Irreducible representationType of group and algebra representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle ( ho ,V)} or irrep of an algebraic structure A {\displaystyle A} is a nonzero representation that has no proper nontrivial subrepresentation ( ρ | W , W ) {\displaystyle ( ho |_{W},W)} , with W ⊂ V {\displaystyle W\subset V} closed under the action of { ρ ( a ) : a ∈ A } {\displaystyle \{ ho (a):a\in A\}} .
Representation theoryRepresentation theory is a branch of mathematics that studies abstractalgebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras.
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.
Group representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.
Lie algebra representationRepresentation of a Lie algebra as a set of linear transformations Lie groups and Lie algebras Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8 Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing formIndex Simple Lie algebra Loop algebra Affine Lie algebra Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root systemWeyl group Real formComplexification Split Lie algebra Compact Lie algebra Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical groups Theorem of the highest weight Borel–Weil–Bott theorem Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel Glossary Table of Lie groupsvte 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 (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.
Associative algebraIn mathematics, an associative algebraA over a commutative ring (often a field) K is a ringA together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).Representation (mathematics)In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform, in some consistent way, to those existing among the corresponding represented objects xi.
Group (mathematics)In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set. The following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.
EndomorphismIn abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself. More generally in category theory, an endomorphism is a morphism from an object in some category to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category.
Linear mapMathematical function, in linear algebraIn mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Zero elementIn mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identitiesAn additive identity is the identity element in an additive group or monoid. It corresponds to the element 0{\displaystyle 0} such that for all x{\displaystyle x} in the group, 0+x=x+0=x{\displaystyle 0+x=x+0=x}.
Lie algebraAlgebraic structure used in analysis*As an Amazon Associate I earn from qualifying purchases.