UDC 519.46

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR I. M. GELFAND, V. A. PONOMAREV

CLASSIFICATION OF INDECOMPOSABLE INFINITESIMAL REPRESENTATIONS OF THE LORENTZ GROUP

Let \(L\) be the Lie algebra of the proper Lorentz group, and let \(L_k\) be its subalgebra corresponding to the maximal compact subgroup, which is isomorphic to the rotation group of three-dimensional space. Recall \((^{2})\) that a module \(M\) is called a Harish-Chandra module over the Lie algebra \(L\) if, regarded as a module over \(L_k\), it is a direct sum of finite-dimensional submodules: \(M=\oplus M_i\). Here each \(M_i\) is an irreducible module over \(L_k\), and for every \(M_{i_0}\) in the sequence \(\{M_i\}\) there exist no more than a finite number equivalent to it.

A module \(M\) over the Lie algebra \(L\) is defined by specifying six operators \(H_+, H_-, H_3, F_+, F_-, F_3\), which satisfy the known commutation relations \((^{1})\). By \(\Delta_1\) and \(\Delta_2\) we shall denote the Laplace operators. They are expressed in terms of the mappings \(H\) and \(F\) as follows:

\[ \Delta_1 = \frac{1}{2}(H_-F_+ + F_-H_+) + H_3F_3 + F_3; \]
\[ \Delta_2 = H_-H_+ - F_-F_+ + H_3^2 - F_3^2 + 2H_3. \tag{1} \]

In the paper the explicit form of the operators \(H\) and \(F\) acting in the space \(M\) of an indecomposable module will be written out.

1. The operators \(H_+, H_-, H_3\) determine in the representation the basis of the subalgebra \(L_k\). It is known that each irreducible representation over \(L_k\) is specified by one positive number \(l\), integer or half-integer. In accordance with this, in the decomposition \(M=\oplus M_i\) we combine those \(M_i\) which correspond to representations with the same \(l\). As a result we obtain a decomposition \(M=\oplus R_l\) such that the representation in \(R_l\) of the subalgebra \(L_k\) is a multiple of an irreducible one. This means that \(R_l\) is a finite-dimensional subspace invariant with respect to the operators \(H_+, H_-, H_3\).

By \(R_{l,m}\) \((R_{l,m}\subset R_l)\) we denote the subspace of eigenvectors of the operator \(H_3\) corresponding to the eigenvalue \(m\):

\[ (H_3)_{l,m}=m I_{l,m} \quad (m=-l,-l+1,\ldots,l-1,l), \tag{2} \]

where \(I_{l,m}\) is the identity operator, and the index \(l,m\) by the operator \(H_3\) means that this operator is considered with domain \(R_{l,m}\).

Using the explicit form of the representation of the algebra \(L_k\) \((^{1})\), one can show that all subspaces \(R_{l,m}\) with one and the same \(l\) have the same dimension and \(R_l=\oplus R_{l,m}\). In this case one can introduce operators \(E_+\) and \(E_-\) with the following properties:

\[ (E_+)_{l,m}: R_{l,m}\to R_{l,m+1}; \qquad (E_-)_{l,m}: R_{l,m}\to R_{l,m-1}, \tag{3} \]

assuming that \(R_{l,l+1}=R_{l,-l-1}=0\) and

\[ (E_-E_+)_{l,m}=I_{l,m}\;(m<l); \qquad (E_+E_-)_{l,m}=I_{l,m}\;(m>-l). \tag{4} \]

The operators \(H_+\) and \(H_-\) are expressed through \(E_+\) and \(E_-\) by the formulas:

\[ (H_+)_{l,m}=\sqrt{(l+m+1)(l-m)}\,(E_+)_{l,m}; \]
\[ (H_-)_{l,m}=\sqrt{(l+m)(l-m+1)}\,(E_-)_{l,m}. \tag{5} \]

Relations (3) and (4) show that the mappings \(E_+\) and \(E_-\) define compatible isomorphisms between the spaces \(R_{l,m}\) and \(R_{l,m+1}\).

