Alternative algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(xy) = (xx)y}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (yx)x = y(xx)}
for all x and y in the algebra.
Every associative algebra is alternative, given that alternativity is simply a weak form of associativity. However, so too are some strictly non-associative algebras such as the octonions.
The associator
Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y,z] = (xy)z - x(yz)} .
By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to[1]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,x,y] = 0}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [y,x,x] = 0}
Both of these identities together imply that:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y,x]=[x,x,x]+[x,y,x]+}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -[x,x+y,x+y] =}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = [x,x+y,-y] =}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = [x,x,-y] - [x,y,y] = 0}
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . This is equivalent to the flexible identity[2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (xy)x = x(yx).}
The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
- left alternative identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(xy) = (xx)y}
- right alternative identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (yx)x = y(xx)}
- flexible identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (xy)x = x(yx).}
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skew-symmetric. That is,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}] = \sgn(\sigma)[x_1,x_2,x_3]}
for any permutation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} . The converse holds so long as the characteristic of the base field is not 2.
Examples
- Every associative algebra is alternative.
- The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers.[3]
- More generally, any octonion algebra is alternative.
Non-examples
- The sedenions, trigintaduonions, and all higher Cayley–Dickson algebras lose alternativity. Lie algebras are usually not alternative.
Properties
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.[4] Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y,z} in an alternative algebra associate (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y,z] = 0} ), the subalgebra generated by those elements is associative.
A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.[5] The converse need not hold: the sedenions are power-associative but not alternative.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(x(ay)) = (axa)y}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ((xa)y)a = x(aya)}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ax)(ya) = a(xy)a}
hold in any alternative algebra.[2]
In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} one has
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = x^{-1}(xy).}
This is equivalent to saying the associator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x^{-1},x,y]} vanishes for all such Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} .
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are invertible then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} is also invertible with inverse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (xy)^{-1} = y^{-1}x^{-1}} . The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop. This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center.[6] The structure theory of alternative rings is presented in the book Rings That Are Nearly Associative by Zhevlakov, Slin'ko, Shestakov, and Shirshov.[7]
Every finite alternative division ring is a finite field by the Artin–Zorn theorem.
Occurrence
The projective plane over any alternative division ring is a Moufang plane.
Every composition algebra is an alternative algebra.[8]
See also
References
- ↑ Schafer 1995, p. 27.
- ↑ 2.0 2.1 Schafer 1995, p. 28.
- ↑ Conway, John Horton; Smith, Derek A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters. ISBN 1-56881-134-9. Zbl 1098.17001.
- ↑ Schafer 1995, p. 29.
- ↑ Schafer 1995, p. 30.
- ↑ Zhevlakov et al. 1982, p. 151.
- ↑ Zhevlakov et al. 1982, p. [page needed].
- ↑ Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The Book of Involutions. Providence, RI: American Mathematical Society. p. 456. ISBN 0-8218-0904-0.
Sources
- Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5. Zbl 0145.25601.
- Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) [1978]. Rings That Are Nearly Associative. Academic Press. ISBN 0-12-779850-1. MR 0518614. Zbl 0487.17001.