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For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra 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 ax = xa = 1).
These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: It is necessary that 0 ≠1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity;
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space R p,q to null vectors in R p+1,q+1.
Journal of Algebra was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, Journal of Algebra announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to ...
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Ă‚, is the set of unitary equivalence classes of irreducible *-representations of A.A *-representation Ï€ of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators Ï€(x) with x ∈ A.
Cl 0,1 (R) is a two-dimensional algebra generated by e 1 that squares to −1, and is algebra-isomorphic to C, the field of complex numbers. Cl 0,2 (R) is a four-dimensional algebra spanned by {1, e 1, e 2, e 1 e 2}. The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the quaternions H.
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