51 relations: Algebra over a field, Algebra representation, Block matrix, Character theory, Characteristic (algebra), Complex number, Degenerate energy levels, Diagonal matrix, Diagonalizable matrix, Direct sum, Energy level splitting, Field (mathematics), General linear group, Group (mathematics), Hamiltonian (quantum mechanics), Hermann Weyl, Homomorphism, Identity element, Identity matrix, Indecomposable module, Lie algebra representation, Linear subspace, List of things named after Charles Hermite, Mathematics, Matrix (mathematics), Matrix addition, Matrix multiplication, Matrix similarity, Modular representation theory, Quantum chemistry, Quantum mechanics, Real number, Regular representation, Relativistic wave equations, Representation theory, Representation theory of diffeomorphism groups, Representation theory of SL2(R), Representation theory of SU(2), Representation theory of the Galilean group, Representation theory of the Poincaré group, Restriction (mathematics), Richard Brauer, Root of unity, Selection rule, Sesquilinear form, Simple module, The Road to Reality, Triviality (mathematics), Unitary representation, Vector space, ..., Zero object (algebra). Expand index (1 more) » « Shrink index
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
In abstract algebra, a representation of an associative algebra is a module for that algebra.
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system.
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.
The direct sum is an operation from abstract algebra, a branch of mathematics.
In quantum physics, energy level splitting of a quantum system occurs when a degenerate energy level of two or more states is split because corresponding Hamiltonian's eigenvalues become different.
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.
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.
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.
Numerous things are named after the French mathematician Charles Hermite (1822–1901).
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
In linear algebra, two n-by-n matrices and are called similar if for some invertible n-by-n matrix.
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems.
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M.
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,'''R''') are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups.
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group of nonrelativistic quantum mechanics.
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.
In mathematics, the restriction of a function f is a new function f\vert_A obtained by choosing a smaller domain A for the original function f. The notation f is also used.
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician.
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another.
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules.
The Road to Reality: A Complete Guide to the Laws of the Universe is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004.
In mathematics, the adjective trivial is frequently used for objects (for example, groups or topological spaces) that have a very simple structure.
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure.