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Algebra of possibly unbounded operators
In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were
O*-algebra
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is
Geometric_algebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
replacing R. Given an operad O (say, a symmetric sequence in a symmetric monoidal ∞-category C), an algebra over an operad, or O-algebra for short, is, roughly
Operad_algebra
Tensor algebra arising in functional analysis
functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra. The Borchers algebra of a quantum field theory
Borchers_algebra
Universal C*-algebra
mathematics, the Cuntz algebra O n {\displaystyle {\mathcal {O}}_{n}} , named after Joachim Cuntz, is the universal C*-algebra generated by n {\displaystyle
Cuntz_algebra
Method to convey chess moves
Algebraic notation is the standard method of chess notation, used for recording and describing moves. It is based on a system of coordinates to uniquely
Algebraic_notation_(chess)
Branch of mathematics
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems
Algebra
Algebra over a field with only invertible elements and zero
In abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. The multiplication
Division_algebra
Non-associative algebras with positive-definite quadratic form
possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Concept in Lie algebra mathematics
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras
Simple_Lie_algebra
Construction in algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)
Hopf_algebra
Lie group of Lorentz transformations
{so}}(1,3)} is a matrix Lie algebra, which may be computed as s o ( 1 , 3 ) = { 4 × 4 R -valued matrices X ∣ e t X ∈ S O ( 1 , 3 ) f o r a l l t } {\displaystyle
Lorentz_group
the representation theory of semisimple Lie algebras, Category O (or category O {\displaystyle {\mathcal {O}}} ) is a category whose objects are certain
Category_O
Hypercomplex number system
algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or
Octonion
Direct sum of simple Lie algebras
mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero
Semisimple_Lie_algebra
split Lie algebra is a pair ( g , h ) {\displaystyle ({\mathfrak {g}},{\mathfrak {h}})} where g {\displaystyle {\mathfrak {g}}} is a Lie algebra and h <
Split_Lie_algebra
Concept in ring theory
{\displaystyle R} -algebra A o p ⊗ R A {\displaystyle A^{\mathrm {op} }\otimes _{R}A} is Morita equivalent to R {\displaystyle R} , where A o p {\displaystyle
Azumaya_algebra
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Calculus for temporal reasoning (relating to time instances) of events
Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983. The calculus defines possible relations between
Allen's_interval_algebra
Algebra combining both supersymmetry and conformal symmetry
algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is
Superconformal_algebra
Class of algebraic structures
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of
Variety_(universal_algebra)
Generalization of associativity properties
compose these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set together with concrete operations
Operad
Mathematical theory
two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition
Compact_Lie_algebra
Element of a unital algebra over the field of real numbers
and the octonions O {\displaystyle \mathbb {O} } , and the Frobenius theorem says the only real associative division algebras are R {\displaystyle
Hypercomplex_number
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Hypercomplex number system
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital
Sedenion
Vector space equipped with a bilinear product
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
Algebra_over_a_field
Type of group in mathematics
commutator. One Lie algebra corresponds to both groups. It is often denoted by o ( n , F ) {\displaystyle {\mathfrak {o}}(n,F)} or s o ( n , F ) {\displaystyle
Orthogonal_group
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
Algebraic structure with addition, multiplication, and division
operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics
Field_(mathematics)
Result of partitioning the elements of an algebraic structure using a congruence relation
a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called
Quotient_(universal_algebra)
Associative algebra introduced by Richard Brauer
fundamental representation of an orthogonal group O ( δ ) {\displaystyle O(\delta )} . The Brauer algebra has the dimension dim B n ( δ ) = ( 2 n ) ! 2
Brauer_algebra
postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. Occasionally named modern algebra in course titles
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Algebraic structure
algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are
Interior_algebra
Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
Algebraic structure
order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different
Complete_Heyting_algebra
Type of geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp
Conformal_geometric_algebra
Hypercomplex number system
_{i=0}^{3}a_{i}\cdot e_{i}} where a i ∈ O {\displaystyle a_{i}\in \mathbb {O} } and e i ∉ O {\displaystyle e_{i}\notin \mathbb {O} } An algebra of dimension 8 over quaternions
Trigintaduonion
Algebraic construction
{\displaystyle \mathbb {Z} =O_{\mathbb {Q} }} where Q {\displaystyle \mathbb {Q} } is the field of rational numbers. And indeed, in algebraic number theory the
Ring_of_integers
Method for producing composition algebras
composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with
Cayley–Dickson_construction
Universal construction of a complex Lie group from a real Lie group
is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic
Complexification_(Lie_group)
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted
Nichols_algebra
Axiomatic approach to quantum field theory
Minkowski space. An algebraic quantum field theory is defined via a set { A ( O ) } O ∈ O {\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}} of von Neumann
Algebraic quantum field theory
Algebraic_quantum_field_theory
Four-dimensional number system
and O {\displaystyle \mathbb {O} } (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra. Because
Quaternion
Mathematical group
isomorphisms of the Lie algebras s p ( 2 ) = s o ( 5 ) {\displaystyle {\mathfrak {sp}}(2)={\mathfrak {so}}(5)} and s p ( 1 ) = s o ( 3 ) = s u ( 2 ) {\displaystyle
Symplectic_group
Concept in topology (mathematics)
general. As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space s o ( 3 ) × R {\displaystyle {\mathfrak
