a scalar exponentiation map \"R\times M\rightarrow M\"\u00a0such that for all \"\alpha,\beta\in R\" and \"x\in M\", \"x^{1}=x\",\u00a0\"x^{\alpha+\beta}=x^{\alpha}x^{\beta}\", and \"x^{\left(\alpha\beta\right)}=\left(x^{\alpha}\right)^{\beta}\".<\/p>\n\n\n\n

These imply that \"x^{0}=1\", \"1^{\alpha}=1\", and \"x^{-1}\" is the\u00a0inverse of \"x\", because \"x\cdot x^{0}=x^{1}x^{0}=x^{1+0}=x^{1}=x\",\u00a0\"1^{\alpha}=\left(1^{0}\right)^{\alpha}=1^{\left(0\alpha\right)}=1^{0}=1\",\u00a0and \"x\cdot x^{-1}=x^{1-1}=1\".<\/p>\n\n\n\n

Just like a \"\mathbb{Z}\"-module is just an abelian group, a nonabelian\u00a0\"\mathbb{Z}\"-module is just a group. Just like a \"\mathbb{Z}/n\mathbb{Z}\"-module\u00a0is an abelian group whose exponent divides \"n\", a nonabelian \"\mathbb{Z}/n\mathbb{Z}\"-module\u00a0is a group whose exponent divides \"n\".<\/p>\n\n\n\n

Exponentiation-like families of operations<\/h3>\n\n\n\n

Perhaps a bit more revealing is what nonabelian modules over free\u00a0rings look like, since then the generators are completely generic\u00a0ring elements. Where \"A\" is the generating set, a \"\mathbb{Z}\left\langle A\right\rangle\"-module\u00a0is an abelian group together with endomorphisms \"\left\{ x\mapsto\alpha x\mid\alpha\in A\right\}\",\u00a0which tells us that modules are about endomorphisms of an abelian\u00a0group indexed by the elements of a ring. Nonabelian modules are certainly\u00a0not about endomorphisms. After all, in a nonabelian group, the map\u00a0\"x\mapsto x^{2}\" is not an endomorphism. I will call the things that\u00a0nonabelian modules are about “exponentiation-like families of operations”,\u00a0and give four equivalent definitions, in roughly increasing order\u00a0of concreteness and decreasing order of elegance. Definition 2 uses\u00a0basic model theory, so skip it if that scares you. Definition 3 is\u00a0the “for dummies” version of definition 2.<\/p>\n\n\n\n

Definition 0: Let \"G\" be a group, and let \"A\" be a family of functions\u00a0from \"G\" to \"G\" (not necessarily endomorphisms). If \"G\" can be\u00a0made into a nonabelian \"\mathbb{Z}\left\langle A\right\rangle\"-module\u00a0such that \"x^{\alpha}=\alpha\left(x\right)\" for \"x\in G\" and \"\alpha\in A\",\u00a0then \"A\" is called an exponentiation-like family of operations on\u00a0\"G\". If so, the nonabelian \"\mathbb{Z}\left\langle A\right\rangle\"-module\u00a0structure on \"G\" with that property is unique, so define \"x^{p}\"\u00a0to be its value according to that structure, for \"p\in\mathbb{Z}\left\langle A\right\rangle\"\u00a0and \"x\in G\".<\/p>\n\n\n\n

Definition 1: \"A\" is an exponentiation-like family of operations\u00a0on \"G\" if for all \"x\in G\", the smallest subgroup containing \"x\"\u00a0which is closed under actions by elements of \"A\" (which I will call\u00a0\"\overline{\left\{ x\right\} }\") is abelian, and the elements of\u00a0\"A\" restrict to endomorphisms of it. Using the universal property\u00a0of \"\mathbb{Z}\left\langle A\right\rangle\", this induces a homomorphism\u00a0\"\mathbb{Z}\left\langle A\right\rangle \rightarrow\text{End}\left(\overline{\left\{ x\right\} }\right)^{\text{op}}\".\u00a0Let \"x^{p}\" denote the action of \"p\" on \"x\" under that map, for\u00a0\"p\in\mathbb{Z}\left\langle A\right\rangle\". By \"\text{End}\left(\overline{\left\{ x\right\} }\right)^{\text{op}}\",\u00a0I mean the endomorphism ring of \"\overline{\left\{ x\right\} }\" with\u00a0composition running in the opposite direction (i.e., the multiplication\u00a0operation given by \"\left(f,g\right)\mapsto g\circ f\"). This is because\u00a0of the convention that nonabelian modules are written as nonabelian\u00a0right-modules by default.<\/p>\n\n\n\n

