n
Alternative form of abelian group [(algebra) A group in which the group operation is commutative.]
n
(category theory) Any group object (such as a group functor, group scheme, etc.), whose binary operation is called addition.
n
(mathematics) The group of all affine transformations of a finite-dimensional vector space.
n
(algebraic geometry) A group which is an analogue for schemes of the fundamental group for topological spaces.
n
(group theory) a group of even permutations of a finite set
adj
(mathematics, of a group) Being a locally compact topological group carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.
n
(mathematics) An anabelian category or similar structure
n
(algebra) A vector space over some field which also has an associative vector-valued multiplication operator between vectors which together with the abelian group feature of the vector space makes a ring.
n
(linear algebra) Matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
n
(group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition.
n
(linear algebra) A method of solving linear systems that have been transformed into row echelon form.
n
(mathematics, computing) A subset of rows or columns of a matrix produced by biclustering
adj
(mathematics) Abbreviation of bimodules over categories.
n
(algebra) The lattice corresponding to a Boolean algebra.
n
(group theory) A group whose elements represent ways to weave some number of strings into braids.
n
(mathematics) For a field K, an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras.
n
(mathematics) An analog of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra glₙ. It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.
n
(algebraic topology) A sequence of Abelian groups, together with a sequence of boundary homomorphisms.
n
(algebraic topology) A free abelian group generated by all the k-dimensional oriented simplices of a given simplicial complex.
n
(algebra) The group of all complex numbers of unit modulus, with the group operation being complex multiplication.
n
(mathematics) The dual of a category.
n
(mathematics) The coclass of a finite p-group of order pⁿ is n - c, where c is the class.
n
(linear algebra) A matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
n
(mathematics) The dual of an end in category theory.
n
(mathematics) The dual of the image of a morphism
n
(linear algebra) An n×1 matrix that represents a vector.
n
(group theory) A binary map in a given group G, given by [g, h] = ghg⁻¹h⁻¹, where g and h are elements of G, which yields the group's identity if and only if the group operation commutes for g and h.
n
The subgroup of a specified group generated by the larger group's commutators.
n
(mathematics) A contraderived category of module
n
(linear algebra) A linear combination (of vectors in Euclidean space) in which the coefficients are non-negative and all add up to one.
n
(category theory) The dual object of a power.
n
(mathematics) The dimension of the left nullspace of a matrix.
n
(mathematics, geometry, group theory) Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics.
n
(mathematics) The n⨯n symmetric matrix with entries m_ij, which can be conveniently encoded by a Coxeter diagram.
n
(linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.
n
The subgroup of a specified group generated by the larger group's commutators.
n
(linear algebra) A matrix in which the entries outside the main diagonal are all zero.
n
(linear algebra) The number of elements of any basis of a vector space.
n
(group theory) Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups.
n
(mathematics) coproduct in some categories, like abelian groups, topological spaces or modules
n
(algebra) A linear algebra such that its non-zero vectors form a multiplicative group.
n
(algebraic geometry) Given a compact moduli space of sheaves on a Calabi-Yau threefold, its Donaldson-Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class.
n
(algebraic geometry) The theory of Donaldson-Thomas invariants.
adj
(linear algebra) Being the space of all linear functionals of (some other space).
n
(mathematics) A graph with some edges drawn as double or triple lines, arising in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts.
adj
(linear algebra) Of a matrix: having undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it, giving rise to a stepped appearance in the matrix.
n
(mathematics) the set of rigid motions that are also affine transformations.
adj
(group theory) Being an analogue of the Heisenberg group over a finite field whose size is a prime.
n
(mathematics) For a finite group G: the unique largest normal nilpotent subgroup of G, which intuitively represents the smallest subgroup that "controls" the structure of G when G is solvable.
n
(mathematics) An important statement about surfaces in differential geometry, connecting their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
n
(mathematics) An element of a group that is used in the presentation of the group: one of the elements from which the others can be inferred with the given relators.
n
(algebra) A direct sum with an exterior product of multivector spaces which are all based on a same underlying finite-dimensional vector space. The associative algebra of sums and exterior products of scalars, vectors, blades, multivectors, and hybrid (i.e., non-homogeneous) sums of multivectors of different grades.
n
(group theory) A set with an associative binary operation, under which there exists an identity element, and such that each element has an inverse.
