40
J. L. Cie´ sli´ nski
One can define following automorphisms and anti-automorphisms in any Clifford
algebra (using also linearity, in the first two cases, or anti-linearity, in the third
case):
• grade involution α α α: α α α(XY ) = α α α(X)α α α(Y ), α α α(v) = −v.
• reversion β β β: β β β(XY ) = β β β(Y )β β β(X) , β β β(v) = v.
• complex conjugation: XY = ¯
X ¯
Y , ¯
v = v.
• Clifford conjugation (complexified): X ∗ := α α α(β β β( ¯
X)).
The reversion can be used to compute the so-called spinor norm:
N(X) = Xβ(X) ,
(9)
which is a real number for X ∈ Γ (V , Q).
3 Spin-Valued Linear Problems
We consider linear problems of the form:
Ψ, μ = U μ Ψ ,
U μ =
j
(10)
where e j are generators of a Clifford algebra, and u jk depend on x μ , λ and are
real for λ ∈ R. However, in the following, we suppress the dependence on x μ and,
sometimes, on λ). Then, Ψ is Spin-valued (provided that the initial condition is
Spin-valued).
Our original motivation came from studying isothermic surfaces (which, by definition, admit conformal parameterization of curvature lines). Using isomorphisms
so(4, 1) ∼ = sp(1, 1) ∼ = spin(4, 1)
(11)
we transformed SO(4, 1)-valued Lax pair into the form (10). Clifford algebras are,
in general, a useful tool in dealing with isothermic surfaces [22].
We point out that Lie algebra su(2) is spanned by σ k σ j , hence SU (2) ∼ = Spin (3)
and all SU (2)-valued linear problems, including (1), belong to the class (10).
The case of isothermic surfaces suggested further restrictions on the form of the
linear problem. We consider two vector spaces, V and W , equipped with quadratic
forms and orthogonal to each other, such that
• dimV = r and e 1 , . . . , e r is an orthonormal basis in V ,
• dimW = q and e r+1 , . . . , e r+q is an orthonormal basis in W .
In this paper we focus on the following class of linear problems [23]:
J. L. Cie´ sli´ nski
One can define following automorphisms and anti-automorphisms in any Clifford
algebra (using also linearity, in the first two cases, or anti-linearity, in the third
case):
• grade involution α α α: α α α(XY ) = α α α(X)α α α(Y ), α α α(v) = −v.
• reversion β β β: β β β(XY ) = β β β(Y )β β β(X) , β β β(v) = v.
• complex conjugation: XY = ¯
X ¯
Y , ¯
v = v.
• Clifford conjugation (complexified): X ∗ := α α α(β β β( ¯
X)).
The reversion can be used to compute the so-called spinor norm:
N(X) = Xβ(X) ,
(9)
which is a real number for X ∈ Γ (V , Q).
3 Spin-Valued Linear Problems
We consider linear problems of the form:
Ψ, μ = U μ Ψ ,
U μ =
j
(10)
where e j are generators of a Clifford algebra, and u jk depend on x μ , λ and are
real for λ ∈ R. However, in the following, we suppress the dependence on x μ and,
sometimes, on λ). Then, Ψ is Spin-valued (provided that the initial condition is
Spin-valued).
Our original motivation came from studying isothermic surfaces (which, by definition, admit conformal parameterization of curvature lines). Using isomorphisms
so(4, 1) ∼ = sp(1, 1) ∼ = spin(4, 1)
(11)
we transformed SO(4, 1)-valued Lax pair into the form (10). Clifford algebras are,
in general, a useful tool in dealing with isothermic surfaces [22].
We point out that Lie algebra su(2) is spanned by σ k σ j , hence SU (2) ∼ = Spin (3)
and all SU (2)-valued linear problems, including (1), belong to the class (10).
The case of isothermic surfaces suggested further restrictions on the form of the
linear problem. We consider two vector spaces, V and W , equipped with quadratic
forms and orthogonal to each other, such that
• dimV = r and e 1 , . . . , e r is an orthonormal basis in V ,
• dimW = q and e r+1 , . . . , e r+q is an orthonormal basis in W .
In this paper we focus on the following class of linear problems [23]: