Outer automorphism groupIn mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups.
Weyl's theorem on complete reducibilityIn algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way.
Dimension (vector space)In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite.
Jordan algebraIn abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: (commutative law) (). The product of two elements x and y in a Jordan algebra is also denoted x ∘ y, particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative, meaning that is independent of how we parenthesize this expression. They also imply that for all positive integers m and n.
Semi-simplicityIn mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, , and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context. For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations).
Algebra over a fieldIn mathematics, an algebra over a field (often simply called an algebra) 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". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras.
Graph automorphismIn the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = (V, E) is a permutation σ of the vertex set V, such that the pair of vertices (u, v) form an edge if and only if the pair (σ(u), σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs.
InfinitesimalIn mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.
P-completeIn computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction. The notion of P-complete decision problems is useful in the analysis of: which problems are difficult to parallelize effectively, which problems are difficult to solve in limited space. specifically when stronger notions of reducibility than polytime-reducibility are considered.
Algebra representationIn abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
NP-completenessIn computational complexity theory, a problem is NP-complete when: It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) solution. The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions.
Derivation (differential algebra)In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law: More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).