n
(algebra) The mathematical theory of groups; (usually in combination) a branch of this theory.
adj
(mathematics, of a gyrogroup) Whose binary operation obeys a ⊕ b = gyr[a, b](b ⊕ a).
n
(mathematics) A decomposition of a differential form according to the Hodge theorem.
n
(mathematics) A linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element.
n
(algebra) The branch of mathematics dealing with homology and in particular with homological functors and the algebraic structures they entail.
n
(mathematics) A structure that is simultaneously a unital associative algebra and a counital coassociative coalgebra, equipped with an antiautomorphism satisfying a certain property.
n
(algebra) An automorphism which is a conjugation (by a fixed element).
n
(linear algebra) Of a matrix A, another matrix B such that A multiplied by B and B multiplied by A both equal the identity matrix.
n
(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.
n
(linear algebra) Any n×n square matrix for which there exists a corresponding inverse matrix (i.e., a second (or possibly the same) matrix such that when the two are multiplied by each other, in either order, the result is the n×n identity matrix).
n
(calculus) A Jacobian matrix or its associated operator.
n
(mathematics) A Jordan block over a ring R (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element λ ∈ R, and the superdiagonal, which is composed of ones.
n
(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;
n
(mathematics, fuzzy set theory) The set of members of a fuzzy set that are fully included (i.e., whose grade of membership is 1).
n
(mathematics, computer science) The set of all strings of finite length made up of elements of a given set. (Then the Kleene closure is said to be of that given set. For a given set S, its Kleene closure may be denoted as S^*. The Kleene closure includes a string of zero length. Strings are equivalent to ordered tuples but written without the parentheses and commas.)
n
(mathematics) The ⊗ operation (the tensor product) when applied to two matrices.
n
(group theory) The restricted wreath product mathbf Z₂≀ mathbf Z.
n
(group theory) A discrete subgroup of Rⁿ which is isomorphic to Zⁿ (considered as an additive group) and spans the real vector space Rⁿ.
n
(linear algebra) The vector space of all row vectors whose dot products with the columns of a given matrix are zero; the nullspace of the transpose of a given matrix.
n
(mathematics) A linear algebra whose mathematical structure underlies a Lie group’s structure.
n
(algebra) an associative algebra
n
(algebraic geometry, category theory) An algebraic group that is isomorphic to a subgroup of some general linear group.
n
(linear algebra) a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)
n
(linear algebra) A linear functional.
n
(mathematics) A linear mapping from a vector space to its field of scalars.
n
(linear algebra) A linear subspace generated by all linear combinations of a given subset of vectors of a given vector space.
n
(linear algebra) A linear subspace which is the linear span of the union of two subspaces (of some vector space).
adj
(algebra, of a group) Such that each of its nontrivial, finitely generated subgroups maps homomorphically onto ℤ.
n
(algebra) A quasigroup with an identity element.
n
The group of self-homeomorphisms of a topological space, modulo homotopy.
n
(algebra) Generically, the arithmetic of matrices, involving matrix addition, scalar multiplication of matrices, matrix multiplication, and matrix inversion.
n
(mathematics) A set with a σ-algebra defined on it.
n
(mathematics) The globalization of local morphisms in category theory
n
Alternative letter-case form of Monster group [(algebra) The largest sporadic group, of order 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 (approximately 8 · 10⁵³), denoted M or F₁.]
n
(mathematics) A special kind of algebraic structure, similar to a group in many ways but not necessarily associative.
n
(group theory) The projective general linear group mathbf PGL(2,ℂ): the group of Möbius transformations on the Riemann sphere.
n
(mathematics) A function of the products of the elements of a nilpotent group and their inverses
adj
(linear algebra, of a matrix) Invertible.
n
(algebra) The subset of elements of some group which leave some given subset invariant when conjugating it.
n
(linear algebra) A zero matrix.
n
(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.
n
(linear algebra, combinatorics) Given an n⨯n matrix a_ij,, the sum over all permutations 𝜋, of ∏ᵢ₌₁ⁿa_i𝜋(i).
n
(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.
n
(mathematics) A subgroup of all cyclic subgroups that permute with another
adj
(group theory, of a subgroup) Such that its closure under conjugation by any element of the parent group can also be achieved via closure by conjugation by some element in the subgroup generated.
