The Automorphisms of a Lie algebra

Transcription

1 Applied Mathematical Sciences Vol no HIKARI Ltd The Automorphisms of a Lie algebra WonSok Yoo Department of Applied Mathematics Kumoh National Institute of Technology Kumi 73-7 Korea Copyright c 24 WonSok Yoo. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract We determine the automorphism groups for each simply connected three-dimensional Lie group. Mathematics Subject Classification: 22E5 53C99 Keywords: Group of automorphisms Three-dimensional Lie groups Introduction A nonzero Borel measure on G invariant under left multiplication is called a left Haar measure on G. If G is a Lie group then a left Haar measure always exists and any two left Haar measures on G are propositional. The Lie group G is unimodular if every left Haar measure is a right Haar measure and vice versa. It is known that G is unimodular if and only if det Ad(t) = for all t G if and only if the trace of ad(x) is zero for all X in its Lie algebra g if and only if g is unimodular. The abelian compact semisimple reductive nilpotent Lie groups are well-known examples of unimodular Lie groups. There are six simply connected three-dimensional unimodular Lie groups: the abelian Lie group R 3 the nilpotent Lie group Nil the special unitary group SU(2) the universal covering group PSL(2 R) of the special linear group the solvable Lie group Sol and the universal covering group Ẽ(2) of the connected component of the Euclidean group. There are uncountably many nonisomorphic it non-unimodular three-dimensional Lie groups. These are all solvable and of the form R 2 ϕ R where R acts on R 2 via a linear map ϕ.

2 22 WonSok Yoo The investigations described here were motivated by the paper [] in which it is classified up to automorphism of the left invariant metrics on the real Heisenberg group. In this paper we determine the group of full automorphisms for each simply connected three-dimensional Lie group. This leads to the classifying up to automorphism of the left invariant metrics on all simply connected threedimensional Lie groups [2]. 2 Main Results 2. The abelian Lie algebra R 3 In the abelian case the Lie algebra is isomorphic to R 3. We choose the canonical basis {X Y Z} in the Lie algebra R 3 where X = ( ) Y = ( ) Z = ( ). Since the Lie bracket is trivial immediately we have: Proposition 2.. The Lie group Aut(R 3 ) is isomorphic to GL(3 R). 2.2 The Heisenberg Lie algebra n In the nilpotent case the Lie algebra is isomorphic to the Heisenberg Lie algebra n of all 3 3 strictly upper triangular real matrices. Choose the canonical basis {X Y Z} in n where X = Y = Z =. Then [X Y ] = Z and [X Z] = [Y Z] =. Proposition 2.2. The Lie group Aut(n) is isomorphic to a c b d ad bc. ad bc Proof. An automorphism on n must map the center Z(n) = Z = R onto itself. Hence it maps the basis {X Y Z} of n as follows: X ax + by + kz Y cx + dy + lz Z rz. Since [X Y ] = Z we have ad bc = r.

3 The automorphisms of a Lie algebra The unimodular solvable Lie algebra R 2 σ R In the unimodular solvable case the Lie algebra [ of (2..3.a) ] is isomorphic to t the semidirect product R 2 σ R where σ(t) =. We can choose the t basis {X X 2 X 3 } of R 2 σ R where X = ([ ] ) X 2 = ([ ] ) X 3 = Then [X X 2 ] = [X 3 X ] = X [X 3 X 2 ] = X 2. ([ ] ). Proposition 2.3. The Lie group Aut(R 2 σ R) is isomorphic to where S = α γ β δ S S αβ γ δ R. Proof. Let ϕ Aut(R 2 σ R). With respect to the basis {X X 2 X 3 } ϕx j = a j X + a 2j X 2 + a 3j X 3 for some a ij. Since ϕ[x i X j ] = [ϕx i ϕx j ] [X X 2 ] = [X 2 X 3 ] = X 2 and [X 3 X ] = X it follows that [ϕ] is of the form [ϕ] = α γ β δ or α γ β δ where α β γ δ are real numbers with αβ. 2.4 The unimodular solvable Lie algebra R 2 so(2) In the unimodular solvable case the Lie algebra of (2..3.b) is isomorphic to the Lie algebra R 2 so(2). We choose a basis {X X 2 X 3 } of s 2 where X = ([ ] [ ]) X 2 = ([ ] [ ]) X 3 = Then [X X 2 ] = [X 3 X ] = X 2 [X 3 X 2 ] = X. ([ ] [ ]).

4 24 WonSok Yoo Proposition 2.4. The Lie group Aut(R 2 so(2)) is isomorphic to S 2 S 2 γ where S 2 = C δ γ δ R. Proof. Let ϕ Aut(R 2 so(2)). With respect to the basis {X X 2 X 3 } ϕx j = a j X + a 2j X 2 + a 3j X 3 for some a ij. As a matrix a a 2 a 3 [ϕ] = a 2 a 22 a 23. a 3 a 32 a 33 Since ϕ[x i X j ] = [ϕx i ϕx j ] [X X 2 ] = [X 3 X ] = X 2 and [X 3 X 2 ] = X it follows that [ϕ] is of the form [ϕ] = α β γ β α δ or α β γ β α δ where α β γ δ are real numbers with (α β) ( ). 2.5 The simple Lie algebra sl(2 R) In the simple case the Lie algebra of (2..4.a) is isomorphic to the Lie algebra sl(2 R) of all 2 2 matrices of trace. We choose a basis {X X 2 X 3 } where [ X = ] X 2 = [ ] [ X 3 = Note that [X X 2 ] = 2X 3 [X 3 X ] = 2X 2 [X 3 X 2 ] = 2X. Proposition 2.5. The Lie group Aut(sl(2 R)) is isomorphic to SO( 2). Proof. Let ϕ Aut(sl(2 R)). With respect to the basis {X X 2 X 3 } ]. ϕx j = a j X + a 2j X 2 + a 3j X 3

