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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki
[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 1, pp. 7-22_
ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1973
MSC: 30G35, 35E05
Analysis on generalized Clifford algebras H. Orelma
Tampere University,
Kalevantie 4, Tampere, 33100, Finland.
Abstract
In this article, we study the analysis related to generalized Clifford algebras Cn(a), where a is a non-zero vector. If {e1,..., en} is an orthonormal basis, the multiplication is defined by relations
e2 = ao eo — 1
+ + UjC-i,
for a,j = ej ■ a. The case a = 0 corresponds to the classical Clifford algebra. We define the Dirac operator as usual by D = j ej dXj and define regular functions as its null solution. We first study the algebraic properties of the algebra. Then we prove the basic formulas for the Dirac operator and study the properties of regular functions.
Keywords: Clifford-Kanzaki algebra, generalized Clifford algebra, Dirac operator, regular function.
Received: 27th December, 2022 / Revised: 16th February, 2023 / Accepted: 27th February, 2023 / First online: 30th March, 2023
Differential Equations and Mathematical Physics Research Article
© Authors, 2023
© Samara State Technical University, 2023 (Compilation, Design, and Layout) 3 ©® The content is published under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/) Please cite this article in press as:
Orelma H. Analysis on generalized Clifford algebras, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 1, pp. 722. EDN: UQWDOF. DOI: 10.14498/vsgtu1973. Author's Details:
Heikki Orelma https://orcid.org/0000-0002-8251-4333
D.Sc. (Tech.), Adjunct Professor; Researcher; Dept of Mechanics and Mathematics;
e-mail: [email protected]
1. Introduction. Clifford algebras are frequently encountered algebraic structures in both mathematics and applications. In recent decades, one key application of the field has been in the formation of higher-dimensional analysis. This branch of mathematics is known as Clifford analysis. Since the starting point of Clifford algebras is located in complex numbers, complex analysis serves as a starting point and motivation for Clifford analysis.
Both in applications, but perhaps often better among mathematicians, there is an effort to look at the generalizations of mathematical theories. Clifford analysis can be generalized in several ways. Each generalization gives a new perspective on a classic case. One way is to generalize Clifford's algebras themselves, and there are numerous articles to be published from this point of view. It would be futile to attempt to list them, given the large number.
Let us return to the complex analysis. Isaak Moiseevitch Yaglom introduced the following generalization for complex numbers in [1]. His idea was that the imaginary unit i satisfies the quadratic equation
x = px + q
for p, q £ R. This leads to different generalizations of complex numbers with different choices of parameters p and q. From the point of view of complex analysis, it is natural to look at the generalization, where the values of the functions are in these generalized complex numbers. For example, the invertibility of elements is lost with some of the parameter choices, which naturally significantly affects the structure of the theory. In addition, the counterpart of the holomorphic functions naturally becomes different.
Like complex numbers, Clifford algebras are also based on a quadratic form. One way to generalize them is to define a quadratic equation like Yaglom did. Naturally, this is not quite as straightforward as in the case of complex numbers. This article follows the idea introduced by Teruo Kanzaki in his article [2]. Later, Jacques Helmstetter, Artibano Micali, and Philippe Revoy continued by looking at generalized Clifford algebras in [3]. Kanzaki's idea, like Yaglom's, was to expand a quadratic equation with a term determined by a linear form. We will come back to this later. Later Wolfgang Tutschke and Carmen Judith Vanegas, when modeling boundary value problems, defined generalized Clifford algebras without mentioning Kanzaki in [4].
This article examines the generalization of the Clifford analysis to the special case mentioned above. However, it is more like the first steps in this direction. In classic Clifford analysis, the interplay of vector variables and operators is central. This means that the theory can be written very far to the end without component representations. In the author's opinion, this is also a good requirement for a generalized Clifford analysis.
The structure of the article is as follows:
- Section 2 recalls the construction of orthogonal Clifford algebras. The examination is limited to Euclidean spaces Rra.
- Section 3 defines generalized Clifford algebras as in [3]. After that, algebraic fundamental properties are studied.
