Clifford theory for group-graded rings.

*(English)*Zbl 0583.16001The best way to treat Clifford’s theory about the effect of a normal subgroup N on the irreducible representations of a finite group H is to consider the group algebra FH of H over any field F as a ring R naturally graded by the factor group \(G=H/N\). Then the \(1_ G\)-component \(R_ 1\) of R is the group algebra FN of N, while the \(\sigma\)-component \(R_{\sigma}\), for any \(\sigma\in G\), is the linear span \(F\sigma\) of the elements of the coset \(\sigma\in H/N\). In particular, \(R_{\sigma}\) contains at least one unit \(u\in \sigma\) of R for each \(\sigma\in G\), i.e. R is a crossed product of G over \(R_ 1.\)

It is well-known that Clifford’s theory extends almost without modification to simple modules over arbitrary crossed products R of the finite group G over rings \(R_ 1\) (it doesn’t work when G is infinite, even for group algebras). The subject of the present paper is the surprising observation that all parts of his theory can be made to work for simple modules over completely arbitrary rings R graded by a finite group G, provided one is willing to redefine (in a natural way) the notion of ”induction” of an \(R_ I\)-module L to an R-module \(L^ R\) whenever I is a subgroup of G and \(R_ I=\sum^{.}_{\sigma \in I}R_{\sigma}\) is the corresponding I-graded subring of R. One must also redefine ”G-conjugacy” for simple \(R_ 1\)-modules. Then our results can be stated as:

(1) The restriction \(M_{R_ 1}\) to \(R_ 1\) of any simple \(R_ 1\)- module M is isomorphic to the direct sum \(m\times (U_ 1\oplus \cdots \oplus U_ n)\) of m copies of \(U_ 1\oplus \cdots \oplus U_ n\), where \(m>0\) and \(U_ 1,\ldots,U_ n\) are representatives for the distinct isomorphism classes in some ”G-conjugacy class” of simple \(R_ 1\)- modules. - (2) If M ”lies over” \(U=U_ 1\) as in (1), then the U-primary component \(M\{\) \(U\}\) of the semi-simple \(R_ 1\)-module \(M_{R_ 1}\) is a simple \(R_{G\{U\}}\)-module lying over U, where \(G\{\) \(U\}\) is the stabilizing subgroup in G of the isomorphism class of the simple \(R_ 1\)-module U. Furthermore, M is isomorphic to the R-module ”induced” from \(M\{\) \(U\}\). Indeed, if L is any simple \(R_{G\{U\}}\)-module lying over U, then the ”induced” R-module \(L^ R\) is simple, lies over U, and has a U-primary component \(L^ R\{U\}\) isomorphic to L. - (3) The \(R_{G\{U\}}\)-endomorphism ring E of the \(R_{G\{U\}}\)-module S ”induced” from the \(R_ 1\)-module U is naturally a crossed product of \(G\{\) \(U\}\) over the division ring \(E_ 1\simeq End_{R_ 1}(U)\). In the case of right modules, any simple \(R_{G(U)}\)-module L lying over U yields a simple E-module \(L<U>=Hom_{R_ G\{U\}}(S,L)\) such that L is isomorphic to the \(R_{G\{U\}}\)-module \(L<U>\otimes_ ES\). Conversely, any simple E-module \(K\) determines a simple \(R_{G\{U\}}\)-module \(K\otimes_ ES\) lying over U such that \(K\simeq (K\otimes_ ES)<U>\). Similar statements can be made for left modules.

Of course, the maps of modules in (2) and (3) can be extended to additive functors forming equivalences of suitable categories. Evidently the combination of (1), (2) and (3) reduces the study of simple modules over rings R graded by a finite group G to that of simple modules over crossed products E of subgroups \(G\{\) \(U\}\) of G over endomorphism rings \(E_ 1\) of simple \(R_ 1\)-modules U.

It is well-known that Clifford’s theory extends almost without modification to simple modules over arbitrary crossed products R of the finite group G over rings \(R_ 1\) (it doesn’t work when G is infinite, even for group algebras). The subject of the present paper is the surprising observation that all parts of his theory can be made to work for simple modules over completely arbitrary rings R graded by a finite group G, provided one is willing to redefine (in a natural way) the notion of ”induction” of an \(R_ I\)-module L to an R-module \(L^ R\) whenever I is a subgroup of G and \(R_ I=\sum^{.}_{\sigma \in I}R_{\sigma}\) is the corresponding I-graded subring of R. One must also redefine ”G-conjugacy” for simple \(R_ 1\)-modules. Then our results can be stated as:

(1) The restriction \(M_{R_ 1}\) to \(R_ 1\) of any simple \(R_ 1\)- module M is isomorphic to the direct sum \(m\times (U_ 1\oplus \cdots \oplus U_ n)\) of m copies of \(U_ 1\oplus \cdots \oplus U_ n\), where \(m>0\) and \(U_ 1,\ldots,U_ n\) are representatives for the distinct isomorphism classes in some ”G-conjugacy class” of simple \(R_ 1\)- modules. - (2) If M ”lies over” \(U=U_ 1\) as in (1), then the U-primary component \(M\{\) \(U\}\) of the semi-simple \(R_ 1\)-module \(M_{R_ 1}\) is a simple \(R_{G\{U\}}\)-module lying over U, where \(G\{\) \(U\}\) is the stabilizing subgroup in G of the isomorphism class of the simple \(R_ 1\)-module U. Furthermore, M is isomorphic to the R-module ”induced” from \(M\{\) \(U\}\). Indeed, if L is any simple \(R_{G\{U\}}\)-module lying over U, then the ”induced” R-module \(L^ R\) is simple, lies over U, and has a U-primary component \(L^ R\{U\}\) isomorphic to L. - (3) The \(R_{G\{U\}}\)-endomorphism ring E of the \(R_{G\{U\}}\)-module S ”induced” from the \(R_ 1\)-module U is naturally a crossed product of \(G\{\) \(U\}\) over the division ring \(E_ 1\simeq End_{R_ 1}(U)\). In the case of right modules, any simple \(R_{G(U)}\)-module L lying over U yields a simple E-module \(L<U>=Hom_{R_ G\{U\}}(S,L)\) such that L is isomorphic to the \(R_{G\{U\}}\)-module \(L<U>\otimes_ ES\). Conversely, any simple E-module \(K\) determines a simple \(R_{G\{U\}}\)-module \(K\otimes_ ES\) lying over U such that \(K\simeq (K\otimes_ ES)<U>\). Similar statements can be made for left modules.

Of course, the maps of modules in (2) and (3) can be extended to additive functors forming equivalences of suitable categories. Evidently the combination of (1), (2) and (3) reduces the study of simple modules over rings R graded by a finite group G to that of simple modules over crossed products E of subgroups \(G\{\) \(U\}\) of G over endomorphism rings \(E_ 1\) of simple \(R_ 1\)-modules U.

##### MSC:

16W50 | Graded rings and modules (associative rings and algebras) |

16S34 | Group rings |

20C15 | Ordinary representations and characters |

16W20 | Automorphisms and endomorphisms |

16D30 | Infinite-dimensional simple rings (except as in 16Kxx) |

16S50 | Endomorphism rings; matrix rings |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |