General Theorems
Lie’s and Engel’s Theorems
VẴ) = {U E VJp(x)v = X(rc)v for all z E g}
In representation theory, if VA(g) is non-zero, the form X is recognized as a weight of the representation p The subspace VA(g) is referred to as the weight subspace, and the nonzero vectors within this subspace are identified as the weight vectors associated with the weight A.
Weight subspaces corresponding to different weights are linearly indepen- dent Thus a finite-dimensional linear representation may have only finitely many weights
The proof of Lie’s theorem is based on the following property of weight subspaces
Lemma 1.1 Let H be a normal subgroup of the group G, x the character of H, and R: G + GL(V) a linear representation Then for any g E G we have
RW’JH) = V,, (HI, where xg(h) = x(g-‘hg) (h E H)
The proof of Theorem 1.1 begins by demonstrating, through induction on the dimension of G, that the representation R has at least one weight in the vector space V For a one-dimensional G, this is straightforward In higher dimensions, the properties of solvable Lie groups indicate the existence of a virtual normal Lie subgroup H within G, which has a codimension of 1.
In the equation G = CH, C represents a connected virtual one-dimensional Lie subgroup According to the inductive hypothesis, the representation V_(H) is non-zero for a certain character x of the group H Lemma 1.1 indicates that the operators R(g), where g belongs to G, permute the weight subspaces associated with the group H Given that G is connected, it follows that V_(H) remains invariant under the action of R(G).
The one-dimensional subgroup C possesses a one-dimensional invariant subspace in V, denoted as H, which remains unchanged under the action of the entire group G Consequently, this establishes the existence of a one-dimensional subspace in V that is invariant under G Furthermore, by employing induction on the dimension of V, we can demonstrate the existence of a complete flag in V that is also invariant under G.
Corollary 1 Any irreducible complex linear representation of a connected solvable Lie group or a solvable Lie algebra is one-dimensional
Corollary 2 Let G c GL(V) b e a connected irreducible complex linear Lie group Then either G is semisimple, or RadG = {cElc E @“}
Proof Suppose that G is not semisimple Consider the vector subspace
W = V,(radG) # 0 L emma 1.1 implies that it is invariant under G Hence
W = V, i.e RadG contains scalar operators only I7
Corollary 3 A Lie algebra g over K = @ or Iw is solvable if and only if the Lie algebra [g,g] = n,(K) is nilpotent
If \( g = tn(K) \), then the commutator \( [g, g] \) represents the nilpotent Lie algebra consisting of all upper diagonal matrices with zeros on the diagonal Generally, one can assume that \( K = c \) by employing the complexification procedure if needed According to Lie's theorem, if \( g \) is solvable, it follows that the Lie algebra \( ad [g, g] \) and \( [ad g, ad g] \) is nilpotent, which in turn implies that \( g \) is also nilpotent.
1.2 Generalizations of Lie’s Theorem First we consider the possibilities of generalizing Lie’s theorem to Lie algebras over an arbitrary field K If a representation p: g -f gr(V) of a Lie algebra g over K has an invariant full flag, then the characteristic numbers of all operators p(x), z E g, must belong to the field K, which is far from being always true if K is not algebraically closed If charK = 0, then the above mentioned property of the operators P(X), z E g, turns out to be also sufficient for the existence of an invariant flag
Theorem 1.2 Let g be a solvable Lie algebra over a field K of characteristic
0 and p: g + gK(V) a li near representation of it over K If all characteristic numbers of all operators p(x), x E g, belong to K, then there is a full flag in
The proof is similar to that of Theorem 1.1, and makes use of the following analogue of Lemma 1.1
Lemma 1.2 Let p: g + gC(V) be a linear representation of a Lie algebra g over a field K of characteristic 0, h an ideal in g, and Vx( 4) a weight subspace of the representation p]b Then the following two equivalent statements hold:
(1) VẴ) is invariant under p(g); (2) X(z) = 0 for any z E [g, b]
Corollary 3 to Theorem 1.1 is extended to the case of an arbitrary field of characteristic 0 If a field of characteristic 0 is algebraically closed, then the analogues of Corollaries 1 and 2 hold
The condition imposed by Theorem 1.2 on the characteristic is essential, as the following example shows
Example If charK = 2, then the Lie algebra gI,(K) is solvable, but its identity representation in K2 has no weight vectors
Lie's theorem can be extended to connected solvable linear algebraic groups over an algebraically closed field of any characteristic, as demonstrated by Borel's fixed point theorem (Springer [1989], Chap 1, Sect 3.5) Additionally, we present a straightforward theorem regarding the representations of abstract solvable groups.