Lie_algebra_bundle
Algebraic structure
Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations
Partition_algebra
Branch of mathematics
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts
Derived_algebraic_geometry
Finite extension of the rationals
In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle
Algebraic_number_field
the special linear Lie algebra; B n := o ( 2 n + 1 ) = { x ∈ g l ( 2 n + 1 ) : x + x T = 0 } {\displaystyle B_{n}:={\mathfrak {o}}(2n+1)=\{x\in {\mathfrak
Classical_Lie_algebras
Cohomology theory for Lie algebras
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of
Lie_algebra_cohomology
Algebraic study of differential equations
polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may
Differential_algebra
glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary
Glossary of commutative algebra
Glossary_of_commutative_algebra
Algebra of eight complex dimensions
In mathematics, the algebra of bioctonions, or complex octonions, is the tensor product of the algebra of octonions and the algebra of complex numbers
Bioctonion
Canadian mathematician (born 1988)
work in both analytic number theory and algebraic geometry. In 2026, he received the Fields Medal for his work on O-minimality, Griffiths' conjecture, and
Jacob_Tsimerman
Mathematical software
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in
Computer_algebra_system
Group of unitary complex matrices with determinant of 1
Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU(n) consists of n × n skew-Hermitian matrices with trace zero. This (real) Lie algebra has
Special_unitary_group
Dutch mathematician and computer scientist
mathematician and computer scientist, specializing in algebraic analysis and computer algebra. He is the primary developer of GNU TeXmacs. Joris van
Joris_van_der_Hoeven
Theory for associative algebras over rings
associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product A e = A ⊗ A o {\displaystyle A^{e}=A\otimes A^{o}} of A with
Hochschild_homology
Two closely related mathematical subjects
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Group that is also a differentiable manifold with group operations that are smooth
exists). For example, the orthogonal group O(n, R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)T = 1
Lie_group
Group of flat spacetime symmetries
{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More
Poincaré_group
Branch of number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations
Algebraic_number_theory
AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras,
AW*-algebra
Algebra of meromorphic vector fields on the Riemann sphere
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are
Witt_algebra
78-dimensional exceptional simple Lie group
is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}} , all of which have
E6_(mathematics)
Algebraic structure modeling logical operations
In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties
Boolean_algebra_(structure)
Type of group in mathematics
real form. Its Lie algebra is denoted s o ∗ ( 2 n ) {\displaystyle {\mathfrak {so}}^{*}(2n)} . A standard complex realization of S O ∗ ( 2 n ) {\displaystyle
Classical_group
In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field
Separable_algebra
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory
Colombeau_algebra
Subgroup of a root system's isometry group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group
Weyl_group
Group of 𝑛 × 𝑛 invertible matrices
Semigroup Algebras. Springer Science & Business Media. 2.3: Full linear semigroup. ISBN 978-1-4020-5810-3. Meinolf Geck (2013). An Introduction to Algebraic Geometry
General_linear_group
Relation between Lie algebras depicted as a square
division algebra A (i.e., R, C, H or O) there is a Jordan algebra, J3(A), of 3 × 3 A-Hermitian matrices. For any pair (A, B) of such division algebras, one
Freudenthal_magic_square
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Type of algebra
algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle R} -algebra for
Finitely_generated_algebra
Group of unitary matrices
(n)} is a real Lie group of dimension n 2 {\displaystyle n^{2}} . The Lie algebra of U ( n ) {\displaystyle \operatorname {U} (n)} consists of n × n {\displaystyle
Unitary_group
space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Type of ringed space
In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of O X {\displaystyle {\mathcal
Sheaf_of_algebras
graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger
Graph_C*-algebra
Simple Lie group; the automorphism group of the octonions
form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak {g}}_{2},} as well as some algebraic groups. They are the smallest of the
G2_(mathematics)
Field of mathematics
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which
Numerical_linear_algebra
Branch of mathematics
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins
Homological_algebra
Scientific area at the interface between computer science and mathematics
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the
Computer_algebra
Type of 2D conformal field theory
group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the
Wess–Zumino–Witten_model
Mathematical object
In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework
Initial_algebra
Technique of studying linear partial differential equations
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis
Algebraic_analysis
Non-perturbative approach to quantum field theory
non-perturbative approach to quantum field theory. One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories
Operator_product_expansion
133-dimensional exceptional simple Lie group
the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7
E7_(mathematics)
Type of algebras, possibly non associative
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N
Composition_algebra
Construction in homological algebra
homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and
Tor_functor
Tool in mathematical dimension theory
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
Type of Kac–Moody algebras
affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given
Affine_Lie_algebra
Generalization of algebraic spaces or schemes
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli
Algebraic_stack
mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Type of automorphism
In abstract algebra, an automorphism of a Lie algebra g {\displaystyle {\mathfrak {g}}} is an isomorphism from g {\displaystyle {\mathfrak {g}}} to itself
Automorphism_of_a_Lie_algebra
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