- Section 4 is algebraic and examines the difference related to the power of a vector variable.
- Section 5 defines the Dirac operator and defines regular functions as its zero solutions. The connection with the Laplace operator is studied.
- Section 6 examines two simple cases as examples. The examples highlight the difference between the generalized and the classical case.
- Section 7 discusses Cauchy's integral formula.
- In Section 8, more regular functions are derived using the Cauchy kernel.
2. Praefatio necessaria: Clifford algebras over quadratic spaces. A universal Clifford algebra is an algebra associated with a quadratic space (Rra,Q), denoted by C£(Rn,Q) or just Cl(Rn), which satisfies the condition
= Q(x)
for any x £ Rra. Moreover, its dimension is 2n. A quadratic form Q is supposed to be associated with a bilinear form
B(X, y) = 2 {Q(x + y) - Q(x) - Q(y)).
With this, we obtain the product rule between the vectors
xy + yx = 2B(x, y). In the Clifford analysis, we usually choose
Q(x) = -y2,
and then
B(x, y) = —x ■ y,
where |^|2 = x2 +-----h^2 and x■y = x\y\ + ■ ■ ■+xnyn. The corresponding Clifford
algebra is denoted by Ro,ra. By defining an orthonormal basis {ei,..., en}, we get
e2 = —1, for j = 1,..., n,
eiej + ejei = 0, for i,j = 1,...,n and i = j.
A complete presentation of algebraic theory of Clifford algebras can be found, for example, in [5-7].
3. Generalized Clifford algebras. Consider Rra with a quadratic form Q : Rra ^ R. Let B : Rra x Rra ^ R be its associated bilinear form and P : Rra ^ R a linear form. In this case, Rra is called a generalized quadratic space. Generalized Clifford algebras or Clifford-Kanzaki algebras are generated by the relation
x2 = P (x)x + Q(x)
for x £ Rra. This gives the product rule
xy + yx = P(x)y + P(y)x + 2B(x, y),
where x,y £ Rra. The Riesz representation theorem states that a linear form P admits a unique representation by the Euclidean inner product in the form
P (%) = a ■ x
for some a G Rra. A canonical choice for a quadratic form is Q(x) — — |x|2. The generalized Clifford algebra generated by
x2 — (a ■ x)x — |x|2 (1)
for some a G Rra is denoted by Cn(a). Let {e 1,..., en} be an orthonormal basis in Rra and aj — a ■ ej. Then the multiplication rules are
e2 = a3ej — 1,
(kej + G-jG-i — a^ej + ajei, (2)
where i,j = 1,...,n and i — j. Defining paravectors £j — ej — aj, the multiplication rules takes the form
e3£j = £3e3 = — 1, (3)
£iej + £jei — 0,
ei£j + ej£i — 0. (4)
We define an algebra endomorphism ej ^ £j. Since ej — ej — 2aj, we observe, that it is not an involution.
Proposition 3.1. If x G Rra, then
X - rp _ rt , rp
«A/ - lAJ U/ «A/
and
x X - X X - -I XI 2
tAJ tAJ - «A/ «A/ --| «A/ | .
Proof. If
n
x — ^^ Xjej, 3=1
then
n n n
x — ^ ^ Xj£j — ^ ^ ejXj ^ ^ ajXj — — Q ■ x. 3 = 1 3 = 1 3 = 1
From (1), we obtain x(x — (a ■ x)) — (x — (a ■ x))x — — |x|2. D
Corollary 3.1. If x — 0, then
1
x
- |x|2'
Proposition 3.2. Let x — x0 + x be a paravector. If xQ + x0(a ■ x) + |x|2 — 0, then
_1 Xo x + & ■ x
x 1 —
Xo +x0(a ■ x) + |x|2 10
Proof. We calculate
x(x0 — gc) = (x0 + x)(x0 — x) = x^ — x0 as + x0x — xx_ =
= x^ — x0(x — a ■ x) + x0x — xx_ = x0 + x0(a ■ x) + Ix\2. □ If a = 0, a generalized Clifford algebra Cn(a) does not have direct sum representation by multivectors. We denote Cn\a) = R and Cn\a) = Rra. Consider the subspace
Cn\o) = Span{eiej : i,j = 1,...,n and i = j}.
