Difference between revisions of "Albert algebra"
From Encyclopedia of Mathematics
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| − | The set of 3×3 [[self-adjoint]] matrices over the [[octonion]]s with binary operation | + | |
| + | The set of 3×3 [[Self-adjoint linear transformation|self-adjoint]] matrices over the [[octonion]]s with binary operation | ||
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x \star y = \frac12 (x \cdot y + y \cdot x) \,, | x \star y = \frac12 (x \cdot y + y \cdot x) \,, | ||
Revision as of 18:48, 6 September 2013
2020 Mathematics Subject Classification: Primary: 17C40 [MSN][ZBL]
The set of 3×3 self-adjoint matrices over the octonions with binary operation $$ x \star y = \frac12 (x \cdot y + y \cdot x) \,, $$ where $\cdot$ denotes matrix multiplication.
The operation is commutative but not associative. It is an example of an exceptional Jordan algebra. Because most other exceptional Jordan algebras are constructed using this one, it is often referred to as "the" exceptional Jordan algebra.
References
- A. V. Mikhalev, Gunter F. Pilz (2002) "The Concise Handbook of Algebra", (Springer) ISBN 0792370724, page 346.
- Richard D. Schafer (1995) [1966], An Introduction to Nonassociative Algebras (Dover) ISBN 0-486-68813-5, page 102.
How to Cite This Entry:
Albert algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Albert_algebra&oldid=30386
Albert algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Albert_algebra&oldid=30386