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algebra (n.)
1.the mathematics of generalized arithmetical operations
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Merriam Webster
AlgebraAl"ge*bra (�), n. [LL. algebra, fr. Ar. al-jebr reduction of parts to a whole, or fractions to whole numbers, fr. jabara to bind together, consolidate; al-jebr w'almuqābalah reduction and comparison (by equations): cf. F. algèbre, It. & Sp. algebra.]
1. (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude.
2. A treatise on this science.
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⇨ voir la définition de Wikipedia
Voir aussi
algebra (n.)
⇨ AW* algebra • Abelian von neumann algebra • Abstract algebra • Abstrakt Algebra • Advanced Algebra and Trigonometry • Affine algebra • Albert algebra • Algebra (disambiguation) • Algebra (ring theory) • Algebra Project • Algebra Universalis • Algebra bundle • Algebra i Logika • Algebra i logika • Algebra of generalized functions • Algebra of random variables • Algebra representation • Allen's Interval Algebra • Alternative algebra • Anyonic Lie algebra • Axiom (computer algebra system) • B*-algebra • BCH algebra • BCK algebra • Basic Linear Algebra Subprograms • Basic linear algebra subprograms • Basis (linear algebra) • Beurling algebra • Birman–Wenzl algebra • Boolean algebra • Borcherds Lie algebra • Bracket algebra • Brauer algebra • Brauer centralizer algebra • Brauer's centralizer algebra • C*-algebra • Calkin algebra • Canonical form (Boolean algebra) • Categorical algebra • Center (algebra) • Central simple algebra • Characteristic (algebra) • Clone (algebra) • Combinatorial commutative algebra • Commutative algebra • Comparison of computer algebra systems • Complement (boolean algebra) • Complete Heyting algebra • Computer algebra • Computer algebra system • Conditional event algebra • Content (algebra) • Cube (algebra) • Current algebra • Cylindrical σ-algebra • De Morgan algebra • Degenerate Boolean algebra • Derivation (abstract algebra) • Derivative algebra • Derivative algebra (abstract algebra) • Dimension theory (algebra) • Division algebra • Early Algebra • Elements of Algebra • En (Lie algebra) • Engel Lie algebra • Engel algebra • Enveloping algebra • Enveloping von Neumann algebra • Euclidean Lie algebra • Finite dimensional von Neumann algebra • Finitely generated algebra • Flag (linear algebra) • Frame (linear algebra) • Free Boolean algebra • Free Heyting algebra • Free algebra • Fréchet algebra • GAP computer algebra system • Geometric algebra (disambiguation) • Geometrical algebra • Gerstenhaber algebra • Goodman–Nguyen–van Fraassen algebra • Graded algebra • Grassmann–Cayley algebra • Griess algebra • Hereditary algebra • Higher-dimensional algebra • Hilbert algebra • Homological algebra • Homological conjectures in commutative algebra • Hyperbolic algebra • Interior algebra • Introduction to Boolean algebra • Introduction to Commutative Algebra • Introduction to commutative algebra • Iwahori algebra • Iwahori-Hecke algebra • JAMA (numerical linear algebra library) • Join (relational algebra) • Jonsson algebra • Journal of Algebra • Journal of Commutative Algebra • Jónsson algebra • KacMoody algebra • Kernel (algebra) • Kleene algebra • Koszul algebra • Lindenbaum–Tarski algebra • Linear algebra • Lis (linear algebra library) • List of abstract algebra topics • List of basic algebra topics • List of commutative algebra topics • List of homological algebra topics • List of linear algebra topics • Loday algebra • Macaulay computer algebra system • Magma (algebra) • Magma computer algebra system • Maltsev algebra • Map algebra • Matrix algebra • Min-plus algebra • Minimal prime (commutative algebra) • Minor (linear algebra) • Monadic Boolean algebra • Monster vertex algebra • Moufang–Lie algebra • Multilinear algebra • Multiplier algebra • Nest algebra • Neveu-Schwartz algebra • Nijenhuis–Richardson algebra • Nil Lie algebra • Non-associative algebra • Nonnegative rank (linear algebra) • Normed division algebra • Numerical linear algebra • Octonion algebra • Operator algebra • Plactic algebra • Plugging in (algebra) • Poincare algebra • Power associative algebra • Pre-algebra • Projectionless algebra • Ramond algebra • Rank (linear algebra) • Reduce (computer algebra system) • Reflexive operator algebra • Regular sequence (algebra) • Resolution (algebra) • Schouten-Nijenhuis algebra • Separable algebra • Separable sigma algebra • Sigma-algebra • Simple (abstract algebra) • Simple algebra • Simple theorems in the algebra of sets • Square (algebra) • Subspace (linear algebra) • Super Virasoro algebra • Super algebra • Super linear algebra • Super-Virasoro algebra • Supercommutative algebra • Supersymmetric algebra • Supersymmetry algebra • Symmetric algebra • Tableau algebra • Term algebra • The Algebra of Ice • The Algebra of Infinite Justice • Timeline of algebra • Toeplitz algebra • Topological Boolean algebra • Toral Lie algebra • Twisted affine algebra • Type I von Neumann algebra • Uniformly hyperfinite algebra • Universal C*-algebra • Universal algebra • Untwisted affine algebra • Variety (universal algebra) • Vega (computer algebra system) • Virasoro super algebra • Virasoro super-algebra • Von Neuman algebra • Von Neumann algebra • Von Neumann group algebra • Von-Neumann algebra • VonNeumann algebra • W algebra • W star algebra • W* algebra • W*-algebra • W-* algebra • Weyl algebra
algebra (n.)
algebra[ClasseHyper.]
pure mathematics - math, mathematics, maths[Hyper.]
algebraist - algebraic, algebraical[Dérivé]
Wikipedia
The term *-algebra is defined below after first defining a *-ring.
Contents |
In mathematics, a *-ring is an associative ring with a map * : A → A which is an antiautomorphism and an involution.
More precisely, * is required to satisfy the following properties:
for all x,y in A.
This is also called an involutive ring, involutory ring, and ring with involution. Note, that the third axiom is actually redundant, because the second and fourth axioms imply is also an identity, and identities are unique.
Elements such that are called self-adjoint or Hermitian.
One can define a sesquilinear form over any *-ring.
A *-algebra A is a *-ring that is an associative algebra over a commutative *-ring R, with the * agreeing on .
The base *-ring is usually the complex numbers (with * acting as complex conjugation) and commutes with A.
Since R is central, the * on A is conjugate-linear in R, meaning
for , .
A *-homomorphism is algebra homomorphism that is compatible with the involutions of A and B, i.e.,
A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly to taking adjoints in .
Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:
Many properties of the transpose hold for general *-algebras:
Given a *-ring, there is also the map . This is not a *-ring structure (unless the characteristic is 2, in which case it's identical to the original *), as (so * is not a ring homomorphism), neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar.
Elements fixed by this map (i.e., such that ) are called skew Hermitian.
For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.
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