Theorem 1.3 (see Merzlyakov [1987]) Let G be a solvable group, and R: G + GL(V) a complex linear representation of it Then there is a full jlag in V invariant under a subgroup of finite index G1 c G
The algebraic closure H = aR(G) of the subgroup R(G) in GL(V) is a solvable linear algebraic group with a finite number of connected components Based on Theorem 1.1, it is established that there exists a full flag within this structure.
V invariant under Ho But then it is also invariant under the subgroup
GI = R-l(H’), w rc is of finite index in G h’ h 0
Theorem 1.3 allows for the selection of the subgroup Gi such that its index is limited by a value that is solely dependent on the dimension of V, as demonstrated by Merzlyakov in 1987.
1.3 Engel’s Theorem and Corollaries to It The cornerstone in the theory of nilpotent Lie algebras and Lie groups is the following theorem first proved by F Engel
Theorem 1.4 (see Bourbaki (19751, Jacobson [1955]) Let ,o:g + @(V) be a linear representation of a Lie algebra g over an arbitrary field K Suppose that for each x E 0 the linear operator p(x) is nilpotent Then there is a basis in V with respect to which the operators p(x), x E 0, are represented by upper triangular matrices with Zeros on the diagonal In particular, the Lie algebra p(g) is nilpotent
Lie's theorem can be proven through induction on the dimension of the vector space V, ultimately demonstrating the existence of a weight vector with weight 0 This proof begins with the case when the dimension is 1, which is straightforward Assuming the theorem holds for all Lie algebras of dimension less than m, we consider a Lie algebra of dimension m According to the theorem and the inductive hypothesis, there exists an ideal h with codimension 1 in g, which can be chosen as any maximal subalgebra of a Consequently, g can be expressed as the sum of h and another subspace Examining the weight subspace Vo(λ), we find it to be non-empty Since h is an ideal in g, it follows from Lemma 1.2 that Vo(b) remains invariant under the action of g The operator p(y) is nilpotent, leading to the existence of a non-zero vector in Vo(h) such that p(y)w = 0.
‘~0 is the desired weight vector with respect to 0 0
Corollary 1 If under the conditions of Theorem 1.4 the representation p is irreducible, then it is trivial and one-dimensional
An application of Engel’s theorem to the adjoint representation easily yields the following corollary
Corollary 2 A Lie algebra 0 is nilpotent if and only if either of the following two conditions is satisfied:
(1) For any x E 0 the operator ads is nilpotent
(2) There is a basis {ei} in 0 such that [ei, e3] is a linear combination of the elements ek, ek+l, , , em, where k = maX(i, j) + 1
A Lie algebra \( \mathfrak{g} \) is considered engelian if all operators \( \text{ad}_z \) for \( x \in \mathfrak{g} \) are nilpotent According to Corollary 2, a finite-dimensional Lie algebra is engelian if and only if it is nilpotent However, this equivalence does not generally apply to infinite-dimensional Lie algebras Nonetheless, if \( \mathfrak{g} \) is finitely generated and \( (\text{ad}_x)^n = 0 \) for some \( n \in \mathbb{N} \) and all \( z \in \mathfrak{g} \), then \( \mathfrak{g} \) is nilpotent.
A stronger version of Engel’s theorem applies to linear representations \( p \) of a Lie algebra \( \mathfrak{g} \) where \( p(\mathfrak{g}) \) is generated by a set of nilpotent operators that are closed under the commutator.
The next theorem lists other important properties of nilpotent Lie algebras proved with the use of Engel’s theorem
Theorem 1.5 (see Bourbaki [1975], Jacobson [1955], Serre [1987]) Let g be a nilpotent Lie algebra Then
(ii) If a is a subspace in g such that g = a + [g, g], then a generates g as a Lie algebra
(iii) If b is an ideal in g, then h n a(g) # 0
(iv) If b is a subalgebra of g, then its normalizer n(b) strictly contains b
Finally, we note the following application of Engel’s theorem to the theory of nilpotent Lie groups
Theorem 1.6 A connected Lie group G is nilpotent if and only if all op- erators Adg (g E G) are unipotent Any compact subgroup of a connected nilpotent Lie group G is contained in Z(G)
The first statement is derived from Corollary 2 of Theorem 1.4, which highlights the relationship between nilpotent Lie groups and Lie algebras, as discussed in Vinberg and Onishchik (1988, Chap 2, Theorem 5.13) To establish the second statement, we examine the restriction R of the Ad representation to a compact subgroup.
