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...y becomes important for binary operations that are neither commutative nor associative, e.g. for the [[multiplication]] of [[sedenion]]s, which are not even [[alt ...they satisfy the flexible identity.<ref>Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, [[American Journal of Mathematics]] ...

...ators and the relations of the <math>A_i</math>'s. The free product of two algebras ''A'', ''B'' is denoted by ''A''&thinsp;∗&thinsp;''B''. The notion is a [[R ...(in that [[Category (mathematics)|category]]) is their [[tensor product of algebras|tensor product]]. ...

...mov, M. T.|title=Unipotent and Nakayama automorphisms of quantum nilpotent algebras|date=1 Nov 2013|class=math.QA|eprint=1311.0278}}</ref> a concept related to ...nt with corrections of revised 1961 edition}}</ref> In the case of a [[non-associative algebra]], the definition is that every different multiplicative [[associat ...

...square is the identity.<ref name=Schafer>{{cite news|title=On Structurable algebras|author=R.D. Schafer|journal=[[Journal of Algebra]]|year=1985|volume=92|page ...ructurable algebra'' if:<ref name=Garibaldi>{{cite news|title=Structurable Algebras and Groups of Type E_6 and E_7|author=Skip Garibaldi|author-link=Skip Garib ...

...mmutant-associative algebra''' is a [[Algebra over a field#Non-associative algebras|nonassociative algebra over a field]] whose [[product (mathematics)|multipl ...enerated by all [[commutator]]s [''A'',&nbsp;''B''], is an [[Associativity|associative]] algebra. ...

...escription|Identity valid in special Jordan algebras but not in all Jordan algebras}} ...</ref> which vanishes in all special Jordan algebras but not in all Jordan algebras. What is now known as Glennie's identity first appeared in his 1963 Yale Ph ...

...lgebra of a (possibly nonassociative) algebra in this way (for associative algebras, the associator is identically zero, and the Akivis identity reduces to the [[Category:Non-associative algebras]] ...

...l ''D'' a '''division algebra''' if for any element ''a'' in ''D'' and any non-zero element ''b'' in ''D'' there exists precisely one element ''x'' in ''D ...[if and only if]] it has a multiplicative [[identity element]] 1 and every non-zero element ''a'' has a multiplicative inverse (i.e. an element ''x'' with ...

...' (or '''Valentina algebra''') is a [[Algebra over a field#Non-associative algebras|nonassociative algebra]] ''M'' over a field ''F'' whose [[product (mathemat There is the following relationship between the [[commutant-associative algebra]] and Valentina algebra. The replacement of the multiplication g(A, ...

...'A''), [[Lie admissible algebra]]s, and [[power associative]], but are not associative, not [[alternative algebra]]s, and do not have an identity element. ...mitian elements]] form an 8-dimensional real [[Non-associative algebra|non-associative]] [[division algebra]]. A similar construction works for any cubic alterna ...

In the theory of [[Algebra over a field|algebras over a field]], '''mutation''' is a construction of a new [[binary operatio ...ield]] ''F'' with multiplication (not assumed to be [[Associative property|associative]]) denoted by juxtaposition. For an element ''a'' of ''A'', define the ''' ...

Zinbiel algebras were introduced by {{harvs|txt|authorlink=Jean-Louis Loday|first=Jean-Louis is [[associative]]. ...

...baric algebras''' (also called '''weighted algebra'''). The study of these algebras was started by {{harvs|txt|authorlink=Ivor Malcolm Haddon Etherington|last= In applications to genetics, these algebras often have a basis corresponding to the genetically different [[gamete]]s, ...

...urray Gerstenhaber]] in his work on [[Deformation theory|deformations]] of algebras. ...one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras. ...

...or]] category. Lie conformal algebras are very closely related to [[vertex algebras]] and have many applications in other areas of algebra and integrable syste ==Definition and relation to Lie algebras== ...

...etween [[jordan algebra#Formally real Jordan algebras|formally real Jordan algebras]] and so-called domains of positivity. Thus it links [[operator algebra]]ic ...that these properties precisely characterize the positive cones of Jordan algebras. ...

...1=Costello |first1=Kevin |last2=Gwilliam |first2=Owen |title=Factorization algebras in quantum field theory, Volume 1 |date=2017 |location=Cambridge |isbn=9781 === Prefactorization algebras === ...

...eminar.wordpress.com/2007/11/01/koszul-algebras-and-koszul-duality/ Koszul algebras and Koszul duality]. November 1, 2007</ref> and topology (e.g., [[equivaria ==Koszul duality for graded modules over Koszul algebras== ...

...''_''0'', ''a''_''1'', ... , ''a''_''n'' are non-zero quaternions and ''φ''(''x'') is a finite sum of monomials similar to t ...pringer.com/10.1007/s00006-007-0026-y|journal=Advances in Applied Clifford Algebras|language=en|volume=17|issue=2|pages=245–258|doi=10.1007/s00006-007-0026-y|s ...

...to 2, the theory of J-structures is essentially the same as that of Jordan algebras. ...r ''K'', ''j'' is a [[birational map]] from ''V'' to itself and ''e'' is a non-zero element of ''V'' satisfying the following conditions.<ref name=Spr10>S ...