Jacobi Algebra
219
Note that the subalgebra K n is abelian and is a Cartan subalgebra of S n .
Furthermore, not only S ±
n , but also G ±
n are its eigenspaces:
[K n , G
±
n ] = G
±
n .
(6)
Thus, K n plays for G n the role that Cartan subalgebras are playing for semi-simple
Lie algebras.
3 Case G 2
Note that the algebra G 1 is isomorphic to the (1+1)-dimensional Schrödinger
algebra (without central extension). The representations of the latter are well known,
cf. [3, 7–9]. Thus, we study the first new case of the G n series, namely, G 2 .
For simplicity, we introduce the following notations for the basis of S 2 :
S
+
: b
+
i ≡ K
+
ii , i = 1, 2; c
+
≡ K
+
12 , d
+
≡ K
0
12
(7a)
S
−
: b
−
i ≡ K
−
ii , i = 1, 2; c
−
≡ K
−
12 , d
−
≡ K
0
21
(7b)
K : h i ≡ K
0
ii , i = 1, 2.
(7c)
Next, using (2) and (3) we give the eigenvalues of the basis of G + w.r.t. K :
h 1 : (b
+
1 , b
+
2 , c
+ , d
+ , a
+
1 , a
+
2 ) : (1, 0,
1
2 ,
1
2 ,
1
2 , 0) ,
(8)
h 2 : (b
+
1 , b
+
2 , c
+ , d
+ , a
+
1 , a
+
2 ) : (0, 1,
1
2 , −
1
2 , 0,
1
2 ) ,
(e.g., [h 1 , b
+
1 ] = b
+
1 , [h 2 , d + ] = −
1
2 d + , etc). Naturally, the eigenvalues of
the basis of G − w.r.t. K are obtained from (8) by multiplying every eigenvalue
by (−1).
Next we introduce the following grading of the basis of G
+
2 :
(b
+
1 , b
+
2 , c
+ , d
+ , a
+
1 , a
+
2 ) : (2δ 1 , 2δ 2 , δ 1 + δ 2 , δ 1 − δ 2 , δ 1 , δ 2 ).
(9)
The grading of the S
+
2 part of the basis follows from the root system of S
+
2 ,
while the grading of the H
+
2 part of the basis is determined by consistency with
commutation relations (2). It is consistent also with formulae (8).
Naturally, the grading of the basis of G − w.r.t. are obtained from (9) by
multiplying every grading by (−1).