5 The automorphisms of a Lie algebra 25 for some a ij. Observe that ϕ[x i X j ] = [ϕx i ϕx j ] if and only if the classical adjoint adj [ϕ] of [ϕ] is adj [ϕ] = a a 2 a 3 a 2 a 22 a 32 a 3 a 23 a 33 = I 2 [ϕ] t I 2 where I 2 =. Since (det[ϕ])i 3 = [ϕ](adj [ϕ]) = [ϕ] I 2 [ϕ] t I 2 we have det[ϕ] = and [ϕ] t I 2 [ϕ] = I 2. Hence ϕ Aut(sl(2 R)) if and only if [ϕ] SO( 2). 2.6 The simple Lie algebra so(3) The simple Lie algebra of (2..4.b) is isomorphic to the Lie algebra so(3) of all 3 3 skew symmetric matrices. We choose the following basis {X X 2 X 3 } where X = X 2 = X 3 = Note that [X X 2 ] = X 3 [X 3 X ] = X 2 [X 3 X 2 ] = X. Proposition 2.6. The Lie group Aut(so(3)) is isomorphic to SO(3). Proof. Let ϕ Aut(so(3)). With respect to the basis {X X 2 X 3 }. ϕx j = a j X + a 2j X 2 + a 3j X 3 for some a ij. As a matrix a a 2 a 3 [ϕ] = a 2 a 22 a 23. a 3 a 32 a 33 Observe that ϕ[x i X j ] = [ϕx i ϕx j ] if and only if a ij = ( ) i+j det[ϕ(i j)] the (i j) cofactor of [ϕ] for all i j = 2 3 if and only if [ϕ] t = adj [ϕ] the classical adjoint of [ϕ]. Since [ϕ][ϕ] t = [ϕ](adj [ϕ]) = (det[ϕ])i 3 we have det[ϕ] = and [ϕ][ϕ] t = I. Hence ϕ Aut(so(3)) if and only if [ϕ] SO(3).

6 26 WonSok Yoo 2.7 The non-unimodular solvable Lie algebras All three-dimensional non-unimodular Lie algebras are solvable. By (2..5) such a Lie algebras is isomorphic to either g I or g c for some c R where g I is the Lie algebra of (2..5.a) and g c is the Lie algebra of (2..5.b). In fact [ ] g I t = R 2 σi R where σ I (t) = ; t [ ] g c ct = R 2 σc R where σ c (t) = t 2t with a basis X = ([ ] ) Y = ([ ] ) Z = ([ ] ). Proposition 2.7. () The Lie group Aut(g I ) is isomorphic to {[ ] } GL(2 R) R 2. (2) For each c R the Lie group Aut(g c ) is isomorphic to β α cα α β + α α β R β 2 + (c )α 2. Proof. () It is easy to[ see that ϕ ] Aut(g) if and only if with respect to the GL(2 R) basis {X Y Z} [ϕ] = where R 2. (2) Let ϕ Aut(g c ). Suppose that with respect to the basis {X Y Z} [ϕ] = a a 2 a 3 a 2 a 22 a 23. Note that ϕ Aut(g c ) if and only if a 3 a 32 a 33 a 32 = (a 2 + 2a 22 )a 3 = ca 3 = (a 2 + 2a 22 )a 33 = 2a 22 ca 2 c(a a 22 a 33 ) = 2a 2 (a + 2a 2 )a 33 (a 3 + 2a 23 )a 3 = a 22 c(a 23 a 3 a 2 a 33 ) = a 2 If a 33 = or c = then the above equations yield that a 3 = a 32 = a 33 = a 2 = ca 2 and a 22 = a + 2a 2. Suppose that a 33 and c. Then the above equations yield that a 2 = ca 2 a 33 a 33 a a 22 = 2a 2 a 33 a 22 = ca 2(a 33 +) 2 a a 33 a 22 = 2a 2 a 33.

7 The automorphisms of a Lie algebra 27 Thus a 33 a = 2a 2a 33 a 33 = a + 22 = ca 2(a 33 +) and c = 4a 33 2 (a 33. Now +) 2 consider the automorphism ϕ 2 and let [ϕ 2 ] = [ ] b ij with respect to the basis {X Y Z}. Then since b 33 = a 2 33 and b 33 we obtain that c = 4b 33 (b 33. Thus +) 2 4a 33 (a 33 = c = 4a2 +) 2 33 and so a (a 2 33 = or a 33 = a contradiction. 33 +)2 ACKNOWLEDGEMENTS. This research was supported by Kumoh National Institute of Technology. References [] M. Goze and P. Piu Classification des métriques invariante à gauche sur le Groupe de Heisenberg Rend. Circ. Mat. Palermo (2) 39 (99) [2] K. Y. Ha and J. B. Lee Left invariant metrics and curvatures on connected and simply connected three-dimensional Lie groups Journal of Geometry and Physics 62(2)(2) [3] N. Jacobson Lie algebras Interscience New York 962. [4] J. Milnor Curvatures of left-invariant metrics on Lie groups Adv. Math. 2 (976) [5] S. Wolfram Mathematica Wolfram Research 993. Received: November 5 24; Published: December 22 24