Definition 2: Let consider the language \"\mathcal{L}_{Rings}\sqcup A\",\u00a0where \"\mathcal{L}_{Rings}:=\left\{ 0,1,+,-,\cdot\right\}\" is the\u00a0language of rings, and each element of \"A\" is used as a constant\u00a0symbol. Closed terms in \"\mathcal{L}_{Rings}\sqcup A\" act as functions\u00a0from \"G\" to \"G\", with the action of \"t\" written as \"x\mapsto x^{t}\",\u00a0defined inductively as: \"x^{0}:=1\", \"x^{1}:=x\", \"x^{\alpha}:=\alpha\left(x\right)\"\u00a0for \"\alpha\in X\", \"x^{t+s}:=x^{t}x^{s}\", \"x^{-t}:=\left(x^{t}\right)^{-1}\",\u00a0and \"x^{ts}:=\left(x^{t}\right)^{s}\" for closed \"\mathcal{L}_{Rings}\sqcup A\"-terms\u00a0\"t\" and \"s\". \"A\" is called an exponentiation-like family of operations\u00a0on \"G\" if \"x^{t}=x^{s}\" whenever \"T_{Rings}\models t=s\", where\u00a0\"T_{Rings}\" is the theory of rings. If \"A\" is an exponentiation-like\u00a0family of operations on \"G\" and \"p\in\mathbb{Z}\left\langle A\right\rangle\"\u00a0is a noncommutative polynomial with variables in \"A\", then for \"x\in G\",\u00a0\"x^{p}\" is defined to be \"x^{t}\" where \"t\" is any term representing\u00a0\"p\".<\/p>\n\n\n\n

Definition 3: Pick a total order on the free monoid on \"A\" (e.g.\u00a0by ordering \"A\" and then using the lexicographic order). The order\u00a0you use won’t matter. Given \"x\in G\" and \"w:=\alpha_{1}...\alpha_{n}\"\u00a0in the free monoid on \"A\", let \"x^{w}=\alpha_{n}\left(...\alpha_{1}\left(x\right)\right)\".\u00a0Where \"p\in\mathbb{Z}\left\langle A\right\rangle\" is a noncommutative\u00a0polynomial, \"p=c_{1}w_{1}+...+c_{n}w_{n}\" for some \"c_{1},...,c_{n}\in\mathbb{Z}\"\u00a0and decreasing sequence \"w_{1},...,w_{n}\" of noncommutative monomials\u00a0(elements of the free monoid on \"A\"). Let \"x^{p}=\left(x^{c_{1}}\right)^{w_{1}}...\left(x^{c_{n}}\right)^{w_{n}}\".\u00a0\"A\" is called an exponentiation-like family of operations on \"G\"\u00a0if for every \"x\in G\" and \"p,q\in\mathbb{Z}\left\langle A\right\rangle\",\u00a0\"x^{pq}=\left(x^{p}\right)^{q}\" and \"x^{p+q}=x^{p}x^{q}\".<\/p>\n\n\n\n

These four definitions of exponentiation-like family are equivalent, and for exponentiation-like families, their definitions of exponentiation by a noncommutative polynomial are equivalent.<\/p>\n\n\n\n

Facts: \"\emptyset\" is an exponentiation-like family of operations\u00a0on \"G\". If \"A\" is an exponentiation-like family of operations on\u00a0\"G\" and \"B\subseteq A\", then so is \"B\". If \"G\" is abelian, then\u00a0\"\text{End}\left(G\right)\" is exponentiation-like. Given a nonabelian\u00a0\"R\"-module structure on \"G\", the actions of the elements of \"R\"\u00a0on \"G\" form an exponentiation-like family. In particular, if \"A\"\u00a0is an exponentiation-like family of operations on \"G\", then so is\u00a0\"\mathbb{Z}\left\langle A\right\rangle\", with the actions being\u00a0defined as above.<\/p>\n\n\n\n