L c G Since R is completely reducible (see below Chap 4, Corollary to Proposition 2.1), Corollary 1 to Theorem 1.4 implies that R is trivial Hence
1.4 An Analogue of Engel’s Theorem in Group Theory The following the- orem can be considered as a group-theoretical analogue of Engel’s theorem
It is not a formal consequence of Engel’s theorem because it applies to groups that are not necessarily Lie groups
The Cartan Criterion
2.1 Invariant Bilinear Forms Let G be a Lie group over a field K A bilinear form b on the tangent algebra g of the group G is said to be invariant if b((Ad gh (Ad g)y31) = b(x> 9) (1) for all g E G, x, y E g It follows from formula (18) in Vinberg and Onishchik
In 1988, it was established that a bilinear form \( b \) on a Lie algebra \( g \) is considered invariant if it fulfills the relation \( b([z, y], 2) + b(y, b, 4) = 0 \) for all elements \( x, y, z \) in \( g \) Furthermore, if the relation holds true, it implies the earlier condition provided that the group \( G \) is connected.
In a three-dimensional Euclidean space E, with a defined scalar product, we can establish an orientation and examine the vector product within E Consequently, E is structured as a Lie algebra over the real numbers, where the scalar product remains invariant.
Example 2 In the Lie algebra gI(V) of linear transformations of a vector space V over K there is an invariant bilinear form b(X,Y) = tr (XY) (3)
Example 3 Let g be a Lie algebra over K, and p: g -f gI(V) a linear representation of it Then the symmetric bilinear form b,(x, Y/) = tr (P(x)P(Y)) is invariant on g In particular, there is an invariant bilinear form
&,(x,Y) = bad (5,~) = tr((adx)(ady)), called the Killing form of the algebra g
In what follows we always assume that an invariant bilinear form b on a Lie algebra g is symmetric The following assertions are proved without difficulty
Proposition 2.1 Let b be an invariant bilinear form on a Lie algebra g and a an ideal in g Then aI = {x E g(b(x, y) = 0 ‘dy E a} is also an ideal in g
If fa = b, al, then a’ 1 ah), and if b is nondegenerate, then a’ = d(g)
The Killing form \( k \) of any Lie algebra \( g \) satisfies the relation \( k(a(x), a(y)) = k(x, y) \) for all \( x, y \in g \) and any automorphism \( a \in \text{Aut} \, g \) Furthermore, if \( a \) is an ideal in \( g \), the restriction of the Killing form \( k \) to \( a \) coincides with \( k \).
2.2 Criteria of Solvability and Semisimplicity In this section we denote by b the invariant bilinear form in gK(V) defined by formula (3)
Theorem 2.1 A subalgebra g c gK(V) is solvable if and only if b([X,Y],Z)=OforallX,Y,ZEg
In the proof of Theorem 2.1, we start by assuming K = @, noting that the real case can be transformed into the complex case through complexification, represented by the Lie algebra g(C) = g + ig and the graded space gr(V(@)) For any element X in gr(V), we can identify its semisimple and nilpotent components through the additive Jordan decomposition, denoted as X = X_s + X_n Here, X_s represents the semisimple operator that shares the same eigenvectors as X_n but has complex conjugate eigenvalues Additionally, we establish that b([X, Y], Z) = 0 for all X, Y, Z in g, and, based on Engel’s theorem, it suffices to demonstrate the necessary conditions for the theorem's validity.
