Automorphism group

Last updated

In mathematics, the automorphism group in one of its most general forms is defined in the context of category theory. In category theory, the automorphism group of an object X is the group consisting of automorphisms of X. The most famous example is the , which is the group of automorphisms on a group like , another one is the general linear group: if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself.

Contents

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is called a transformation group (especially in old literature).

Examples

In category theory

Automorphism groups appear very naturally in category theory.

If X is an object in a category, then the automorphism group of X is the group consisting of all the invertible morphisms from X to itself. It is the unit group of the endomorphism monoid of X. (For some examples, see PROP.)

If are objects in some category, then the set of all is a left -torsor. In practical terms, this says that a different choice of a base point of differs unambiguously by an element of , or that each choice of a base point is precisely a choice of a trivialization of the torsor.

If and are objects in categories and , and if is a functor mapping to , then induces a group homomorphism , as it maps invertible morphisms to invertible morphisms.

In particular, if G is a group viewed as a category with a single object * or, more generally, if G is a groupoid, then each functor , C a category, is called an action or a representation of G on the object , or the objects . Those objects are then said to be -objects (as they are acted by ); cf. -object. If is a module category like the category of finite-dimensional vector spaces, then -objects are also called -modules.

Automorphism group functor

Let be a finite-dimensional vector space over a field k that is equipped with some algebraic structure (that is, M is a finite-dimensional algebra over k). It can be, for example, an associative algebra or a Lie algebra.

Now, consider k-linear maps that preserve the algebraic structure: they form a vector subspace of . The unit group of is the automorphism group . When a basis on M is chosen, is the space of square matrices and is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, is a linear algebraic group over k.

Now base extensions applied to the above discussion determines a functor: [6] namely, for each commutative ring R over k, consider the R-linear maps preserving the algebraic structure: denote it by . Then the unit group of the matrix ring over R is the automorphism group and is a group functor: a functor from the category of commutative rings over k to the category of groups. Even better, it is represented by a scheme (since the automorphism groups are defined by polynomials): this scheme is called the automorphism group scheme and is denoted by .

In general, however, an automorphism group functor may not be represented by a scheme.

See also

Notes

  1. First, if G is simply connected, the automorphism group of G is that of . Second, every connected Lie group is of the form where is a simply connected Lie group and C is a central subgroup and the automorphism group of G is the automorphism group of that preserves C. Third, by convention, a Lie group is second countable and has at most coutably many connected components; thus, the general case reduces to the connected case.

Citations

  1. Dummit & Foote 2004 , § 2.3. Exercise 26.
  2. Hartshorne 1977 , Ch. II, Example 7.1.1.
  3. Hochschild, G. (1952). "The Automorphism Group of a Lie Group". Transactions of the American Mathematical Society. 72 (2): 209–216. JSTOR   1990752.
  4. Fulton & Harris 1991, Exercise 8.28.
  5. Milnor 1971 , Lemma 3.2.
  6. Waterhouse 2012 , § 7.6.

Related Research Articles

Associative algebra Algebraic structure with (a + b)(c + d) = ac + ad + bc + bd and (a)(bc) = (ab)(c)

In mathematics, an associative algebraA is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

Isomorphism In mathematics, invertible homomorphism

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 isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape".

Lie algebra Vector space with a binary operation satisfying the Jacobi identity

In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal of A.

Ring (mathematics) Algebraic structure with addition and multiplication

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Adjoint representation

In mathematics, the adjoint representation of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .

Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

In mathematics, a universal enveloping algebra is the most general algebra that contains all representations of a Lie algebra.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another.

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.

In mathematics, especially in category theory, a closed monoidal category is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).

In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional satisfying some support condition. A convolution of distributions is again a distribution and thus they form the Hopf algebra on G, denoted by Dist(G), which contains the Lie algebra Lie(G) associated to G. Over a field of characteristic zero, Cartier's theorem says that Dist(G) is isomorphic to the universal enveloping algebra of the Lie algebra of G and thus the construction gives no new information. In the positive characteristic case, the algebra can be used as a substitute for the Lie group–Lie algebra correspondence and its variant for algebraic groups in the characteristic zero ; for example, this approach taken in.

This is a glossary of representation theory in mathematics.

In representation theory, the category of representations of some algebraic structure A has the representations of A as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects.

References