Multiplication rule (2) states that in addition to the bivectors, the set contains vectors. We can represent it defining
->(2)
Cn (a) = Span{eiej : i < j}
and then
C^(a)= C™(a) © Rra
(2)
Indeed, if B £ C Kn2)(a), using (2) we obtain the representation
B = hje-i&j = — bji)eiej + bji(ajei + aiej)■
i=j i<j i<j
Similarly, for any k = 2,... ,n, we can represent
c (nk) (a) = e (a) ©■■■© c^ (a) © Rra,
where C^ (a) is spanned by all products of k basis vectors and C^ (a) is spanned by all products of basis vectors in increasing order.
Another consideration is that the vector a = 0 can be used to divide space by
Rra = V (a) © Span {a},
where
V(a) = Span{a}x = {x £ Rra : a ■ x = 0}. If x £ V(a), then a; = x and x2 = — Ix\2. We have
C£(V (a)) = R0,„-i.
4. Powers of vectors. Let us look at algebraic differences a bit more. In a Clifford algebra, the powers xk for k £ N, are easily calculated and they are always either scalars or vectors. In the generalized case, the situation is very different. From the definition of multiplication, we have
x2 = —Ix\2 + (a ■ x)x. Proposition 4.1. Let A, B £ R and x £ Rra. Hence
(A + Bx)x = —BIx\2 + (A + B(a ■ x))x,
that is, all the powers xk are proper paravectors, that is, they have a non-zero scalar and vector part. We calculate
□
(A + Bx)x — Ax + Bx2 — Ax + B(—|x|2 + (a ■ x)x) —
— Ax — B|x|2 + B(a ■ x)x — —B|x|2 + (A + B(a ■ x))x.
We get the following recursive representation for the powers. Proposition 4.2. If x G Rra, then
xk — Pk (x) + Qk (x)x,
where
p3 (x) — —Qj-^xM2,
Qj(x) — Pj-1 (x) + Qj-1(x)(a ■ x),
starting from P1(x) — 0 and Q1 (x) — 1. Proof. The first step is
P2(x) — — Ql(x)|x|2 — —|x|2, Q2(x) — P1 (x) + Q1(x)(a ■ x) — a ■ x,
and we obtain
x2 — P2(x) + Q2(x)x — —x|2 + (a ■ x)x.
Assume
xk — Pk (x) + Qk (x)x. Using the preceding proposition, we calculate
xk+1 — (Pk (x) + Qk (x)x)x —
— —Qk(x)|x|2 + {Pk(x) + Qk(x)(a ■ x))x,
that is
Pk+1(x) — —Qk (x)|x|2, Qk+1(x) — Pk (x) + Qk (x)(a ■ x).
□
We observe, that the homogeneous polynomials Pk and Qk are generated by |x|2 and a ■ x. For example,
P2 (x) — —N2, P3(x) — —(a ■ x)|x|2, P4(x) — |x|4 — (a ■ x)2|x|2, Q2(x) — a ■ x,
Q3(x) = \x\2 + (a • x)2, Q4(x) = -2(a • x)\x\2 + (a • x)3.
5. Dirac operators and regular functions. We define the Dirac operator
by
n
D = ^ e3dX]. 3=1
Let Q C Rra be an open subset and f : Q ^ Cn(a) a differentiable function. If Df = 0 in Q, the function f is called (left) regular, and respectively fD = 0 is called right regular. We define
n
D = ^ ^ =D — a D, 3=1
where a • D is the directional derivative along a.
Remark 5.1 (Monogenic functions). If a = 0, we consider functions f : Q ^ Mo,ra. This is the Clifford analysis case. Then the solutions Df = 0 (or fD = 0) are called left (or right) monogenics.
