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A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $mathbb{L}$ of $mathbb{F}$: (1) finite-dimensional central division algebras over $mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups over $mathbb{L}$, up to graded-isomorphism; (3) $mathbb{F}$-forms of certain graded matrix algebras with coefficients in $Deltaotimes_{mathbb{L}}mathcal{C}$ where $Delta$ is as in (1) and $mathcal{C}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.
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We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.
Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian subgroup $H$, a positive integer $d$ and an element of $Hom(M(H), k^times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main theorem is the converse: Given an extension $1rightarrow Hrightarrow Grightarrow G/Hrightarrow 1$, where $H$ is abelian, a positive integer $d$, and an element of $Hom(M(H), k^times)^G$, there is a division algebra with center containing $k$ that realizes these data. We apply this result to classify the $G$-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field.
Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. We consider graded Verbally prime $T$-ideals in the free $G$-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime $G$-graded $T$-ideals and strongly verbally prime $T$-ideals. At first, following Kemers ideas, we classify $G$-graded verbally prime $T$-ideals. The main bulk of the paper is devoted to the stronger notion. We classify $G$-graded strongly verbally prime $T$-ideals which are $T$-ideal of affine $G$-graded algebras or equivalently $G$-graded $T$-ideals that contain a Capelli polynomial. It turns out that these are precisely the $T$-ideal of $G$-graded identities of finite dimensional $G$-graded, central over $F$ (i.e. $Z(A)_{e}=F$) which admit a $G$-graded division algebra twisted form over a field $k$ which contains $F$ or equivalently over a field $k$ which contains enough roots of unity (e.g. a primitive $n$-root of unity where $n = ord(G)$).
We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.
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