[The following paragraph has been edited since this comment<\/a>.]<\/p>\n\n\n\n

For an abelian group \"A\", the endomorphisms of \"A\" form a ring \"\text{End}\left(A\right)\",\u00a0and an \"R\"-module structure on \"A\" is simply a homomorphism \"R\rightarrow\text{End}\left(A\right)\".\u00a0Can we say a similar thing about exponentiation-like families of operations\u00a0of \"G\"? Let \"\text{Exp}\left(G\right)\" be the set of all functions \"G\rightarrow G\" (as sets). Given \"\alpha,\beta\in\text{Exp}\left(G\right)\",\u00a0let multiplication be given by composition: \"x^{\left(\alpha\beta\right)}=\left(x^{\alpha}\right)^{\beta}\",\u00a0addition be given by \"x^{\alpha+\beta}=x^{\alpha}x^{\beta}\", negation\u00a0be given by \"x^{-\alpha}=\left(x^{\alpha}\right)^{-1}\", and \"0\"\u00a0and \"1\" be given by \"x^{0}=1\" and \"x^{1}=x\". This makes \"\text{Exp}\left(G\right)\" into a near-ring. A nonabelian \"R\"-module structure on \"G\"\u00a0is a homomorphism \"R\rightarrow\text{Exp}\left(G\right)\", and a set\u00a0of operations on \"G\" is an exponentiation-like family of operations\u00a0on \"G\" if and only if it is contained in a ring which is contained\u00a0in \"\text{Exp}\left(G\right)\".<\/p>\n\n\n\n

Some aimless rambling<\/h3>\n\n\n\n

What are some interesting examples of nonabelian modules that are\u00a0not abelian? (That might sound redundant, but “nonabelian module”\u00a0means that the requirement of abelianness has been removed, not that\u00a0a requirement of nonabelianness has been imposed. Perhaps I should\u00a0come up with better terminology. To make matters worse, since the\u00a0requirement that got removed is actually stronger than abelianness,\u00a0there are nonabelian modules that are abelian and not modules. For\u00a0instance, consider the nonabelian \"\mathbb{Z}\left[\alpha\right]\"-module\u00a0whose underlying set is the Klein four group (generated by two elements \"a,b\") such that \"a^{\alpha}=a\", \"b^{\alpha}=b\", and \"\left(ab\right)^{\alpha}=1\".)<\/p>\n\n\n\n

In particular, what do free nonabelian modules look like? The free\u00a0nonabelian \"\mathbb{Z}\"-modules are, of course, free groups. The\u00a0free nonabelian \"\mathbb{Z}/n\mathbb{Z}\"-modules have been studied\u00a0in combinatorial group theory; they’re called Burnside groups. (Fun\u00a0but tangential fact: not all Burnside groups are finite (the Burnside\u00a0problem), but despite this, the category of finite nonabelian \"\mathbb{Z}/n\mathbb{Z}\"-modules\u00a0has free objects on any finite generating set, called Restricted Burnside\u00a0groups.)<\/p>\n\n\n\n