Proposition 5.1. If x E Rra, then
Dx. = Dx = —n, (5)
Dx = —n + a,
and if x = 0, then
Dx-1 =
n-2
lx\2 '
Proof. Using (3), we calculate
n n
Dx = ^ ei£jdXixj = ej£j = —n-
i,3 = l 3 = 1
Similarly, we have Dx = —n. Since x = 5 + a • x, we have
Dx = Dx + D(a • x) = —n + a.
If x = 0, then we have
^ x Dx ^ 1 _ n xx n — 2
Dx-1 = —D~ =__= — D_x =__+ 2 — =_.
_ lx\2 ixxl2 lx\2~ ixxl2 ix4 ixxl2 '
□
We call the constant —n+a an abstract dimension of the generalized quadratic space Rra.
□
Proposition 5.2. Dx2 = (—n + a)(a ■ x) + (—2 + a)x. Proof. Since x2 = (a ■ x)x — lx\2 and Dx = —n + a, we calculate
Dx2 = —Dlx\2 + D(a ■ x)x + (a ■ x)Dx = = —2x + ax + (a ■ x)(—n + a) = = (—2 + a)x + (a ■ x)(—n + a).
Recall that the Euler operator is defined by
n
E = J2xidxi.
3 = 1
Then we can prove the following product rule for the Dirac operator.
Proposition 5.3. If f is a differentiable function taking values in Cn(a), then
D(xf) = (—n + a)f — gcDf — 2Ef + x(a ■ D) f, D (xf ) = —nf — xDf — 2Ef, D(xf ) = —nf — xDf — 2Ef.
Proof. We calculate
n
D(M_f) = (Dx)f ei£3x3dxi f.
i,3 = l
Using (3), (4) and (5), we obtain
n n
D(xf) = —nf —Y1 ei£ixidxif + 2 Y1 e3£3x3dXjf =
i,3 = l 3 = 1
= —nf — xDf — 2Ef.
Since x = x + a ■ x, we have
D(xf) = D((x + a ■ x)f) = D(xf) + D((a ■ x)f) = = (—n + a)f — xDf — 2Ef + (a ■ x)Df = = (—n + a)f — x.Df — 2Ef + x(a ■ D) f.
Moreover,
D(xf) = D(xf) — af — x(a ■ D)f = —nf — xDf — 2Ef. Using the preceding operators, we can factorize the Laplacian
n
2
X
3=1
as usual.
□
Proposition 5.4. DD = DD = —A.
Proof. Let be a twice differentiable function. We calculate
n
DDf = ^ ei£jdxidXjf
e í£ j^xi^xj,
i,j=1
n
2
Y^ ei£jdXidXjf + Y ejSjdl.f + ^ ei£jdXidXjf i<j j=1 i>j
n
= Y1 ei£jdXidXjf — Yd2X]f + Y ej£idXidXjf = i<j j=l i<j
= J2( e i£3 + ej£i)dXidXjf — Af = —Af, i<j
where we use (3) and (4). Similarly, we calculate DD = —A. □
This property allows us to prove the following classical results. Proposition 5.5. If f : Q ^ Cn(a) is regular, its component functions are harmonic.
Proposition 5.6. If f : Q ^ Cn(a) is harmonic, then
Df — (a -D)f
is regular in Q.
From Proposition 5.3, we obtain the following results. Proposition 5.7. If f : Q ^ Cn(a) is regular, then
(a) D(xf) = —nf — 2Ef,
(b) A(xJ) = 0, that is, xf is harmonic.
Proposition 5.8. If f : Q ^ Cn(a) satisfies Df = 0, then
(a) D(xif) = —nf — 2Ef,
(b) A(xf) = 0, that is, xf is harmonic.
6. Vector and paravector-valued solutions. Let us look at two examples in this section. The examples illustrate the role of the vector a among the regular functions.
Proposition 6.1. Consider a vector valued differentiable function
n
f (x) = J2 ejfj (x).
3=1
Then
Df = ^ (kej (dXi fj — dXj fi) + (a •D)f — D • f.
i<j
Hence, f is regular if and only if
dXifj = dXjfi, (a •D)f = 0, D -f = 0.
