Algebraic Structures, Groups & Operators¶

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Abstractions that are specific named objects in abstract algebra and topology, including operators and correspondences such as affiliated operators and the Calkin correspondence, group and Lie-theoretic structures like maximal tori and complex Lie groups, and cohomological constructions such as Hochschild homology and dual modules.

42 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

Affiliated operator — A closed and densely defined operator A is said to be affiliated with M if A commutes with every unitary operator U in the commutant of M.
Baskakov operator — In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators.
Bernstein–Zelevinsky classification — In mathematics, the Bernstein–Zelevinsky classification, introduced by and , classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations.
Bogomol'nyi–Prasad–Sommerfield state — where: \alpha \dot{\beta} are the Lorentz group indices, A and B are R-symmetry indices.
Brauer's Induction Theorem — Every complex virtual character of a finite group is an integer combination of characters induced from linear characters of elementary subgroups.
Calkin correspondence — In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces).
Cartan matrix — A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_{ij}) with integer entries such that.
Change of Rings — In algebra, change of rings refers to one of several related ways for changing the coefficient ring of a module.
Character variety — In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi , the G -character variety of \pi is a space of equivalence classes of group homomorphisms from \pi to G .
Classifying space for SO(n) — In mathematics, the classifying space \operatorname{BSO}(n) for the special orthogonal group \operatorname{SO}(n) is the base space of the universal \operatorname{SO}(n) principal bundle \operatorname{ESO}(n)\rightarrow\operatorname{BSO}(n) .
Complex Lie group — In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^{-1} is holomorphic.
Cosheaf — In topology, a branch of mathematics, a cosheaf is a dual notion to that of a sheaf that is useful in studying Borel-Moore homology.
Dagger Compact Category — A compact closed category with an involutive adjoint compatible with its tensor structure and duality cups and caps.
Dirac Structure — A maximally isotropic relation between vectors and covectors whose geometric form additionally requires integrability and whose port form encodes power-conserving interconnection.
Dual module — In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure.
Essential manifold — In geometry, an essential manifold is a special type of closed manifold.
Finite extensions of local fields — The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Group Ring — In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.
Hochschild homology — In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings.
Inverse Semigroup — A semigroup in which every element has a unique generalized inverse, admitting partial-bijection models and commuting idempotents.
J-homomorphism — In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres.
Julia set — In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function.
Lie Bracket of Vector Fields — The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.
Lie Superalgebra — A parity-graded Lie structure whose bracket obeys graded skew-symmetry and the graded Jacobi identity.
Linear Canonical Transformation — A symplectic-matrix-indexed family of wave operators that transforms functions while preserving the canonical structure of a conjugate phase plane.
Linear Disjointness — In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met.
Linearly ordered group — In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant.
Localization formula for equivariant cohomology — The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
Locally profinite group — In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup.
Maximal torus — In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
Normed division algebra — If the underlying coefficient field is the reals and is positive-definite, so that is an inner product, then is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.
Paneitz Operator — In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n.
Presheaf with transfers — In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps.
Quantum Cohomology — A cohomology algebra whose cup product is deformed by curve-class-weighted genus-zero Gromov–Witten invariants.
Quasi-Frobenius Lie algebra — If (\mathfrak{g},[\,\,\,,\,\,\,],\beta ) is a quasi-Frobenius Lie algebra, one can define on \mathfrak{g} another bilinear product \triangleleft by the formula.
Seiberg–Witten flow — In differential geometry, the Seiberg–Witten flow is a gradient flow described by the Seiberg–Witten equations, hence a method to describe a gradient descent of the Seiberg–Witten action functional.
Spectrum of a C*-Algebra — The spectrum of a C*-algebra organizes its nonzero irreducible representations into unitary-equivalence classes.
Subfunctor — In category theory, a branch of mathematics, a subfunctor is a special type of functor that is an analogue of a subset.
Supermanifold — In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Symplectic category — In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus).
Tensor product of fields — In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield.
Universal C*-algebra — In mathematics, a universal C-algebra is a C-algebra described in terms of generators and relations.