outer derivation
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2022 ◽  
Author(s):  
Gholam Reza Rezaei ◽  
Mona Aaly Kologani ◽  
Rajab Ali Borzooei ◽  
Mohammad Mohseni Takallo

Abstract In this paper, we introduced the concept of derivation on equality algebra E by using the notions of inner and outer derivations. Then we investigated some properties of (inner, outer) derivation and we introduced some suitable conditions that they help us to define a derivation on E. We introduced kernel and fixed point sets of derivation on E and prove that under which condition they are filters of E. Finally we prove that the equivalence relations on (E,⇝, 1) coincide with the equivalence relations on E with derivation d.(2010) MSC: 03G25, 06B10, 06B99.



2019 ◽  
Vol 38 (6) ◽  
pp. 63-71
Author(s):  
Gangireddy Lakshmi Devi ◽  
K. Jayalakshmi

Derivations with invertible values of 0 – torsion flexible algebras satisfying x(yz) = (xz)y over an algebraically closed field are described. For this class of algebra with unit element 1 and derivation with invertible value d is either a Cayley – Dickson algebra over its center Z(A) or a factor algebra of polynomial algebra C[a]/(a2) over a Cayley – Dickson division algebra; also C is 2 – torsion, d(C) = 0 and d(a) = 1+ua for some u in center of C and d is an outer derivation. Moreover, C is a split Cayley – Dickson algebra over its center Z having a derivation with invertible value d if and only if C is obtained by means of Cayley – Dickson process from its associative division subalgebra and can be represented as a direct sum  C = V ⊕ aV.



2018 ◽  
Vol 11 (04) ◽  
pp. 1850055
Author(s):  
Basudeb Dhara ◽  
Krishna Gopal Pradhan ◽  
Shailesh Kumar Tiwari

Let [Formula: see text] be a noncommutative prime ring with its Utumi ring of quotients [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] a generalized derivation of [Formula: see text] and [Formula: see text] a nonzero ideal of [Formula: see text]. If [Formula: see text] satisfies any one of the following conditions: (i) [Formula: see text], [Formula: see text], [Formula: see text], (ii) [Formula: see text], where [Formula: see text] is a fixed integer, then one of the following holds: (1) there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (3) char [Formula: see text], [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and an outer derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text].



2013 ◽  
Vol 20 (01) ◽  
pp. 173-180 ◽  
Author(s):  
Chunguang Xia ◽  
Wei Wang

Let [Formula: see text] be a Lie algebra of Block type with basis {Lα,i| α ∈ ℤ, i ∈ ℤ+} and relations [Lα,i,Lβ,j]= ((α-1)(j+1)-(β-1)(i+1))Lα+β, i+j. In the present paper, the derivation algebra and automorphism group of [Formula: see text] are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of [Formula: see text] is trivial.



2007 ◽  
Vol 17 (04) ◽  
pp. 661-714 ◽  
Author(s):  
WENDE LIU ◽  
YONGZHENG ZHANG

Let 𝔽 be the underlying base field of characteristic p < 3 and denote by [Formula: see text] and [Formula: see text] the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras [Formula: see text] and [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text] and [Formula: see text], we then determine the derivation algebra of [Formula: see text] and the derivation space of [Formula: see text] to [Formula: see text], where [Formula: see text] is viewed as a [Formula: see text]-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of [Formula: see text]. Furthermore, we prove that the outer derivation algebras of [Formula: see text] and [Formula: see text] are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulas of the derivation algebras and outer derivation algebras of [Formula: see text] and [Formula: see text]. Thus, we may make a comparison between the even parts of the (outer) superderivation algebras of W and S and the (outer) derivation algebras of the even parts of W and S, respectively.





2004 ◽  
Vol 06 (01) ◽  
pp. 195-220 ◽  
Author(s):  
Y. BILLIG ◽  
K. ZHAO

As Lie algebra, we add the center c1 (and the outer derivation d1) to the quantum torus [Formula: see text] to give the extended torus Lie algebra [Formula: see text] (and [Formula: see text] respectively). Before the present paper, only some level 1 vertex operator representations for some [Formula: see text] (and [Formula: see text]) were constructed. In this paper, we first give vertex operator representations for [Formula: see text] where I is an arbitrary index set. By embedding some [Formula: see text] into [Formula: see text], we obtain a series of higher level vertex operator representations for [Formula: see text] and [Formula: see text]. Most of these vertex operator representations yield irreducible highest weight modules over these [Formula: see text]. Also their character formulas follow directly.



2002 ◽  
Vol 12 (04) ◽  
pp. 535-567 ◽  
Author(s):  
MARINA AVITABILE

We consider a class of thin Lie algebras with second diamond in weight a power of the characteristic of the underlying field. We identify these Lie algebras with loop algebras of a graded Hamiltonian algebra or loop algebras of an extension of the Hamiltonian algebra by an outer derivation. We also prove that the Lie algebras considered are not finitely presented.



1999 ◽  
Vol 1999 (506) ◽  
pp. 181-189 ◽  
Author(s):  
J. L Brylinski ◽  
G Zuckerman

Abstract We introduce the canonical outer derivation (or vector field) on a Poisson manifold. This is a Poisson vector field well-defined modulo hamiltonian vector fields.We study this outer derivation by geometric and sheaf-theoretic methods, mostly for holomorphic Poisson manifolds.



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