Proof. We substitute ej = £j + aj and we have
f = £j + a) & = Y1 (x) + a ■ f = f + a ■ f.
3=1 3=1
Hence, using (4),
Df = ei£3d^ih = ^ ei£j9Xi fj + ej£j9*i fj + Y1 ei£j9xih i,j=\ i<j j=l i>j
n
= ^2 ei£ j9xi fj + ej£i9%j fi — 9xi ¿3 = i<j i<j j=l
ei£3dXifj — ei£3d^j fi — D ■ f i<j i<j
ei£3 (9*i fj — 9Xjfi) — D ■ f =
i<j
ei(ej — a)(9Xi fj — dXj fi) —D ■ f =
i<j
e*e3 (9*i fj— 9xjfi) — Y1 eiaj (9xi fj— 9xi f'i) — D ■ f.
i<j i<j On the other hand,
D(a ■ f) = YaiDfi = ^ ejai9Xj f,
Cjat9xj Ji i=l i,j=l
n n
y] ejai9Xj fi + ^ ejai9Xj fi + ^ ejaj9Xj fj = i<j i>j j=l n n
= E ejai9x, fi + E ^iaj9Xi fj + E ejaj9xifj, i<j i<j j=l
and we obtain
Df = Y1 eiej (9xi fj— 9xi f'i) — Y1 eiaj9*i fj + Y1 ^j9*! f^— D ■ i<j i<j i<j
n n
&jai9Xj fi + eiaj9Xifj + ejaj9xj fj =
i<j i<j j=l
= Y1 eiej (9xi fj— 9xi f'i) + Y1 ^j9*! fi + i<j i<j
&iai9Xi fi + y^ e-iaj9Xi fj — D ■ f. i=l i>j
The middle sum terms are obtained in the form
n
y] e ia,jdXj fi + ^ eiai9xi fi + ^ h i<j i=l i>j
fl fl fl ^ eiajdXj fi = ( ^ ajdx^ ( ^ af^j = (a ■ D) f.
i,j=l j=l i=l We conclude
Df = ^ (dXi fj - dXj fi) + (a ■D)f — D ■ f. Q
i<j
When a = 0, the solution is a vector-valued monogenic function. Therefore, a regular vector-valued function is a monogenic function whose directional derivative in the direction a vanishes, i.e. the function is constant in this direction. Corollary 6.1. A paravector-valued differentiable function
f(x) = fo(x)+l(x),
where
'n
¿(X) = Y ej fj (x), j=l
is regular if and only if
dxifj = 9xjfi, for i,j = l,...,n, (a ■D)l + Dfa = 0, D ■f = 0.
Thus, a regular paravector-valued function f = f0 + f is a monogenic vector-valued function f whose directed derivative in the direction a is —Df0.
7. Cauchy's integral formula. In some situations, the generalized theory and the Clifford analysis are exactly the same in form and proof. One such example will be presented next. It is assumed that the reader knows the structure of the proof of Cauchy's formula in the Clifford analysis case (see e.g. [7,8]). We calculate the Cauchy kernel as usual.
Proposition 7.1 (Cauchy kernel). The Cauchy kernel is of the form
E (x) = -
1 x-1 üf-1 \x\n-2
and it is left and right regular for x = 0. In the kernel, un-l is the surface area of the unit sphere in Rn.
Proof. We start from the Newton potential
N (x) = 1
(2 -n)un-i\x\n-2'
which defines the fundamental solution for the Laplace equation, that is, AN = 5. We calculate
9*N (x) = — — ^
J un-l \x\n
and
1 T
DN (x) = —
un-l \x\n' We define the Cauchy kernel by
E(x) = —DN(x) = —(D — a ■ D)N(x) =
1 x 1 a x
= —DN(x) + a ■ DN(x) =
un-l lxln un-l \x\
Since
we have
ry _ /7 . rv< - /y>
•A/ U/ tXy - «A/ «
E(x) = —__IL =__1__x l
Un-l \x\n Un-l \x\n-2'
□
Although the Cauchy kernel looks formally the same as in the classical case, it is nevertheless of paravector valued.