X, = 0 for any X E [g, g] Write X = 5 [Xi, k;], where Xi, Y, E g Then i=l b&X,X,) = ~b([-W’J,~,) = &(L[X’,,XiI) i=l 2=1
The relation (adX)(g) c [g, g] and the equality adz, = (adX), imply that
(adds) c [g,gl H ence b(X,X,) = tr (XXI;,) = 0, whence X, = 0 The converse statement easily follows from Lie’s theorem 0
Corollary A Lie algebra g is solvable if and only if lc,( [x, y/l, z) = 0 for all x, y, z E g or if k,(z, y) = 0 for all z, y E [g, g]
Theorem 2.2 A Lie algebra g is semisimple if and only if the Killing form k, is nondegenerate
In this proof, let \( u \) be defined as the set of \( x \) in \( g \) such that \( k(z, y) = 0 \) for all \( y \) in \( g \) According to Proposition 2.1, \( u \) forms an ideal within \( g \), and Theorem 2.1 indicates that the Lie algebra \( \text{ad}u \) is solvable Given that \( \text{ad}u \) is isomorphic to \( u \), it follows that \( u \) must equal zero if \( g \) is semisimple Conversely, if \( g \) is not semisimple and \( a \) represents its nonzero abelian ideal, then \( a \) is contained within \( u \).
Remark A similar proof yields the following assertion: if p is a faithful linear representation of a semisimple Lie algebra g, then the form b, (see Example 3) is nondegenerate on g
Corollary If g is semisimple, then g = [g, g]
Proposition 2.3 If g is semisimple, and a is an ideal in g, then g = g@a’ Any ideal a c g and the quotient algebra g/a are semisimple
Proof As in the proof of Theorem 2.2, one can verify that a n aI is a solvable ideal in g 0
The following theorem is now derived without difficulty
Theorem 2.3 A Lie algebra g is semisimple if and only if g can be decom- posed into the direct sum
Complete reducibility in the context of cohomology involves the expression of any ideal within the algebra g as a sum of simple noncommutative ideals, denoted as gi This decomposition is not only fundamental but also unique, highlighting the distinct nature of simple ideals in the structure of the algebra.
Theorem 2.3 can be reformulated in the context of Lie groups, stating that a Lie group G is considered decomposable into the locally direct product of its normal subgroups G1, G2, , Gn if it can be expressed as G = G1G2 Gn, with all intersections of these subgroups being trivial.
Theorem 2.4 A connected Lie group G is semisimple if and only if it is decomposable into the locally direct product of connected simple normal Lie subgroups: G = G1 G, Given such a decomposition, any connected normal Lie subgroup of G is the product of some of the subgroups G,
3 3 Complete Reducibility of Representations and Triviality of the Cohomology of Semisimple Lie Algebras
3.1 Cohomological Criterion of Complete Reducibility Let g be a Lie algebra over K As usual, a g-module is a vector space V over K together with a given linear representation p: g -+ gI(V) As before, the space V is assumed to be finite-dimensional
Proposition 3.1 A linear representation of a Lie algebra g is completely reducible if and only if H1(g, V) = 0 for any g-module V
The proof of the proposition relies on a well-established bijection between the space H1(g, Hom(V, W)), where V and W are g-modules, and the classes of extensions of the g-module V by W In this context, the zero cohomology class is associated with the class of split (nonessential) extensions, as detailed in Feigin and Fuks [1988], Chapter 2, Section 2.1 D.
3.2 The Casimir Operator Let b be a nondegenerate invariant bilinear form on a Lie algebra g Using the isomorphism g* -f g generated by the form b, one can identify b E g* @ g* with an element of the space g @ g The corresponding element c of the universal enveloping algebra U(g) (see Vinberg and Onishchik [1988], Chap 3, Sect 1) is said to be the Casimir element corresponding to the form b Since b is invariant, the element c belongs to the centre of the algebra U(g) The element c can be found as follows Let el, , e, be an arbitrary basis of the algebra g, and fl, , fn the basis dual to it with respect to b (i.e b(ei, fj) = &j) Then c = 2 eifi For any linear i=l representation p: g -f gK(V) one defines the operator
C = p(c) = 2 AeiMfi) i=l in the space V, which is called the Casimir operator of the representation p and commutes with all p(z), II: E g
Let \( g \) be a semisimple Lie algebra, and \( p \) its faithful representation As noted after Theorem 2.2, the invariant form \( b \) is nondegenerate on \( g \) We denote the corresponding Casimir operator in the vector space \( V \).