2. Operators \(F_+, F_-, F_3\). In order to write expressions for these operators, we introduce the mappings \(D_+, D_-, \Delta\)

\[ (D_+)_{l,m}: R_{l,m}\to R_{l+1,m}; \qquad (D_-)_{l,m}: R_{l,m}\to R_{l-1,m}; \tag{6} \]

\[ (\Delta)_{l,m}: R_{l,m}\to R_{l,m}. \]

We shall write out these operators later. Now, with their help, we write expressions for the operators \(F\):

\[ (F_3)_{l,m}=\sqrt{l^2-m^2}(D_-)_{l,m}-m d_l(\Delta)_{l,m}-\sqrt{(l+1)^2-m^2}(D_+)_{l,m}; \tag{7} \]

\[ \begin{aligned} (F_+)_{l,m}={}&\sqrt{(l-m)(l-m-1)}(D_-)_{l,m+1}(E_+)_{l,m}\\ &-d_l\sqrt{(l-m)(l+m+1)}(E_+)_{l,m}(\Delta)_{l,m}\\ &+\sqrt{(l+m+1)(l+m+2)}(E_+)_{l+1,m}(D_+)_{l,m}; \end{aligned} \tag{8} \]

\[ \begin{aligned} (F_-)_{l,m}={}&-\sqrt{(l+m)(l+m-1)}(D_-)_{l,m-1}(E_-)_{l,m}\\ &-d_l\sqrt{(l+m)(l-m+1)}(E_-)_{l,m}(\Delta)_{l,m}\\ &-\sqrt{(l-m+1)(l-m+2)}(E_-)_{l+1,m}(D_+)_{l,m}, \end{aligned} \tag{9} \]

where \(d_l=-l^{-1}(l+1)^{-1}\).

We shall require of the operators \(D_+, D_-, \Delta\) that the diagrams be commutative

\[ \begin{array}{ccccc} R_{l-1,m+1} & \xleftarrow{D_-} & R_{l,m-1} & R_{l,m+1} & \xleftarrow{\Delta} & R_{l,m+1} & \xrightarrow{D_+} & R_{l+1,m+1} \\[-2pt] \uparrow E_+ && \uparrow E_+ && \uparrow E_+ && \uparrow E_+ && \uparrow E_+ \\[-2pt] R_{l-1,m} & \xleftarrow{D_-} & R_{l,m} & R_{l,m} & \xleftarrow{\Delta} & R_{l,m} & \xrightarrow{D_+} & R_{l+1,m} \end{array} \tag{10} \]

\[ -l+1\le m<l-1 \qquad\qquad -l\le m<l . \]

In addition, the same diagrams with the mapping \(E_-\) instead of \(E_+\) must also be commutative. Note that, by definition, the mapping \((E_+)_{l,m}\) is an isomorphism (for \(m<l\)). Consequently, the commutativity of diagrams (10) means that the mappings \(D_+, D_-, \Delta\) do not depend on the index \(m\). Therefore, in what follows we shall write \((D_+)_l\) instead of \((D_+)_{l,m}\), etc., wherever this cannot cause misunderstanding.

Next we shall require that the operators \(D_+, D_-, \Delta\) satisfy the relations

\[ (D_-)_{l}(\Delta)_l=(\Delta)_{l-1}(D_-)_{l}; \tag{11} \]

\[ (D_+)_{l}(\Delta)_l=(\Delta)_{l+1}(D_+)_{l}; \tag{12} \]

\[ (2l-1)(D_+)_{l-1}(D_-)_{l}-(2l+3)(D_-)_{l+1}(D_+)_{l} = I+l^{-2}(l+1)^{-2}(\Delta)_l^2. \tag{13} \]

Let us note that of all the relations which the operators \(D_+, D_-, \Delta\) must satisfy, essentially all except (13) are trivial.

Assertion 1. Let \(M\) be a Harish-Chandra module over the Lie algebra \(L\) of the proper Lorentz group. Then the space \(M\) is representable in the form of a direct sum of subspaces \(R_{l,m}\). Moreover, one can choose mappings \(E_+, E_-, D_+, D_-, \Delta\) such that the operators \(H\) and \(F\) are expressed through them by formulas (5), (7)—(9). Such mappings \(D, E, \Delta\) must satisfy relations (3)—(4), (10)—(13).

The converse is also true.

Assertion 2. If in the space \(M=\displaystyle\bigoplus_{l,m} R_{l,m}\) \((m=-l,-l+1,\ldots,l)\) operators \(D_+, D_-, \Delta, E_+, E_-\) are given which satisfy relations (3), (4), (10)—(13), and through these operators the mappings \(H_+, H_-, H_3, F_+, F_-, F_3\) are expressed by formulas (2), (5), (7)—(9), then the operators \(H\) and \(F\) define a representation of the Lie algebra \(L\) of the proper Lorentz group.