The proof for the Clifford-Stokes formula is identical:
Jan
since in
fdag = i ((fD)g + f(Dg))dV, Jn
d(fdag) = ((fD)g + f(Dg ))dV
we use only the product rule of the exterior derivative d.
In the proof of the Cauchy formula, it is important to evaluate the integral
I E (y — x)n(y)f(y)dS (y),
J a Br (x)
where Br (x) is the r-ball centered at x and n(y) the outward pointing unit vector on the boundary. The unit normal is as usual
i ^ y — x
n(y) = ^r~
and hence
f 1 f (y — x)-1 y — x
E(y — x)n(y)f (y)dS(y) = — - -n-2~r ~f(y)dS(y) =
JdBr(x) - - - - Un-lJdBr(X) \y — x\ 2 r - -
n
= ,, rn-l f (y)dS(y) ^ f(x)
Wn-lrn l JdBr(x) - -
when r ^ 0. So, this part of the proof is exactly the same as in the classical case.
Theorem 7.1 (Cauchy integral formula). Let Q C Rn be an open set with a smooth boundary, let f : U ^ Cn(a) be a regular function, and Q C U. Then
f(x)=í E(y - x)da(y)f(y) Jdn
for any x G Q.
We conclude that in above the only difference is the interpretation of the Cauchy kernel and the proof itself is identical. A more detailed treatment of this issue is naturally unnecessary.
8. Regular functions generated by the Cauchy kernel. Let us use the
classic multi-index notation, i.e. let a = (al}... ,an), aj G N U {0} for all j = 1,... ,n, |a| := al + ■ ■ ■ + an, a! := al! ■ ■ ■ an!, xa := xf1 ■ ■ ■ x^n and d™ := d^1 ■ ■ ■ dX^. We define paravector valued regular functions
Ua(x) = dg x
x
Indeed, if Ua = u0a) + U(a), we have
U0a) (x) = —d" ^, U(a) (x) = 6^. 0 w x lx\n ~ w x lx\n
Remark 8.1. These functions are useful, when we want to find Taylor series, using
1 ^ (—1)\^ 1
E^ ■D,)
ly — x\n-2 ¿=0 k! v" y-J lyln-2:
see e.g. [9, p. 34], and the Cauchy formula in the above. The multi-index Leibniz rule is
6x(fg)=Eia) (d*f )(9*-i3g).
Since
61 (x — a ■ x) = 0
for ^ 2, we obtain
Ua(x) = %= E (0)64(x — a ■ x)d«-
xx \xn feW xv---\xlf
= (x - a ■ x)dl+ E (dx3 (y - a ■ x)dX~ij
1
= (x — a ■ x)9% ^ + E ( (e3 — a )91 ^
where ej = (0,..., 1,..., 0) is a unit multi-index. We define polynomials pa by
9a 1 = Pa(x) - \x\n = \x\n+2lal.
Hence, the regular functions are of the form
P (x) Pa e■ (x)
Ua(x) = (x — a ■ x)+ E a,(e3 — a) ^+2^-2.
Let us take a closer look at the polynomial. We have
1 Pa+€j (x)
9X 3
\x\\x\n+2|a+^ and
Mj Pa(x) _ ( + 2, h Pa(x) QXjpa(x) _
dx \x|n ~°Xi \x|n+2|a| _ (n + Z\a\)X3 \x|n+2|a|+2 + \x|n+2|a| _
_ -(n + 2\a\)Xjpa(x) + \x|2dxjPa(x)
\ x I n+2W+ej | '
and by comparing these, we get the differential-recurrence relations
Pa+ej (x) _ -(n + 2 \ a )xjPa(x) + \x\2&Xj Pa(x) and po(x) _ 1- Multiplying the recursion both sides by \x|-2|a|-n-2, we get
\ xl ^^Pa+e, (x) _ -(n + 2\a )yj\x\-2a-n-2Va(x) + \x| ^^^ Pa(x), that is,
\ x|^W--2^. (x) _ dx. (| x| ^^(x))
or
Pa+e, (x) _ \x|2|a|+n+2дx] (|x|-2H-nPa(x)) ■ Let us consider linear operators
Lj f(x) _ \x|2\a\+n+2dXj (|x|^^f (x)),
satisfying
Lj(1)_ -(2\a\ + n)xj
and
Lj (xf) _ xf-1(m|x|2 - (2\a\ +n)x2) _
= mx\x™-1 + ■■■ + (m - 2\a\ - n)xf+l + ■■■ + mx2nxf-1.
