Lecture 3 Hypercomplex Linear Representations

1. Traditionally only complex valued characters of the subgroup N are considered, they are:

   χτℂ

   ⎛ ⎜ ⎝         1 0         t 1 ⎞ ⎟ ⎠

   =ei τ t,     where τ∈ℝ. (10)

   A direct calculation shows that:

   r

   ⎛ ⎜ ⎝         a b c d ⎞ ⎟ ⎠

   =

   ⎛ ⎜ ⎜ ⎜ ⎜ ⎝         1 0 c d 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

   ∈ N′.

   Thus:

   r(g−1*s(u,v))=

   ⎛ ⎜ ⎜ ⎜ ⎜ ⎝         1 0 cv d+cu 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

   ,   where  g−1=

   ⎛ ⎜ ⎝         a b c d ⎞ ⎟ ⎠

   . (11)

   A substitution of this value into the character (10) together with the Möbius transformation (1) we obtain the next realisation of (5):

   ℂτ(g) f(w)= exp

   ⎛ ⎜ ⎜ ⎝

   i

   τ c v cu+d

   ⎞ ⎟ ⎟ ⎠

   f

   ⎛ ⎜ ⎜ ⎝

   aw+b cw+d

   ⎞ ⎟ ⎟ ⎠

   ,    where  w=u+ε v,   g−1=

   ⎛ ⎜ ⎝ a b c d ⎞ ⎟ ⎠

   .

   The representation acts on the space of complex valued functions on the upper half-plane ℝ₊², which is a subset of dual numbers as a homogeneous space SL₂(ℝ)/N′. The mixture of complex and dual numbers in the same expression is confusing.
2. The parabolic character χ_τ with the algebraic flavour is provided by multiplication (2) with the dual number:

   χτ

   ⎛ ⎜ ⎝         1 0         t 1 ⎞ ⎟ ⎠

   =eε τ t=1+ε τ t,     where τ∈ℝ.

   If we substitute the value (11) into this character, then we receive the representation:

   τ(g) f(w)=

   ⎛ ⎜ ⎜ ⎝

   1+ε

   τ c v cu+d

   ⎞ ⎟ ⎟ ⎠

   f

   ⎛ ⎜ ⎜ ⎝

   aw+b cw+d

   ⎞ ⎟ ⎟ ⎠

   ,

   where w, τ and g are as above. The representation is defined on the space of dual numbers valued functions on the upper half-plane of dual numbers. Thus expression contains only dual numbers with their usual algebraic operations. Thus it is linear with respect to them.

1. Subgroups K, N′ and play a similar rôle in the structure of the group SL₂(ℝ) and its representations.
2. The subgroups shall be swapped simultaneously with the respective replacement of hypercomplex unit ι.

1. Let X=Z be a generator of the subgroup K (3). Since this is a compact subgroup the corresponding eigenspaces Z′ v_k=i k v_k are parametrised by an integer k∈ℤ. The raising/lowering or ladder operators L^± [240]*§ VI.2 [321]*§ 8.2 are defined by the following commutation relations:

   [Z′,L±]=λ±L±.  (15)

   In other words L^± are eigenvectors for operators adZ of adjoint representation of sl₂ [240]*§ VI.2.
   Remark 8  The existence of such ladder operators follows from the general properties of Lie algebras if the element X∈sl₂ belongs to a Cartan subalgebra. This is the case for vectors Z and B, which are the only two non-isomorphic types of Cartan subalgebras in sl₂. However the third case considered in this paper, the parabolic vector B+Z/2, does not belong to a Cartan subalgebra, yet a sort of ladder operators is still possible with dual number coefficients. Moreover, for the hyperbolic vector B, besides the standard ladder operators an additional pair with double number coefficients will also be described.

   From the commutators (15) we deduce that L⁺ v_k are eigenvectors of Z′ as well:

   Z′(L+ vk) = (L+Z′+λ+L+)vk=L+(Z′vk)+λ+L+vk =i k L+vk+λ+L+vk   = (i k+λ+)L+vk.