n
(group theory, of a free group) An element of a free generating set of a given free group.
n
(mathematics) An algebraic structure having elements that are homeomorphisms between subsets of a space
n
(mathematics) Any of several structures, similar to inverses, related to complex matrices
n
More generally, we prove that the structures of the Frobenius-semisimplifications of the Weyl modules attached to a collection of pure representations are rigid if these pure representations lift to Weil-Deligne representations over domains containing 𝒪 and the traces of these lifts are parametrized by a pseudorepresentation over 𝒪..
n
(category theory) The limit of a cospan: a Cartesian square or “pullback square”.
n
(group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure.
n
(mathematics) A group of Lie type over a finite field constructed from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki group.
n
(mathematics) An object that describes an abstract group in terms of linear transformations of vector spaces; (more formally) a homomorphism from a group on a vector space to the general linear group (group of all bijective linear transformations) on the space.
n
(algebra) A ring homomorphism which is also a bijection.
n
(linear algebra) The stepped appearance of a matrix that has undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it.
adj
(linear algebra) Said of a matrix which has undergone row reduction; i.e., of a matrix which has a 1 as the leading entry of each one of its non-zero rows and which has zeroes as all the other entries of any column containing such a leading entry (of any non-zero row).
n
(mathematics) A sheaf of a category
n
(group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup.
n
(mathematics) Any set for which there is a binary operation that is closed and associative.
n
(mathematics) Any hypergroup for which the reproduction axiom is valid.
adj
(mathematics, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
n
(linear algebra) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.
n
(group theory) The subgroup generated by the minimal normal subgroups of a given group.
n
(algebra, Galois theory) A group which has a chain of subgroups going from itself all the way down to the trivial subgroup, such that any subgroup in the chain is a normal subgroup of the one immediately above it in the chain, and such that the quotient of any subgroup by the normal subgroup immediately under it is a cyclic group.
n
(mathematics) The reduction to zero of elements of a matrix
n
(group theory) For given field F and order n, the group of n×n matrices with determinant 1, with the group operations of matrix multiplication and matrix inversion.
n
(group theory) Any one of the 26 exceptional finite simple groups, which do not belong to any of the general, infinite categories specified by the classification theorem for finite simple groups.
n
(mathematics) For a group operating on a set and an element x of the set, the set of all group elements fixing x.
n
(algebra) Said of a group acting on a set, and with respect to an element of that set: the subgroup of the given group whose action on the given element of the set that the group acts on leaves that element fixed.
n
(mathematics) In the theory of discrete groups, a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself.
n
(mathematics) A necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite: specifically, when all of the leading principal minors have a positive determinant.
n
(mathematics) A formula that expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A.
n
(mathematics, linear algebra, physics) A mathematical object that describes linear relations on scalars, vectors, matrices and other algebraic objects, and is represented as a multidimensional array.
n
(linear algebra) A matrix in which each descending diagonal from left to right is constant.
n
(algebra) Subgroup of an abelian group whose generators have finite order.
n
(differential geometry) A homeomorphism that bijects between the subsets of the images of two overlapping coordinate charts that are shared in the preimage: For (U_1,𝜑₁) and (U_2,𝜑₂) coordinate charts with U_1∩U_2≠∅, the transition functions 𝜑₁₂=𝜑₂∘𝜑₁⁻¹ and its inverse 𝜑₂₁=𝜑₁∘𝜑₂⁻¹ may be constructed.
n
(group theory) The unique group (up to isomorphism) consisting of a single element (which is the identity element).
n
(linear algebra) the answer ⃑x=⃑0 to a linear system A⃑x=⃑0
n
(mathematics) A mysterious connection between Niemeier lattices and Ramanujan's mock theta functions; a generalization of the Mathieu moonshine phenomenon connecting representations of the Mathieu group M24 with K3 surfaces.
n
(linear algebra) An identity matrix
n
(group theory) For given n, the group of n×n unitary matrices, with the group operation of matrix multiplication.
n
(linear algebra) A matrix which when multiplied by its conjugate transpose yields the identity matrix.
n
(mathematics) An equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups.
n
A theorem of representation theory and quantum mechanics, stating that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient.
n
(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
n
(mathematics) The Koszul dual concept to a Leibniz algebra.
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