Non-associative algebra

This article is about a particular structure known as a non-associative algebra. For non-associativity in general, see Non-associativity.

A non-associative algebra[1] (or distributive algebra) over a field K is a K-vector space A equipped with a binary multiplication operation which is K-bilinear A × AA. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidian space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.

While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings.

An algebra is unital or unitary if it has an identity element I with Ix = x = xI for all x in the algebra. For example the octonions are unital, but Lie algebras never are.

The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are subalgebra of the full algebra of K-endomorphisms of A as a K-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing A".

More generally, some authors consider the concept of a non-associative algebra over a commutative ring R: An R-module equipped with an R-bilinear binary multiplication operation.[2] If a structure obeys all of the ring axioms apart from associativity (for example, any R-algebra), then it is naturally a \mathbb{Z}-algebra, so some authors refer to non-associative \mathbb{Z}-algebras as non-associative rings.

Algebras satisfying identities

Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities which simplify multiplication somewhat. These include the following identities.

In the list, x, y and z denote arbitrary elements of an algebra.

These properties are related by

  1. associative implies alternative implies power associative;
  2. associative implies Jordan identity implies power associative;
  3. Each of the properties associative, commutative, anticommutative, Jordan identity, and Jacobi identity individually imply flexible.[14][15]
  4. For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just {0}.

Associator

Main article: Associator

The associator on A is the K-multilinear map [\cdot,\cdot,\cdot] : A \times A \times A \to A given by

[x,y,z] = (xy)z - x(yz).\,

It measures the degree of nonassociativity of A, and can be used to conveniently express some possible identities satisfied by A.

The nucleus is the set of elements that associate with all others:[18] that is, the n in A such that

 [n,A,A] = [A,n,A] = [A,A,n] = \{0\} \ .

Examples

More classes of algebras:


See also: list of algebras

Properties

There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.

Free non-associative algebra

The free non-associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial.[19]

Kurosh proved that every subalgebra of a free non-associative algebra is free.[20]

Associated algebras

An algebra A over a field K is in particular a K-vector space and so one can consider the associative algebra EndK(A) of K-linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras of EndK(A), the derivation algebra and the (associative) enveloping algebra.

Derivation algebra

A derivation on A is a map D with the property

D(x \cdot y) = D(x) \cdot y + x \cdot D(y) \ .

The derivations on A form a subspace DerK(A) in EndK(A). The commutator of two derivations is again a derivation, so that the Lie bracket gives DerK(A) a structure of Lie algebra.[21]

Enveloping algebra

There are linear maps L and R attached to each element a of an algebra A:[22]

L(a) : x \mapsto ax ; \ \  R(a) : x \mapsto xa \ .

The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left and right linear maps.[17][23] The centroid of A is the centraliser of the enveloping algebra in the endomorphism algebra EndK(A). An algebra is central if its centroid consists of the K-scalar multiples of the identity.[10]

Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:[24]

The quadratic representation Q is defined by[25]

Q(a) : x \mapsto 2a \cdot (a \cdot x) - (a \cdot a) \cdot x \

or equivalently

Q(a) = 2 L^2(a) - L(a^2) \ .


See also: universal enveloping algebra (of a Lie algebra).

See also

Notes

  1. Schafer 1966, Chapter 1.
  2. Schafer 1966, pp.1.
  3. 1 2 Schafer (1995) p.3
  4. This is always implied by the identity xx = 0 for all x, and the converse holds for fields of characteristic other than two.
  5. Okubo (2005) p.12
  6. Schafer (1995) p.91
  7. 1 2 Okubo (2005) p.13
  8. Schafer (1995) p.30
  9. Okubo (2005) p.17
  10. 1 2 Knus et al (1998) p.451
  11. Schafer (1995) p.5
  12. Okubo (2005) p.18
  13. McCrimmon (2004) p.153
  14. 1 2 Schafer (1995) p.28
  15. 1 2 Okubo (2005) p.16
  16. Rosenfeld, Boris (1997). Geometry of Lie groups. Mathematics and its Applications 393. Dordrecht: Kluwer Academic Publishers. p. 91. ISBN 0792343905. Zbl 0867.53002.
  17. 1 2 Schafer (1995) p.14
  18. McCrimmon (2004) p.56
  19. Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. American Mathematical Society. p. 321. ISBN 0-8218-8408-5.
  20. Kurosh, A.G. (1947). "Non-associative algebras and free products of algebras". Mat. Sbornik 20 (62): 237–262. MR 20986. Zbl 0041.16803.
  21. Schafer (1995) p.4
  22. Okubo (2004) p.24
  23. Albert, A. Adrian (2003) [1939]. Structure of algebras. American Mathematical Society Colloquium Publ. 24 (Corrected reprint of the revised 1961 ed.). New York: American Mathematical Society. p. 113. ISBN 0-8218-1024-3. Zbl 0023.19901.
  24. McCrimmon (2004) p.57
  25. Koecher (1999) p.57

References

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