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Register a new userOn nonassociative graded-simple algebras over the field of real numbers
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We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence.
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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.
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.
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.
We study the general properties of commutative differential graded algebras in the category of representations over a reductive algebraic group with an injective central cocharacter. Besides describing the derived category of differential graded modules over such an algebra, we also provide a criterion for the existence of a t-structure on the derived category together with a characterization of the coordinate ring of the Tannakian fundamental group of its heart.
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