The free nonabelian \"\mathbb{Z}\left[\alpha\right]\"-modules are monstrosities.\u00a0They can be constructed in the usual way of constructing free objects\u00a0in a variety of algebraic structures, but that construction seems\u00a0not to be very enlightening about their structure. So I’ll give a\u00a0somewhat more direct construction of the free nonabelian \"\mathbb{Z}\left[\alpha\right]\"-module\u00a0on \"d\" generators, which may also not be that enlightening, and which\u00a0is only suspected to be correct. Define an increasing sequence of\u00a0groups \"G_{n}\", and functions \"\alpha_{n}:G_{n}\rightarrow G_{n+1}\",\u00a0as follows: \"G_{0}\" is the free group on \"d\" generators. Given \"G_{n}\",\u00a0and given a subgroup \"X\leq G_{n}\", let the top-degree portion of \"X\" be \"\alpha_{n-1}^{k}\left(X\right)\" for the largest \"k\" such\u00a0that this is nontrivial. Let \"H_{n}\" be the free product of the top-degree\u00a0portions of maximal abelian subgroups of \"G_{n}\". Let \"G_{n+1}\" be the free product of \"G_{n}\" with \"H_{n}\" modulo commutativity\u00a0of the maximal abelian subgroups of \"G_{n}\" with the images of their\u00a0top-degree portions in \"H_{n}\". Given a maximal abelian subgroup \"X\leq G_{n}\", let \"\alpha_{n}\restriction_{X}\" be the homomorphism\u00a0extending \"\alpha_{n-1}\restriction_{X\cap G_{n-1}}\" which sends\u00a0the top-degree portion identically onto its image in \"H_{n}\". Since\u00a0every non-identity element of \"G_{n}\" is in a unique maximal abelian\u00a0subgroup, this defines \"\alpha_{n}\". \"G:=\bigcup_{n}G_{n}\" with\u00a0\"\alpha:=\bigcup_{n}\alpha_{n}\" is the free nonabelian \"\mathbb{Z}\left[\alpha\right]\"-module\u00a0on \"d\" generators. If \"A\" is a set, the free nonabelian \"\mathbb{Z}\left\langle A\right\rangle\"-modules\u00a0can be constructed similarly, with \"\left|A\right|\" copies of \"H_{n}\" at each step. Are these constructions even correct? Are there nicer\u00a0ones?<\/p>\n\n\n\n

A nonabelian \"\mathbb{Z}\left[\frac{1}{2}\right]\"-module would be\u00a0a group with a formal square root operation. As an example, any group\u00a0of odd exponent \"n\" can be made into a \"\mathbb{Z}\left[\frac{1}{2}\right]\"-module\u00a0in a canonical way by letting \"x^{\frac{1}{2}}=x^{\frac{n+1}{2}}\".\u00a0More generally, any group of finite exponent \"n\" can be made into\u00a0a \"\mathbb{Z}\left[\left\{ p^{-1}|p\nmid n\right\} \right]\"-module\u00a0in a similar fashion. Are there any more nice examples of nonabelian\u00a0modules over localizations of \"\mathbb{Z}\"?<\/p>\n\n\n\n

In particular, a nonabelian \"\mathbb{Q}\"-module would be a group\u00a0with formal \"n\"th root operations for all \"n\". What are some nonabelian\u00a0examples of these? Note that nonabelian \"\mathbb{Q}\"-modules cannot\u00a0have any torsion, for suppose \"x^{n}=1\" for some \"n\neq0\". Then\u00a0\"x=\left(x^{n}\right)^{\frac{1}{n}}=1^{\frac{1}{n}}=1\". More generally,\u00a0nonabelian modules cannot have any \"n\"-torsion (meaning \"x^{n}=1\implies x=1\")\u00a0for any \"n\" which is invertible in the scalar ring.<\/p>\n\n\n\n

The free nonabelian \"\mathbb{Z}\left[\frac{1}{m}\right]\"-modules\u00a0can be constructed similarly to the construction of free nonabelian \"\mathbb{Z}\left[\alpha\right]\"-modules above, except that when constructing \"G_{n+1}\" from \"G_{n}\" and \"H_{n}\", we also mod out by elements\u00a0of \"G_{n}\" being equal to the \"m\"th powers of their images in \"H_{n}\".\u00a0Using the fact that \"\mathbb{Q}\cong\mathbb{Z}\left\langle \left\{ p^{-1}|\text{primes }p\right\} \right\rangle\",\u00a0this lets us modify the construction of free nonabelian \"\mathbb{Z}\left\langle A\right\rangle\"-modules\u00a0to give us a construction of free nonabelian \"\mathbb{Q}\"-modules.\u00a0Again, is there a nicer way to do it?<\/p>\n\n\n\n