Similarly,
Lj (xa) =xa1 ■■■ xaj-1Lj (x™j )xai+
j
= xai ■■■xa>-1 (ajxjx^-1 + ■■■ +
and
Xj x i x j
+ a - 2\a\ - n)xa/+1 + ■ ■ ■ + a]x2nxa/-1)xai+1 ■■■xan = = aj xa+2"1-"i + ■■■ + ajxa+2e*-1-e* + (aj - 2\a\ - n)xa+"i +
+ aj xa+2"i+— + ■■■ + ajxa+2"n-"^
Pa+ej (x) = Lj (pa (x)) .
Conclusion. This paper considers analysis with generalized Clifford algebras. The central point of the analysis is the effect of the direction vector a, which determines the input on the theory. Most of the results of the classical Clifford analysis can be converted almost as is to the generalized case. The biggest differences come in situations where powers of a vector variable are needed. The effect of the vector a on the class of regular functions still needs to be examined further.
Competing interests. I declare that I have no competing interests.
Author's Responsibilities. I take full responsibility for submitting the final manuscript
in print. I approved the final version of the manuscript.
Acknowledgments. The author is grateful to his family, whose understanding and patience has helped in the writing this article.
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Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2023. Т. 27, № 1. С. 7-22 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1973
EDN: UQWDOF
УДК 512.646.7:517.95
Анализ обобщенных алгебр Клиффорда H. Orelma
Tampere University,
Kalevantie 4, Tampere, 33100, Finland.
Аннотация
Изучается вопрос, связанный с обобщенными алгебрами Клиффорда Сп(а), где а — ненулевой вектор. Если {е\,..., еп} —ортонормированный базис, операция умножения определяется соотношениями
е ^ = ajej — 1,
где aj = ej ■ а. Случай а = 0 соответствует классической алгебре Клиффорда. Определяется оператор Дирака D = Y1 j ej9Xj и регулярные функции как его нулевое решение. Изучаются алгебраические свойства рассматриваемой алгебры. Доказываются основные формулы для оператора Дирака и изучаются свойства регулярных функций.
Ключевые слова: алгебра Клиффорда-Канзаки, обобщенная алгебра Клиффорда, оператор Дирака, регулярная функция.
Получение: 27 декабря 2022 г. / Исправление: 16 февраля 2023 г. / Принятие: 27 февраля 2023 г. / Публикация онлайн: 30 марта 2023 г.
Конкурирующие интересы. Я заявляю, что у меня нет конкурирующих интересов в отношении данной статьи.
Авторская ответственность. Я несу полную ответственность за представление окончательной рукописи в печатном виде. Я одобрил окончательный вариант рукописи.
Благодарности. Автор благодарен своей семье, чье понимание и терпение помогло в написании этой статьи.
Дифференциальные уравнения и математическая физика Научная статья
© Коллектив авторов, 2023 © СамГТУ, 2023 (составление, дизайн, макет)
3 ©® Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru) Образец для цитирования
Orelma H. Analysis on generalized Clifford algebras, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 1, pp. 722. EDN: UQWDOF. DOI: 10.14498/vsgtu1973.
Сведения об авторе
Heikki Orelma https://orcid.org/0000-0002-8251-4333
D.Sc. (Tech.), Adjunct Professor; Researcher; Dept of Mechanics and Mathematics;
e-mail: [email protected]
