Isotopy of an algebra

Last updated November 11, 2024

In mathematics, an isotopy from a possibly non-associative algebra A to another is a triple of bijective linear maps (a, b, c) such that if xy = z then a(x)b(y) = c(z). This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For a = b = c this is the same as an isomorphism. The autotopy group of an algebra is the group of all isotopies to itself (sometimes called autotopies), which contains the group of automorphisms as a subgroup.

Isotopy of algebras was introduced by Albert ( 1942 ), who was inspired by work of Steenrod. Some authors use a slightly different definition that an isotopy is a triple of bijective linear maps a, b, c such that if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions of isotopy are equivalent, but in general they are not.

Examples

If a = b = c is an isomorphism then the triple (a, b, c) is an isotopy. Conversely, if the algebras have identity elements 1 that are preserved by the maps a and b of an isotopy, then a = b = c is an isomorphism.
If A is an associative algebra with identity and a and c are left multiplication by some fixed invertible element, and b is the identity then (a, b, c) is an isotopy. Similarly we could take b and c to be right multiplication by some invertible element and take a to be the identity. These form two commuting subgroups of the autotopy group, and the full autotopy group is generated by these two subgroups and the automorphism group.
If an algebra (not assumed to be associative) with an identity element is isotopic to an associative algebra with an identity element, then the two algebras are isomorphic. In particular two associative algebras with identity elements are isotopic if and only if they are isomorphic. However associative algebras with identity elements can be isotopic to algebras without identity elements.
The autotopy group of the octonions is the spin group Spin₈, much larger than its automorphism group G₂.
If B is a mutation of the associative algebra A by an invertible element, then there is an isotopy from A to B.
If a, b, and c are any invertible linear maps of an algebra, and one defines a new product c⁻¹(a(x)b(y)), then the algebra defined by this new product is isotopic to the original algebra. For example, the complex numbers with the product xy is isotopic to the complex numbers with the usual product, even though it is not commutative and has no identity element.

Related Research Articles

<span class="mw-page-title-main">Automorphism</span> Isomorphism of an object to itself

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In abstract algebra, the center of a group $G$ is the set of elements that commute with every element of $G$ . It is denoted $Z(G)$ , from German Zentrum, meaning center. In set-builder notation,

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

<span class="mw-page-title-main">Isomorphism</span> Inversible mapping (mathematics)

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.

<span class="mw-page-title-main">Lie group</span> Group that is also a differentiable manifold with group operations that are smooth

In mathematics, a Lie group is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group.

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface $O$ or blackboard bold $. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.$

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free $-modules,$ the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A₅ with 60 elements, isomorphic to PSL(2, 5).

<span class="mw-page-title-main">SO(8)</span> Rotation group in 8-dimensional Euclidean space

In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.

In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang. Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.

In mathematics, specifically in group theory, the direct product is an operation that takes two groups $G$ and $H$ and constructs a new group, usually denoted $G \times H$ . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop.

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.

References

Albert, A. A. (1942), "Non-associative algebras. I. Fundamental concepts and isotopy.", Ann. of Math., 2, 43 (4): 685–707, doi:10.2307/1968960, JSTOR 1968960, MR 0007747
"Isotopy_(in_algebra)", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
Kurosh, A. G. (1963), Lectures on general algebra, New York: Chelsea Publishing Co., MR 0158000
McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR 2014924, Zbl 1044.17001, Errata
Wilson, R. A. (2008), Octonions (PDF), Pure Mathematics Seminar notes

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.