Topological nonabelian modules<\/h3>\n\n\n\n

It is also interesting to consider topological nonabelian modules\u00a0over topological rings; that is, nonabelian modules endowed with a\u00a0topology such that the group operation and scalar exponentiation are\u00a0continuous. A module over a topological ring has a canonical finest\u00a0topology on it, and the same remains true for nonabelian modules.\u00a0For finite-dimensional real vector spaces, this is the only topology.\u00a0Does the same remain true for finitely-generated nonabelian \"\mathbb{R}\"-modules?\u00a0Finite-dimensional real vector spaces are complete, and topological\u00a0nonabelian modules are, in particular, topological groups, and can\u00a0thus be made into uniform spaces, so the notion of completeness still\u00a0makes sense, but I think some finitely-generated nonabelian \"\mathbb{R}\"-modules\u00a0are not complete.<\/p>\n\n\n\n

A topological nonabelian \"\mathbb{R}\"-module is a sort of Lie group-like\u00a0object. One might try constructing a Lie algebra for a complete nonabelian \"\mathbb{R}\"-module \"M\" by letting the underlying set be \"M\", and\u00a0defining \"x+y=\lim_{\varepsilon\rightarrow0}\left(x^{\varepsilon}y^{\varepsilon}\right)^{\left(\varepsilon^{-1}\right)}\"\u00a0and \"\left[x,y\right]=\lim_{\varepsilon\rightarrow0}\left(x^{\varepsilon}y^{\varepsilon}x^{-\varepsilon}y^{-\varepsilon}\right)^{\left(\varepsilon^{-2}\right)}\".\u00a0One might try putting a differential structure on \"M\" such that this\u00a0is the Lie algebra of left-invariant derivations. Does this or something\u00a0like it work?<\/p>\n\n\n\n

A Lie group is a nonabelian \"\mathbb{R}\"-module if and only if its\u00a0exponential map is a bijection between it and its Lie algebra. In\u00a0this case, scalar exponentiation is closely related to the exponential\u00a0map by a compelling formula: \"x^{\alpha}=\exp\left(\alpha\exp^{-1}\left(x\right)\right)\".\u00a0As an example, the continuous Heisenberg group is a nonabelian \"\mathbb{R}\"-module\u00a0which is not abelian. This observation actually suggests a nice class\u00a0of examples of nonabelian modules without a topology: given a commutative\u00a0ring \"R\", the Heisenberg group over \"R\" is a nonabelian \"R\"-module.<\/p>\n\n\n\n

The Heisenberg group of dimension \"2n+1\" over a commutative ring \"R\" has underlying set \"R^{n}\times R^{n}\times R^{1}\", with the\u00a0group operation given by \"\left(\boldsymbol{a}_{1},\boldsymbol{b}_{1},c_{1}\right)*\left(\boldsymbol{a}_{2},\boldsymbol{b}_{2},c_{2}\right):=\left(\boldsymbol{a}_{1}+\boldsymbol{a}_{2},\boldsymbol{b}_{1}+\boldsymbol{b}_{2},c_{1}+c_{2}+\boldsymbol{a}_{1}\cdot\boldsymbol{b}_{2}-\boldsymbol{a}_{2}\cdot\boldsymbol{b}_{1}\right)\".\u00a0The continuous Heisenberg group means the Heisenberg group over \"\mathbb{R}\".\u00a0Scalar exponentiation on a Heisenberg group is just given by scalar\u00a0multiplication: \"\left(\boldsymbol{a},\boldsymbol{b},c\right)^{\alpha}:=\left(\alpha\boldsymbol{a},\alpha\boldsymbol{b},\alpha c\right)\".<\/p>\n","protected":false},"excerpt":{"rendered":"

This is a rough overview of my thoughts on a thing I’ve been thinking about, and as such is incomplete and may contain errors. Proofs have been omitted when writing them out would be at all tedious. Edit: It has been pointed out to me that near-ring modules have already been defined, and the objects … Continue reading Nonabelian modules<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-25","post","type-post","status-publish","format-standard","hentry","category-math"],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/posts\/25","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/comments?post=25"}],"version-history":[{"count":17,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/posts\/25\/revisions"}],"predecessor-version":[{"id":305,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/posts\/25\/revisions\/305"}],"wp:attachment":[{"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/media?parent=25"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/categories?post=25"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/alexmennen.com\/index.php\/wp-json\/wp\/v2\/tags?post=25"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}