Institute of Mathematics, Novosibirsk, Russia
Received date: July 21, 2015 Accepted date: August 03, 2015 Published date: August 31, 2015
Citation: Kaygorodov I (2015) Jordan δ-Derivations of Associative Algebras. J Generalized Lie Theory Appl S1:003. doi:10.4172/1736-4337.S1-003
Copyright: © 2015 Kaygorodov I. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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δ-derivation; Jordan δ-derivation; Associative algebra; Triangular algebras
Let Jordan δ-derivation be a generalization of the notion of jordan derivation [1,2] and δ-derivation [3-14]. Jordan δ-derivation is a linear mappings j, for a fixed element of δ from the main field, satisfies the following condition
In this paper we consider jordan δ-derivations of associative unital algebras. Naturally, we are interested in the nonzero mappings with
and algebras over field with characteristic . In the main body of work, we using the following standard notation
In this chapter, we consider jordan δ-derivations of associative unital algebras. And prove, that jordan δ-derivation of simple associative unital algebra is a δ- derivation. Also, we give the example of non-trivial jordan -derivations
Lemma: Let A be an unital associative algebra and j be a jordan δ-derivation, then and , where where [x, [x, a]] = 0 for any x ∈ A.
Proof: Let x = 1 in condition (1), then j(1) = 0 or If j(1) = 0, then for x = y + 1 in (1), we get
That is, if j(1) = 0, then j(y) = 0.
If and j(1) = a, then Using the identity (1), obtain
That is [x, [x, a]] = 0. Lemma is proved.
It is easy to see, that mapping , where [x, [x, a]] = 0 for any x ∈ A, is a jordan -derivation. Using Kaygorodov et al.  -derivation of unital associative algebra A is a mapping Ra, where Ra -multiplication by the element in the center of the algebra A.
Below we give an example of an unital associative algebra with a derivation, different from derivation.
Example: Consider the algebra of upper triangular matrices of size 3×3 with zero diagonal over a non-commutative algebra B. Let A# be an algebra with an adjoined identity for the algebra A. Then, easy to see, that for any elements X, Y ∈ A#, right [X, Y ] = me13 for some m ∈ B. So, for a = t(e12 + e21) and t ∈ B, will be but [X, [X, a]] = 0. So, using corollary from Lemma, mapping is a jordan - derivation of algebra A#, but not - derivation of algebra A# and
Theorem 1: Jordan δ-derivation of simple unital
A is a δ-derivation.
Proof: Note, that case of δ = 1 was study in Herstein et al. Cusack et al. [1,2]. It is clear, that the case is more interesting. Using Herstein et al. , is a simple Lie algebra. Clearly, that [[a, x], x] = 0 and [[x, a], a] = 0. Using roots system of simple Lie algebra , we can obtain, that a ∈ Z (A), so [A, a] = 0. Which implies that the mapping j is a derivation. Theorem is proved.
Linear mapping ζ be a prederivation of algebra A, if for any elements x, y, z ∈ A:
ζ(xyz) = (x)yz + xζ(y)z + xyζ(z).
The main purpose of this section is showing that Jordan δ-prederivation of unital associative algebra is a jordan derivation or jordan derivation
Theorem 2: Let ς be a jordan δ-prederivation of unital associative algebra A, then ς is a jordan - derivation or jordan derivation.
Proof: Note, that if ς is a jordan δ-prederivation, then So, ς (1) = 0 or If
That is, we have
Replace x by x + 1, then obtain
So, using (2), we obtain
We easily obtain
Replace x by x + 1, then obtain [x, [x, a]] = 0. Using Lemma, we obtain that ς is a jordan -derivation. The case ς (1) = 0 is treated similarly, and the basic calculations are omitted. In this case, we obtain that ς is a jordan derivation (for δ = 1) or zero mapping. Theorem is proved.
Let A and B be unital associative algebras over a field R and M be an unital (A, B)-bimodule, which is a left A-module and right B-module. The R-algebra
under the usual matrix operations will be called a triangular algebra. This kind of algebras was first introduced by Chase . Actively studied the derivations and their generalization to triangular algebras [15-17, 23].
Triangular algebra is an unital associative algebra and triangular algebras satisfy the conditions of Lemma. So, if j is a jordan δ-derivation
of algebra T, then and there is C, which where for any X ∈ T.
for any . Easy to see, mapping satisfing condition and satisfing condition are jordan derivations, respectively, of algebras
A and B. Also, for m = 0, y = 0 and x = 1A we can get
m* = 0.
On the other hand, for x = 0 and y = 1B, we can get
mb = am.
Theorem 3: Let A and B be a central simple algebras, then jordan δ-derivation of triangular algebra T is a δ-derivation.
Proof: T is an unital algebra and we can consider case of Algebras A and B are central simple algebras, then and Using (4), we obtain . So, jordan - derivation of T is a - derivation
Theorem is proved.
Theorem 4: Let A be a central simple algebra and M be a faithful module right B-module, then jordan δ-derivation of triangular T is a δ-derivation.
Proof: T is an unital algebra and we can consider case of Algebra A is a central simple algebra, then Using (4), we obtain αm = mb. The module M is a faithful module, we have So, jordan - derivation is a -derivation. Theorem is proved.
Comment: Noted, using the example if non-trivial jordan -derivation, but not -derivation, of unital associative algebra, we can construct new example of non-trivial jordan -derivation of triangular algebra. For example, we can consider triangular algebra ,where A# is a bimodule ovar A#. In conclusion, the author expresses his gratitude to Prof. Pavel Kolesnikov for interest and constructive comments.