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Mutation (algebra)

释义

Definitions
Properties
Jordan algebras
References

In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original.

Definitions

Let A be an algebra over a field F with multiplication (not assumed to be associative) denoted by juxtaposition. For an element a of A, define the left a-homotope to be the algebra with multiplication

Similarly define the left (a,b) mutation

Right homotope and mutation are defined analogously. Since the right (p,q) mutation of A is the left (−q, −p) mutation of the opposite algebra to A, it suffices to study left mutations.^[1]

If A is a unital algebra and a is invertible, we refer to the isotope by a.

Properties

If A is associative then so is any homotope of A, and any mutation of A is Lie-admissible.
If A is alternative then so is any homotope of A, and any mutation of A is Malcev-admissible.^[1]
Any isotope of a Hurwitz algebra is isomorphic to the original.^[1]
A homotope of a Bernstein algebra by an element of non-zero weight is again a Bernstein algebra.^[2]

Jordan algebras

A Jordan algebra is a commutative algebra satisfying the Jordan identity . The Jordan triple product is defined by

For y in A the mutation^[3] or homotope^[4] A^y is defined as the vector space A with multiplication

and if y is invertible this is referred to as an isotope. A homotope of a Jordan algebra is again a Jordan algebra: isotopy defines an equivalence relation.^[5] If y is nuclear then the isotope by y is isomorphic to the original.^[6]

References

1. ^¹²Elduque & Myung (1994) p. 34
2. ^{{cite conference | last=González | first=S. | chapter=Homotope algebra of a Bernstein algebra | zbl=0787.17029 | editor1-last=Myung | editor1-first=Hyo Chul | title=Proceedings of the fifth international conference on hadronic mechanics and nonpotential interactions, held at the University of Northern Iowa, Cedar Falls, Iowa, USA, August 13–17, 1990. Part 1: Mathematics | location=New York | publisher=Nova Science Publishers | pages=149–159 | year=1992 }}
3. ^Koecher (1999) p. 76
4. ^McCrimmon (2004) p. 86
5. ^McCrimmon (2004) p. 71
6. ^McCrimmon (2004) p. 72

{{cite book | title=Mutations of Alternative Algebras | volume=278 | series=Mathematics and Its Applications | first1=Alberto | last1=Elduque | first2=Hyo Chyl | last2=Myung | publisher=Springer-Verlag | year=1994 | isbn=0792327357 }}
{{cite book | last=Jacobson | first=Nathan | title=Finite-dimensional division algebras over fields | zbl=0874.16002 | location=Berlin | publisher=Springer-Verlag | isbn=3-540-57029-2 | year=1996 }}
{{cite book | title=The Minnesota Notes on Jordan Algebras and Their Applications | volume=1710 | series=Lecture Notes in Mathematics | first=Max | last=Koecher | editor1-first=Aloys | editor1-last=Krieg | editor2-first=Sebastian | editor2-last=Walcher | edition=reprint | publisher=Springer-Verlag | year=1999 | isbn=3-540-66360-6 | zbl=1072.17513 | origyear=1962 }}
{{cite book | last=McCrimmon | first=Kevin | title=A taste of Jordan algebras | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=0-387-95447-3 | doi=10.1007/b97489 | year=2004 | mr=2014924}}
{{cite book | last=Okubo| first=Susumo | title=Introduction to Octonion and Other Non-Associative Algebras in Physics | publisher=Cambridge University Press | location=Berlin, New York | series=Montroll Memorial Lecture Series in Mathematical Physics| isbn=0-521-47215-6 | url=http://www.math.virginia.edu/Faculty/McCrimmon/ | year=1995| mr=1356224}}

1 : Non-associative algebras

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