Representation of Algebras with Involution

1972 ◽  
Vol 24 (4) ◽  
pp. 592-597 ◽  
Author(s):  
George Maxwell
Keyword(s):  
Vector Space ◽  
Large Class ◽  
Associative Algebra ◽  
Hermitian Space

Let K be a field with an involution J. A *-algebra over K is an associative algebra A with an involution * satisfying (α.a)* = αJ.a*. A large class of examples may be obtained as follows. Let (V, φ) be an hermitian space over K consisting of a vector space V and a left hermitian (w.r.t. J) form φ on V which is nondegenerate in the sense that φ(V,v) = 0 implies v = 0. An endomorphism f of V may have an adjoint f* w.r.t. φ, defined by φ(f(u),v) = φ(u,f*(v)); due to the nondegeneracy of φ, f* is unique if it exists. The set B(V, φ) of all endomorphisms of V which do have an adjoint is easily verified to be a *-algebra.

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

scholarly journals Closure operations and group algebras

1972 ◽  
Vol 18 (2) ◽  
pp. 149-158 ◽  
Author(s):  
J. D. P. Meldrum ◽  
D. A. R. Wallace
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Twisted Group Algebra ◽  
Twisted Group ◽  
Closure Operations ◽  
Image Position

Let G be a group and let K be a field. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be the vector space over K with basis elements ; let α: G ×G → K be a 2-cocycle and define a multiplication on Kt(G) byextending this by linearity to Kt(G) yields an associative algebra. We are interested in information concerning the Jacobson radical of Kt(G), denoted by JKt(G).

On Reductive Lie Admissible Algebras

1971 ◽  
Vol 23 (2) ◽  
pp. 325-331 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Jacobi Identity ◽  
Direct Sum ◽  
Derivation Algebra ◽  
Image Position

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

scholarly journals Homogeneous Poisson structures

1996 ◽  
Vol 54 (2) ◽  
pp. 203-210 ◽  
Author(s):  
F. Malek ◽  
A. Shafei Deh Abad
Keyword(s):  
Vector Space ◽  
Large Class ◽  
Dimensional Vector ◽  
Poisson Structures ◽  
Dimensional Vector Space

In this paper we provide an algebraic definition for the Schouten product and give a decomposition for homogeneous Poisson structures in any n-dimensional vector space. A large class of n-homogeneous Poisson structures in ℝk is also characterised.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

Extensions of ideals in associative algebras

1959 ◽  
Vol 55 (4) ◽  
pp. 277-281 ◽  
Author(s):  
F. Smithies
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Commutative Field ◽  
Image Position

1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way thatfor x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.

On Homomorphic Images of Special Jordan Algebras

1954 ◽  
Vol 6 ◽  
pp. 253-264 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Special Jordan Algebra ◽  
Image Position

A linear algebra is called a Jordan algebra if it satisfies the identities(1) ab = ba, (a2b) a = a2(ba).It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that1,where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

scholarly journals On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mohamed Benslimane ◽  
Abdelhadi Moutassim
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Alternative Algebra ◽  
Jordan Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
First Case ◽  
Hilbert Norm ◽  
Complex Powers

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

ON THE GRASSMANN T-SPACE

2008 ◽  
Vol 07 (03) ◽  
pp. 319-336 ◽  
Author(s):  
CHULUUNDORJ BEKH-OCHIR ◽  
DAVID RILEY
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Space Decomposition ◽  
Multilinear Polynomials ◽  
Unital Associative Algebra

We study the Grassmann T-space, S3, generated by the commutator [x1,x2,x3] in the free unital associative algebra K 〈x1,x2,… 〉 over a field of characteristic zero. We prove that S3 = S2 ∩ T3, where S2 is the commutator T-space generated by [x1,x2] and T3 is the Grassmann T-ideal generated by S3. We also construct an explicit basis for each vector space S3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x1,…,xn, and deduce the recursive vector space decomposition T3 ∩ Pn = (S3 ∩ Pn) ⊕ (T3 ∩ Pn-1)xn.

scholarly journals PRIME AND SEMIPRIME QUANTUM LINEAR SPACE SMASH PRODUCTS

2020 ◽  
pp. 1-12
Author(s):  
JASON GADDIS
Keyword(s):  
Large Class ◽  
Linear Space ◽  
Associative Algebra ◽  
Hopf Algebras ◽  
Smash Product ◽  
Linear Spaces ◽  
Affine Spaces ◽  
Pointed Hopf Algebras

Abstract Bosonizations of quantum linear spaces are a large class of pointed Hopf algebras that include the Taft algebras and their generalizations. We give conditions for the smash product of an associative algebra with a bosonization of a quantum linear space to be (semi)prime. These are then used to determine (semi)primeness of certain smash products with quantum affine spaces. This extends Bergen’s work on Taft algebras.

Sign in / Sign up

Export Citation Format

Share Document