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I have a semisimple algebra $R$ over a field $k$ that looks like a group algebra $k[G]$ except that it's deformed slightly. That is, instead of a basis $e_1 ..., e_n$ closed under multiplication, I have a set of $1$-dimensional subspaces $v_1,...,v_n$, such that their direct sum is $R$ (that is, a choice of one generator from each subspace forms a basis) which is also closed under multiplication (that is, for each $i$ and $j$, there is some $k$ such that $v_i v_j = v_k$.)

Does this concept, or a slight variation on it, appear in the literature?

This is not a Hopf algebra. There is a natural "diagonal" subalgebra of $R \otimes_k R$, but it is not necessarily or naturally isomorphic to $R$.

The main reason I can't just view this as a semisimple algebra when I work with it is the "diagonal" subalgebra structure. There is some basic representations theory involving this that I would not like to duplicate.

$R \otimes_k \bar{k}$ is not necessarily a group algebra. For instance, this is false for $R$ a quaternion algebra.

Other examples include field extensions generated by $n$th roots, and a group algebra modulo the ideal identifying a $2$-torson central element with $-1$. Nonexamples include field extensions not generated by $n$th roots.

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there is something called "twisted group algebra". – Dima Pasechnik Sep 15 '12 at 6:06
$G$-graded ring?… – Simon Wadsley Sep 15 '12 at 7:12
up vote 3 down vote accepted

Yes, Dima is right, these are twisted group algebras. They are discussed in this MathOverflow Question

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