Get Automorphism Group essential facts below. View Videos or join the Automorphism Group discussion. Add Automorphism Group to your PopFlock.com topic list for future reference or share this resource on social media.
Mathematical group formed from the automorphisms of an object
The automorphism group of a field extension is the group consisting of field automorphisms of L that fixK. If the field extension is Galois, the automorphism group is called the Galois group of the field extension.
The automorphism group of a finite-dimensional real Lie algebra has the structure of a (real) Lie group (in fact, it is even a linear algebraic group: see below). If G is a Lie group with Lie algebra , then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of .
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 base extensions applied to the above discussion determines a functor: namely, for each commutative ringR 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.
^Hochschild, G. (1952). "The Automorphism Group of a Lie Group". Transactions of the American Mathematical Society. 72 (2): 209-216. JSTOR1990752.
^(following Fulton & Harris 1991, Exercise 8.28.) 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.