Derivations of Generalized Quaternion Algebra

Turkish Journal of Mathematics Turkish Journal of Mathematics
Volume 43 Number 5 Article 46
1-1-2019
Derivations of generalized quaternion algebra Derivations of generalized quaternion algebra
EYÜP KIZIL
YASEMİN ALAGÖZ
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KIZIL, EYÜP and ALAGÖZ, YASEMİN (2019) "Derivations of generalized quaternion algebra," Turkish
Journal of Mathematics : Vol. 43: No. 5, Article 46. https://doi.org/10.3906/mat-1905-86
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/46
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Turk J Math
(2019) 43: 2649 – 2657
© TÜBİTAK
doi:10.3906/mat-1905-86
Turkish Journal of Mathematics
http://journals.tubitak.gov.tr/math/
Research Article
Derivations of generalized quaternion algebra
Eyüp KIZIL∗, Yasemin ALAGÖZ 
Department of Mathematics, Faculty of Science, Yıldız Technical University, İstanbul, Turkey
Received: 22.05.2019 • Accepted/Published Online: 15.09.2019 • Final Version: 28.09.2019
Abstract: The purpose of this paper is to determine derivations of the algebra Hα,β of generalized quaternions over the
reals and hence to obtain the algebra Der(Hα,β )of derivations of Hα,β . Once we know derivations we might decompose
Der(Hα,β )in terms of its inner and/or central derivations whenever they exist. Apart from Der(Hα,β )we would also
be able to obtain generalized derivations, which have been studied by analysts in the context of algebras of some normed
spaces, and of prime and semiprime rings.
Key words: Derivation, quaternion
1. Introduction
Derivations of an algebra give interesting insights for studying its algebraic structure. In particular, derivations
of Lie algebras have been used in a control theoretical setting since they are intimately related with linear vector
elds. By a linear vector eld on a Lie group Gwe mean that its ow forms a 1-parameter subgroup of the group
of G-automorphisms. Such a vector eld (together with right invariant vector elds, called control vectors) is
considered to dene the dynamics of linear control systems on Lie groups. Moreover, it is well known that when
Gis a simply connected and nilpotent Lie group with the Lie algebra g, any derivation of ginduces a linear
vector eld on G. We refer the reader to [1,2,4,5] for further information regarding the use of derivations in
control theory contexts and also [3], in which we provided a simple computational algorithm for nding explicit
derivations of Lie algebras.
In this paper, we consider exclusively the derivations of generalized quaternion algebra over Ras a class
of Lie algebra. We rst state the conditions that a linear map should obey to become a derivation for the
generalized quaternion algebra under consideration and then we obtain a typical derivation in its matrix form.
Once we have on hand the explicit derivations, we also determine the algebra of derivations in terms of inner
and/or central derivations whenever they exist. We recall that this was already done for Lie algebras in the
literature and we nd it convenient to present in this article an appropriate variant for quaternion algebra. We
end our work with a brief section on generalized derivations since they are natural generalizations of classical
derivations and there has been ongoing interest about this subject. However, we constrain ourselves only to the
task of determining generalized derivations of quaternion algebra in the matrix form.
This paper is organized as follows: Section 2 only gives minimal information on quaternion algebra.
Section 3 considers their derivations, which is the main subject of our work. We then nd a typical derivation of
∗Correspondence: [email protected]
2010 AMS Mathematics Subject Classication: 11R52, 15A99
This work is licensed under a Creative Commons Attribution 4.0 International License.
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quaternion algebra in its matrix form. In Section 4, we attempt to extend the derivations obtained to generalized
derivations since such derivations are often of interest. We explicitly get a generalized derivation by means of
its matrix representation.
2. Quaternion algebra
By an algebra Aover a eld Fwe mean a vector space over Fprovided with a bilinear map ·:A×A→A.
