# modular theory

A representation  of a group/algebra  into $GL(V,k)$, $End_{k}V$, $\mathfrak{gl}_{k}(V)$, etc. is called modular if the characteristic  of $k$ is positive. An algebra $A$ is called modular if the characteristic of the field $k$ is positive (thus prime for some $p$.) As the algebra can be represented in $End_{k}A$ by left or right multiplication  (often called the regular   or adjoint representation  ) this definition matches the requirements for a modular representation.

Typically theory for representations over fields of characteristic 0 cannot be transplanted directly to modular representations. For example, Maschke’s theorem, Lie’s theorem, the use of the Killing form   for a Lie algebra  all fail in various ways in modular representations. There are two common obstructions in modular representations.

1. 1.
2. 2.

There are also two common workarounds for these obstructions.

1. 1.

Treat the rational coefficients as formal coefficients. For example, start in characteristic 0 and define an integer subalgebra  with basis elements of the form $\frac{x^{n}}{n!}$ (or similarly useful combinations   .) Then tensor (over $\mathbb{Z}$) the $\mathbb{Z}$-subalgebra with $k$. Thus the fractions of the form $1/n!$ are not actually coefficients in $k$ but formal coefficients and so they do not cause division by 0. This technique is used in the theory of Chevalley groups over arbitrary fields as it allows for version of the exponential of a nilpotent element of a Lie algebra.

2. 2.

Make restrictions   on the dimensions  of the representation, for instance, $(p,\dim V)=1$ or $x^{n}=0$ for $n etc. Thus the division by 0 and polynomial oddities are avoided. For example, with Maschke’s theorem the solution is to assume that the characteristic $p$ does not divide the order of the group $G$ which is being represented. The second approach to insist on large $p$ often excludes $p=2,3$ but provides workable results for $p\geq 5$, or 7, etc. Then the small prime cases are studied as exceptional examples.

There is a third problem which can arise for modular representations which has no obvious work around. This is when the definition and/or theorems still work but their implications  are useless.

For example, with the Killing form of a Lie algebra we take the trace of a linear transformation. However, over characteristic $p$ it is possible for the identity matrix  to have trace 0, for example, in dimension $d=mp$ for any integer $m$. This situation cannot be be avoided as easily as with Maschke’s theorem by assuming restrictions on the dimension of the representation. For we can use $d=2$ and the diagonal matrix  $Diag(1,p-1)$ to obtain a similar  problem with the trace. And this problem embeds into all higher dimensions thus a straight forward use of trace would work only in 1 dimensional representations.

There is no problem with the definition of the Killing form for modular representations but the results may no longer be applicable. In these situations usually entirely new approaches are required.

Title modular theory ModularTheory 2013-03-22 16:26:45 2013-03-22 16:26:45 Algeboy (12884) Algeboy (12884) 6 Algeboy (12884) Definition msc 17B99 RestrictedLieAlgebra modular theory modular representation