   Thus action of ladder operators on respective eigenspaces can be visualised by the diagram:

   1 …  <.4ex>[r]L+         Vi k−λ   <.4ex>[l]L−<.4ex>[r]L+         Vi k  <.4ex>[l]L− <.4ex>[r]L+         Vi k+ λ <.4ex>[l]L−  <.4ex>[r]L+         …<.4ex>[l]L− (16)

   Assuming L⁺=aA′+bB′+cZ′ from the relations (13) and defining condition (15) we obtain linear equations with unknown a, b and c:

   c=0,    2a=λ+ b,    −2b=λ+ a.

   The equations have a solution if and only if λ₊²+4=0, and the raising/lowering operators are

   L±=±i A′+B′. (17)

2. Consider the case X=2B of a generator of the subgroup  (5). The subgroup is not compact and eigenvalues of the operator B′ can be arbitrary, however raising/lowering operators are still important [140]*§ II.1 [251]*§ 1.1. We again seek a solution in the form L_h⁺=aA′+bB′+cZ′ for the commutator [2B′,L_h⁺]=λ L_h⁺. We will get the system:

   4c=λ a,   b=0,   a=λ c.

   A solution exists if and only if λ²=4. There are obvious values λ=± 2 with the ladder operators L_h^±=±2A′+Z′, see [140]*§ II.1 [251]*§ 1.1. Each indecomposable sl₂-module is formed by a one-dimensional chain of eigenvalues with a transitive action of ladder operators.

   Admitting double numbers we have an extra possibility to satisfy λ²=4 with values λ=±2є. Then there is an additional pair of hyperbolic ladder operators L_є^±=±2єA′+Z′, which shift eigenvectors in the “orthogonal” direction to the standard operators L_h^±. Therefore an indecomposable sl₂-module can be parametrised by a two-dimensional lattice of eigenvalues on the double number plane, see Fig. 3.2

   ---

   =2.5em@C=1.5em@M=.5em  …  <.4ex>[d]^L_є+  …  <.4ex>[d]^L_є+  …  <.4ex>[d]^L_є+
   …  <.4ex>[r]⁻L_h⁺  V_{(n−2)+є (k−2)}  <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{n+є (k−2)}  <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{(n+2)+є (k−2)} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  …<.4ex>[l]⁻L_h⁻
   …  <.4ex>[r]⁻L_h⁺  V_{(n−2)+є k}  <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{n+є k}  <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{(n+2)+є k} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  …<.4ex>[l]⁻L_h⁻
   …  <.4ex>[r]⁻L_h⁺  V_{(n−2)+є (k+2)}  <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{n+є (k+2)}  <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  V_{(n+2)+є (k+2)} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+  …<.4ex>[l]⁻L_h⁻
    …  <.4ex>[u]^L_є−  …  <.4ex>[u]^L_є−  …  <.4ex>[u]^L_є−

   Figure 3.2: The action of hyperbolic ladder operators on a 2D lattice of eigenspaces. Operators Lh± move the eigenvalues by 2, making shifts in the horizontal direction. Operators Lє± change the eigenvalues by 2є, shown as vertical shifts.

   ---

3. Finally consider the case of a generator X=−B+Z/2 of the subgroup N′ (6). According to the above procedure we get the equations:

   b+2c=λ a,   −a=λ b,

   a 2

   =λ c,

   which can be resolved if and only if λ²=0. If we restrict ourselves with the only real (complex) root λ=0, then the corresponding operators L_p^±=−B′+Z′/2 will not affect eigenvalues and thus are useless in the above context. However the dual number roots λ =± ε t, t∈ℝ lead to the operators L_ε^±=± ε tA′−B′+Z′/2. These operators are suitable to build an sl₂-modules with a one-dimensional chain of eigenvalues.

• the introduction of complex numbers is a necessity for the existence of ladder operators in the elliptic case;
• in the parabolic case we need dual numbers to make ladder operators useful;
• in the hyperbolic case double numbers are not required neither for the existence or for the usability of ladder operators, but they do provide an enhancement.

Then raising/lowering operators L^± satisfying to the commutation relation:

are:

Here Y∈sl₂ is a linear combination of B and Z with the properties:

• Y=[A,X].
• X=[A,Y].
• Killings form K(X,Y) [159]*§ 6.2 vanishes.

Any of the above properties defines the vector Y∈span{B,Z} up to a real constant factor.