Proposition 3.3 We have tr C, = dim g If p is irreducible, then the oper- ator C, is nondegenerate
3.3 Theorems on the Triviality of Cohomology
Theorem 3.1 Let p: g f gK(V) be a nonzero irreducible representation of a semisimple Lie algebra g Then HP(g, V) = 0 for all p > 0
In this proof, we define \( g_2 \) as the kernel of \( p \) and \( g_1 \) as the ideal in \( g \) that complements \( g_2 \) We denote \( p_1 = p|_{g_1} \) and use \( e_1, \ldots, e_n \) as the basis in \( g_1 \), while \( f_1, \ldots, f_n \) represent the dual basis with respect to a certain form The Casimir operator of the representation \( p_1 \) is denoted as \( C \in GL(V) \) We then define a mapping \( k : \mathcal{K}(g, V) \to \mathcal{CP}^{-1}(g, V) \) for \( p \geq 1 \) using the formula \( h(c)(z_1, \ldots, z_{p-1}) = e_p(e_i)c(f_i, z_1, \ldots, z_{p-1}) \) for \( i = 1 \).
Then d o h + h o d = C on CP(g, V), p 2 1, where d is the coboundary operator in the complex C*(g, V), and the Casimir operator C acts on the cochain values In other words, there is a chain homotopy between C and 0
It follows from this and the equality C o d = d o C that Hp(g, V) = 0 for p > 0 The case p = 0 is evident q
Theorem 3.2 If g is a semisimple Lie algebra, then H1(g, V) = H2(g, V) 0 for any g-module V
Using the exact cohomology sequence as outlined by Feigin and Fuks (1988), one can apply induction on the dimension of V to simplify the proof to the case where V is irreducible If V is nontrivial, the conclusion is supported by Theorem 2.1 Additionally, the result H1(g, K) = 0 is derived from the corollary of Theorem 1.2, while the result H2(g, K) = 0 is established through Theorem 1 from Feigin.
Corollary If g is semisimple, then Der g = ad g and Int g = (Aut 8)‘
Levi Decomposition
In the context of the representation plradg, the weight subspace VA in V is invariant under the operator p, leading to the decomposition of V into VA and V’, where V’ is also invariant under p This process can be iteratively applied, resulting in a further refined decomposition of the vector space.
V = @ VA&, where Vxz are the weight subspaces of plradg invariant under p i=l
Note that Theorem 3.1 is easily generalized to the case of reductive g This implies the following theorem
Theorem 3.5 Let g be a reductive Lie algebra and p: g f gK(V) a completely reducible representation of it Then for any p > 0 we have
HP(g, V) = HP(g, VO) E HP(g, K) @ V”, where V” = H’(g,V) = {v E Vlp(x)v = 0 Vx E g} fj 4 Levi Decomposition
4.1 Levi’s Theorem Let g be a reductive Lie algebra over a field K A subalgebra I c g is said to be a Levi subalgebra if g can be decomposed into the semidirect sum g=radg ZB I (5)
The decomposition (5) is called the Levi decomposition of g
A Levi subalgebra I within a Lie algebra g is inherently semisimple due to the natural homomorphism that maps g onto the semisimple Lie algebra g/rad g, establishing an isomorphism with I Consequently, I is recognized as a maximal semisimple subalgebra Additionally, as discussed in Section 4.2, the converse of this statement holds true as well.
Theorem 4.1 (Levi) In any Lie algebra g over K there exists a Levi sub- algebra
The proof of the theorem is reduced to the case where the radical, radg, is commutative, utilizing induction on dimradg In this context, the adjoint representation of the Lie algebra g in radg establishes a representation of the semisimple Lie algebra g/radg According to Theorem 3.2, the second cohomology group H2(g/radg, radg) equals zero This, combined with the established relationship between extensions and cohomology, indicates that the extension 0 → rad g → g → g/rad g → 0 is split Consequently, there exists a subalgebra of g that complements radg.
Levi's theorem can be proven without relying on cohomology by utilizing the complete reducibility of representations of a semisimple Lie algebra For detailed discussions on this proof, refer to Serre (1987) and Onishchik and Vinberg (1990).
Corollary Let G be a connected Lie group There exists a virtual connected Lie subgroup L c G such that
A connected virtual Lie subgroup L c G having the property (6) is said to be a Levi subgroup of the group G, and the decomposition (6) is said to be a Levi decomposition of this group
In a simply-connected group G, both the radical RadG and the Levi subgroup L are also simply-connected, leading to a semidirect Levi decomposition This is demonstrated in the proof of Theorem 4.2 Notably, in this scenario, L qualifies as a Lie subgroup of G; however, it is important to recognize that this does not universally apply in all cases.