In paper (²) it was shown that in an indecomposable module the Laplace operators \(\Delta_1\) and \(\Delta_2\) have one eigenvalue each, \(\lambda_1\) and \(\lambda_2\). Moreover, for each indecomposable module one can specify a number \(l_0\), integer or half-integer \((l_0 \geq 0)\), such that the numbers \(\lambda_1,\lambda_2,l_0\) are related by

\[ l_0^4+(1+\lambda_2)l_0^2-\lambda_1^2=0. \tag{14} \]

The pair \((\lambda_1,\lambda_2)\) is called special if one can find a real number \(l_1\) satisfying the identities

\[ l_1^2l_0^2=-\lambda_1^2;\qquad l_1^2+l_0^2=-1-\lambda_2 \tag{15} \]

and such that the difference \((|l_1|-l_0)\) is a positive integer. A module \(M\) with such a pair \((\lambda_1,\lambda_2)\) will be called special. Otherwise we shall say that the pair \((\lambda_1,\lambda_2)\) and the module \(M\) with such a pair are nonspecial.

1. The operators \(D_+,D_-,\Delta\) in a nonspecial indecomposable module. In paper (²) it was shown that to every indecomposable nonspecial module there corresponds, and moreover uniquely, a finite-dimensional space \(P\) with a linear nilpotent mapping \(A\), where the matrix \([A]\) in some basis has the form of a single Jordan cell.

Theorem 3. Let \(M\) be a nonspecial indecomposable Harish-Chandra module. Then the spaces \(R_{l,m}\) \((l_0 \leq l)\) have the same dimension, equal to \(\dim P\). At the same time the mappings \((D_-)_{l,m}\) \((l\ne l_0)\) and \((D_+)_{l,m}\) are isomorphisms. In the spaces \(R_{l,m}\) one can choose a basis such that the matrices of the mappings \(D_+,D_-,\Delta\) have the form

\[ [D_+]_{l,m}=[I];\qquad [\Delta]_{l,m}=[A]+\lambda_1[I]; \]
\[ [D_-]_{l,m}=(l^2-l_0^2)(4l^2-1)^{-1}\bigl((1+\lambda_1l^{-2}l_0^{-2})[I]+ \tag{16} \]
\[ {}+l^{-2}l_0^{-2}(2[A]+[A]^2)\bigr), \]

where \([I]\) is the identity matrix, and the matrix \([A]\) has the form of a Jordan cell with zeros on the diagonal.

1. The operators \(D_+,D_-,\Delta\) in a special indecomposable module. In paper (²) it was shown that in this case the situation is radically different from that just described. A special indecomposable module, in comparison with an irreducible one, is described by a set of integers, and this set may be arbitrarily long; moreover, in some cases, also by an additional complex number \(\mu\). It was shown there that to indecomposable special modules there corresponds, and moreover bijectively, a pair of finite-dimensional spaces \(P_1\) and \(P_2\) with mappings \(d_+\colon P_1\to P_2;\ d_-\colon P_2\to P_1;\ \delta\colon P_2\to P_2\) such that the mappings \(d_+d_-\) and \(\delta\) are nilpotent and \(d_-\delta=\delta d_+=0\). Knowing the matrices of the mappings \(d_+,d_-,\delta\), one can find the matrices of the mappings \(D_+,D_-,\Delta\) in the module \(M\). It turns out that in a special indecomposable module \(M\) the subspaces \(R_{l,m}\) \((l_0\leq l\leq l_1-1)\) have the same dimension, equal to \(\dim P_1\), and the subspaces \(R_{l,m}\) \((l_1\leq l)\) also have the same dimension, equal to \(\dim P_2\). At the same time, in the subspaces one can choose a basis such that the matrices of the mappings \(D_+,D_-,\Delta\) have the following form:

\[ [D_+]_{l,m}=[I]\quad (l\ne l_1-1);\qquad [D_+]_{l_1-1,m}=[d_+]; \tag{17} \]

\[ [\Delta]_{l,m}=-l_1l_0\bigl((4l_1^2-1)(l_1^2-l_0^2)^{-1}[d_-][d_+]+[I]\bigr)^{1/2}\quad (l\leq l_1-1); \]

\[ [\Delta]_{l,m}=-il_1l_0\bigl((4l_1^2-1)(l_1^2-l_0^2)^{-1}[d_+][d_-]+[\delta]+[I]\bigr)^{1/2}\quad (l\geq l_1); \tag{18} \]

