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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the classification of simple antiflexible algebras


Author: Mahesh Chandra Bhandari
Journal: Trans. Amer. Math. Soc. 173 (1972), 159-181
MSC: Primary 17A20
MathSciNet review: 0313334
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Abstract: In this paper, we begin a classification of simple totally antiflexible algebras (finite dimensional) over splitting fields of characteristic $ \ne 2,3$. For such an algebra $ A$ let $ P$ be the largest associative ideal in $ {A^ + }$ and let $ N$ be the radical of $ P$. We say that $ A$ is of type $ (m,n)$ if $ N$ is nilpotent of class $ m$ with $ \dim A = n$. Define $ {N_i} = {N_{i - 1}} \cdot N,{N_1} = N$, then $ A$ is said to be of type $ (m,n,{d_1},{d_2}, \cdots ,{d_q})$ if $ A$ is of type $ (m,n),\dim ({N_i} - {N_{i - 1}}) = {d_i}$ for $ 1 \leqslant i \leqslant q$ and $ \dim ({N_i} - {N_{i + 1}}) = 1$ for $ q < i < m$. We then determine all nodal simple totally antiflexible algebras of types $ (n,n),(n - k,n,k + 1),(n - 2,n)$ (over fields of characteristic $ \ne 2,3$) and of type (3, 6) (over the field of complex numbers). We also give preliminary results for nodal simple totally antiflexible algebras of type $ (n - k,n,k,2)$ and of type $ (m,n,{d_1}, \cdots ,{d_q})$ in general with $ m > 2$ (the case $ m = 2$ has been classified by D. J. Rodabaugh).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0313334-3
PII: S 0002-9947(1972)0313334-3
Keywords: Simple algebras, antiflexible algebras, totally antiflexible algebras
Article copyright: © Copyright 1972 American Mathematical Society