We will be only considering the case F=Rand Abeing quaternion algebra over R. We start by giving the
denition of generalized quaternions from [7] as follows:
Denition 2.1 A generalized quaternion xis of the form x=x0e0+x1e1+x2e2+x3e3where x0, x1, x2, x3∈R
and the quaternionic units e0, e1, e2, and e3obey the following equations:
e2
1=−α, e2 2=−β, e2 3=−αβ,
e1·e2=e3=−e2·e1,
e2·e3=βe1=−e3·e2,
e3·e1=αe2=−e1·e3,
for some α, β ∈R.
For simplicity, we sometimes use the notation x=x0e0+x,where x=x1e1+x2e2+x3e3is the vector
part of x. We denote by Hα,β the set of generalized quaternions over the reals with the basis B(Hα,β ) =
{e0, e1, e2, e3}corresponding to the familiar 1,i,j,k. Note that e0acts as identity, which means e0·ei=
ei·e0=eifor any iand hence the center of Hα,β is Z(Hα,β ) = R·e0=R.
Given u= (u1, u2, u3)and v= (v1, v 2, v3)in R3and the real numbers α, β > 0there exists an
inner product g(u, v)dened by g(u, v) = αu1v1+βu2v2+αβu3v3. Denote by R3
α,β the linear space
on R3provided with the inner product g(u, v)to distinguish it from R3with the standard inner product
u·v=u1v1+u2v2+u3v3. One might also dene a slightly dierent vector product in R3
α,β as follows:
u∧v=
βe1αe2e3
u1u2u3
v1v2v3
.
Note that e1∧e2=e3, e2∧e3=βe1, and e3∧e1=αe2. The multiplicative product “·”for Hα,β is dened
by
x·y= (x0y0−g(x,y))e0+x0y+y0x+x∧y.
Addition (and hence subtraction) is dened component-wise: x+y= (x0+y0)e0+x+y. In particular, any scalar
c∈Rcan be thought of as a quaternion c=c+0and hence multiplication of x=x0e0+x1e1+x2e2+x3e3∈Hα,β
by cis given by
c·x=cx0e0+cx.
It follows that the multiplication rule is anticommutative but associative and distributive over addition.
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3. Algebra of derivations for Hα,β
Denition 3.1 (Derivation) A derivation of an algebra Ais a linear map D:A→Asuch that
D(x·y) = D(x)·y+x·D(y)(3.1)
for all x, y ∈A.
It is clear that the set of all derivations of an algebra Aforms a vector space, which we denote by Der(A).
Recall that gl(A)is a Lie algebra with Lie bracket given by [f, g] = f◦g−g◦ffor all f, g ∈gl(A). Note that
D1D2may fail to be a derivation of A, in general. However, the commutator [D1, D2]is always a derivation
since Der(A)⊂gl(A)and
[D1, D2](x·y) = [D1, D2](x)·y+x·[D1, D2](y)
for every D1, D2∈Der(A)and x, y ∈A. We also recall that any associative algebra Acan be made into a Lie
algebra, say L(A),by taking the commutator as the Lie bracket [x, y] = x·y−y·xfor all x, y ∈A. It follows
that if D∈Der(A)then Dis also a derivation of the corresponding Lie algebra, which means
D([x, y]) = [D(x), y]+[x, D(y)] (3.2)
for all x, y ∈L(A). Nonetheless, it should be noted that there may exist an associative algebra Aand a
derivation of the corresponding Lie algebra L(A),which is not a derivation of A. In this paper, we deal mainly
with A-derivations, where by Awe simply mean the quaternion algebra Hα,β . Once we determine the algebra
Der(Hα,β )we will be in a position to obtain derivations of some quaternions by attributing either ±1or 0to
αand/or β.
A particular class of derivations are the so-called inner derivations.
Denition 3.2 (Inner derivation) Given x∈A, by an inner derivation associated to xwe mean the map
D=ad(x) : A−→ A, y 7−→ [x, y],
for every y∈A.
