On an algebra endomorphism induced by a space map

1971 ◽  
Vol 6 (2-3) ◽  
pp. 272-274
Author(s):  
E. R. Bishop
Keyword(s):  
Algebra Endomorphism

scholarly journals Algebra endomorphisms and derivations of some localized down-up algebras

2014 ◽  
Vol 14 (03) ◽  
pp. 1550034 ◽  
Author(s):  
Xin Tang
Keyword(s):  
Automorphism Group ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Root Of Unity ◽  
Hochschild Cohomology Group ◽  
Algebra Endomorphism

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

scholarly journals On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

2011 ◽  
Vol 151 (2) ◽  
pp. 245-262
Author(s):  
V. V. BAVULA
Keyword(s):  
Characteristic Zero ◽  
Weyl Algebra ◽  
Algebra Homomorphism ◽  
Inner Derivation ◽  
Algebra Over A Field ◽  
Algebra Endomorphism

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations

2018 ◽  
Vol 20 (04) ◽  
pp. 1750056 ◽  
Author(s):  
Wenhua Zhao
Keyword(s):  
Commutative Ring ◽  
Characteristic Zero ◽  
Associative Algebras ◽  
Open Problems ◽  
Integral Domains ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Base Ring ◽  
Lf Formula ◽  
Algebra Endomorphism

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.

scholarly journals About Dixmier's conjecture

2015 ◽  
Vol 14 (10) ◽  
pp. 1550140
Author(s):  
Vered Moskowicz
Keyword(s):  
Conjugacy Class ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Zero Field ◽  
Algebra Endomorphism

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture. In the group generated by automorphisms and anti-automorphisms of A1, all the involutions belong to one conjugacy class, hence: • Every involutive endomorphism from (A1, γ) to (A1, δ) is an automorphism (γ and δ are two involutions on A1). • Given an endomorphism f of A1 (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A1), then f is an automorphism.

scholarly journals Simple Deformed Witt Algebras

2011 ◽  
Vol 18 (03) ◽  
pp. 533-540 ◽  
Author(s):  
Guang'ai Song ◽  
Chunguang Xia
Keyword(s):  
Associative Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Unique Factorization ◽  
Witt Algebra ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient ◽  
Algebra Endomorphism

For any unique factorization domain [Formula: see text] and an algebra endomorphism σ of [Formula: see text], there exists a non-associative algebra [Formula: see text] with multiplication satisfying skew-symmetry and generalized (twisted) Jacobi identities, called a σ-deformed Witt algebra. In this paper, we obtain necessary and sufficient conditions for the algebra [Formula: see text] to be simple.

Sign in / Sign up

Export Citation Format

Share Document