\[ [D_-]_{l,m}=(l^2-l_0^2)l^{-2}(4l^2-1)^{-1}\bigl(l_1^2(4l_1^2-1)(l_1^2-l_0^2)^{-1}[d_-][d_+]+ \]

\[ {}+(l_1^2-l^2)[I]\bigr)\quad (l_0\leq l\leq l_1-1); \tag{19} \]

\[ [D_-]_{l_1,m}=[d_-]; \]

\[ [D_-]_{l,m}=(l^2-l_0^2)l^{-2}(4l^2-1)^{-1}\bigl(l_1^2(4l_1^2-1)(l_1^2-l_0^2)[d_+][d_-]+ \]

\[ {}+(l_1^2-l^2)[I]+l_0^2(l^2-l_1^2)(l^2-l_0^2)[\delta]\bigr)\quad (l>l_1), \]

where \([I]\) is the identity matrix.

Assertion 4. Let \(M\) be an indecomposable special Harish-Chandra module over the Lie algebra \(L\). Then the spaces \(R_{l,m}\) \((l_0 \leq l \leq l_1-1)\) have the same dimension, and the spaces \(R_{l,m}\) \((l \geq l_1)\) also have the same dimension. Moreover, the mappings \((D_+)_{l,m}\) \((l \ne l_1-1)\) and \((D_-)_{l,m}\) \((l \ne l_0;\ l \ne l_1)\) are isomorphisms. The matrices of all mappings \(D_+\), \(D_-\), \(\Delta\) are expressed in terms of the matrices \([d_+]\), \([d_-]\), \([\delta]\) by formulas (17)—(19). The matrices \([d_+]\), \([d_-]\), \([\delta]\) correspond to operators on the spaces \(P_1\) and \(P_2\):

\[ P_1 \xrightarrow{d_+} P_2 \xrightarrow{\delta} P_2 \xrightarrow{d_-} P_1 \]

so that the mappings \((d_+d_-)\) and \((\delta)\) are nilpotent, and \(\delta d_+ = d_-\delta = 0\).

The canonical form of an indecomposable object which is a pair of spaces with mappings \(d_+\), \(d_-\), \(\delta\), is described in paper \((^2)\). It is of two types. The first type is called an open object, and its invariants are a set of positive integers \([s,n_1,m_1;\ n_2,m_2;\ldots;\ n_k,m_k]\), where \(s=0\) or \(1\); \(n_1 \geq 0\); \(n_i>0\); \(m_i>0\); \(m_k \geq -1\). The second type is called a closed object, and its invariants are a set of numbers \([n_1,m_1;\ n_2,m_2;\ldots;\ n_k,m_k;\ \mu;\ N]\), where \(n_i\), \(m_i\), \(N\) are positive integers, and \(\mu\) is an arbitrary complex number. We shall not write out the canonical form of the matrices of the mappings \(d_+\), \(d_-\), \(\delta\). We note only that, in the case when these matrices correspond to an open chain, they consist of zeros and ones. If, however, the matrices correspond to a closed chain, then \([d_+]\) and \([d_-]\) have approximately the same form as in the preceding case, while in the matrix \([\delta]\) the number \(\mu\) occurs in certain positions, and in the remaining positions there are zeros and ones.

Let us describe what the matrices \([d_+]\), \([d_-]\), \([\delta]\) look like in the simplest closed object, to which the set of numbers \([1,1;\ \mu;\ 1]\) corresponds. In this case the space \(P_1\) has dimension 1, and the space \(P_2\) has dimension 2. In them one can choose a basis \(l_2\) and \(f_1,f_2\) such that

\[ d_- f_1 = l_1;\qquad d_- f_2 = 0;\qquad \delta f_1 = \mu f_2;\qquad \delta f_2 = 0;\qquad d_+ l_1 = f_2. \]

Thus the matrices \([d_+]\), \([d_-]\); \([\delta]\); \([d_-][d_+]\); \([d_+][d_-]\) have the form

\[ [d_+] = \begin{pmatrix} 0\\ 1 \end{pmatrix}; \qquad [d_-] = (1\ 0); \qquad [\delta] = \begin{pmatrix} 0 & 0\\ \mu & 0 \end{pmatrix}; \qquad [d_-][d_+] = (0); \qquad [d_+][d_-] = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix}. \]

Received
6 VI 1967

CITED LITERATURE

\(^1\) I. M. Gel'fand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation Group and the Lorentz Group, Moscow, 1958. \(^2\) I. M. Gel'fand, V. A. Ponomarev, DAN, 176, No. 2 (1967).