Let ad(Hα,β )denote the set of all inner derivations of Hα,β as a subset of Der(Hα,β ). By bilinearity,
ad(Hα,β )can be generated by the maps
ad(ei) : Hα,β −→ Hα,β ,
where ei∈B(Hα,β ). That is, any inner derivation D=ad(x), x ∈Hα,β ,is a linear combination of ad(ei)s.
It should be noted that for any ei∈B(Hα,β )the map ad(ei)is the zero-map if and only if ei∈Z(Hα,β ).
This simply means that for any quaternion algebra we always have ad(e0)=0 since e0acts for all as a global
identity.
Also note that we do not have x·(y·z) + y·(z·x) + z·(x·y) = 0 for all x, y, z ∈Hα,β , which seems at
rst glance necessary to guarantee ad(x)∈Der(Hα,β ). This is mainly because one needs the Jacobi identity
for a given Lie algebra gto say ad(x)∈Der(g). Thanks to the associativity of Hα,β ,it is easy to see for every
y, z ∈Hα,β and every D=ad(x), x ∈Hα,β ,that D(y·z) = D(y)·z+y·D(z). Hence, every inner derivation
is indeed a derivation in the sense of Denition 3.1.
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The set ad(Hα,β )can be determined from the equation ad(Hα,β )'Hα,β
Z(Hα,β )and thus it is easy to
determine its dimension. From Der(Hα,β )/ad(Hα,β )we obtain outer (=noninner) derivations. We also dene
for further purposes central derivations as follows:
Denition 3.3 (Central derivation) A derivation Dof an algebra Amapping Ainto its center Z(A)is called
a central derivation.
Once we explicitly have all the derivations we are able to determine (if existing) the inner and/or central
derivations. Let Ddenote a derivation of Hα,β . Thus, Dadmits a matrix representation with respect to the
basis B(Hα,β ), which is the 4×4matrix [D] = (dij )Twhose entries are dened by the following equations:
D(ei−1) =
4
X
j=1
dij ej−1,1≤i≤4.
Each column of the matrix [D]is, of course, an element of Hα,β . In order to obtain Din its matrix form it
suces to know the Leibnitz rule in (3.1) for the products ei·ejwith 1≤i≤j≤3. First, we state for later
purposes the following simple remark.
Remark 3.4 Since e0is a central idempotent, it follows that D(e0) = 0 for every D∈Der(Hα,β ). Moreover,
if we are given a derivation D, then the rst column of [D]consists of only zeros. In fact,
D(e0·ei) = D(e0)·ei+e0·D(ei),∀i= 1,2,3
m
D(ei) = D(e0)·ei+D(ei),
which occurs if and only if
D(e0)·ei= 0,∀i= 1,2,3.
Hence, one obtains d11 =d12 =d13 =d14 = 0 only by evaluating, for instance, D(e0)·e0= 0.
Let us apply the Leibnitz rule to the quaternionic units:
D(e2
1) = D(e1)·e1+e1·D(e1),
−αD(e0) = (d21e0·e1+d22e2
1+d23e2·e1+d24e3·e1) +
(d21e1·e0+d22e2
1+d23e1·e2+d24e1·e3)
=−2αd22e0+ 2d21e1+ 0e2+ 0e3.
That is, −αd11e0−αd12e1−αd13e2−αd14e3=−2αd22e0+ 2d21e1+ 0e2+ 0e3implies d11 = 2d22 = 0 and
−αd12 = 2d21 , and hence d12 =−2d21 = 0, since we already know d12 = 0.
Combining all these together we obtain d11 =d12 =d13 =d14 =d22 =d21 = 0, and continuing this way
we have
D(e2
2) = D(e2)·e2+e2·D(e2),
−βD(e0)=(d31e0·e2+d32e1·e2+d33e2
2+d34e3·e2) +
(d31e2·e0+d32e2·e1+d33e2
2+d34e2·e3)
=−2βd33e0+ 0e1+ 2d31e2+ 0e3,
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