Let A be a simply-connected covering group for SLz(Iw), where Z(A) is a subgroup of Z A generator of Z(A) is denoted as to, and a rotation through an angle incommensurable with X is represented as to E ‘lT = SOa The discrete normal subgroup r = ((t 0, 20)) is formed in T x A, leading to the quotient group G = (Ti’ x A)/r The image L of subgroup A under the natural homomorphism from T x A to G is identified as a Levi subgroup of G Since the subgroup (to) is dense in ?T, it follows that L is also dense in G, indicating that it is not a Lie subgroup.
4.2 Existence of a Lie Group with a Given Tangent Algebra Let g be a Lie algebra over a field K Our aim is to prove the existence of a Lie group G over
K whose tangent algebra is isomorphic to g In the case where g is solvable, this fact is established by induction on dimg (see Chap 2, Theorem 5.10)
On the other hand, if g is a Lie algebra with trivial centre, then p z ad g, and one can take for G the group Int g of inner automorphisms of the algebra g
Theorem 4.2 (Cartan) For any Lie algebra g over the field K there exists a simply-connected Lie group G over K whose tangent algebra is isomorphic to g
The Levi decomposition reveals the existence of simply-connected Lie groups R and L, corresponding to the tangent algebras rad g and I, respectively The adjoint representation of the subalgebra I in rad g is denoted as p: I → Der(rad g) A homomorphism B: L → Aut(rad g) exists such that dB = β According to Proposition 4.2 from Vinberg and Onishchik (1988, Chap 2), the semidirect product G = R ×B L, defined by this homomorphism, is the desired simply-connected Lie group.
The proof presented aligns closely with the original proof by I Cartan from 1930 Direct proofs of Theorem 4.2 that do not rely on Levi’s theorem are available in the works of Gorbatsevich (1974b) and van Est (1988) Additionally, Theorem 4.2 can be easily derived from Ado’s theorem, as discussed in Section 5.3.
4.3 Malcev’s Theorem It turns out that a Levi subalgebra of an arbi- trary Lie algebra is unique up to conjugacy This is implied by the following assertion
Theorem 4.3 (Malcev) Let I be a Levi subalgebra of a Lie algebra g For any semisimple subalgebra 5 c g there exists cp E Int g such that (p(5) c I
Various proofs of this theorem can be found in Onishchik and Vinberg
[1990], Bourbaki [1975], and Serre [1987] Note that the automorphism ‘p can be chosen in the form cp = expadz, where z belongs to the nilpotent radical of the algebra g
Corollary 1 Any two Levi subalgebras of a Lie algebra can be taken into one another by an inner automorphism of this algebra
Corollary 2 Any maximal semisimple subalgebra of a Lie algebra is a Levi subalgebra
Corollary 3 Let G be a connected Lie group, and L a Levi subgroup of it Any connected semisimple virtual Lie subgroup of G is conjugate to a subgroup of L Any two Levi subgroups of G are conjugate A connected virtual Lie subgroup S of G is a Levi subgroup if and only if S is a maximal semisimple subgroup of G
4.4 Classification of Lie Algebras with a Given Radical Given a solvable Lie algebra t over the field K = @ or KC, consider the problem of classifying (up to an isomorphism) all Lie algebras g for which rad g = r
According to Theorem 4.1, a Lie algebra \( g \) with a radical \( r \) can be expressed as \( g = I \oplus r \), where \( I \) is a semisimple subalgebra A Lie algebra is termed faithful if the adjoint representation of the subalgebra \( I \) in \( g \) is faithful, meaning it contains no nonzero semisimple ideals Theorem 4.3 supports this by stating that any semisimple ideal \( s \) of \( g \) is included in any Levi subalgebra \( I \), leading to the conclusion that \( [s, t] = 0 \) Consequently, a Lie algebra with radical \( t \) can be decomposed into the direct sum of a semisimple Lie algebra and a faithful Lie algebra sharing the same radical, allowing us to conclude that \( g \) can be assumed to be a faithful Lie algebra.
Let 10 be a fixed Levi subalgebra of the Lie algebra Derr We call the faithful Lie algebra go = Ia B id t, where id is the identity representation of the Lie algebra IO in r, the universal Lie algebra with radical r
Theorem 4.4 (Malcev [1944]) Any faithful Lie algebra with radical r is isomorphic to a Lie algebra of the form, I B id t, where 11, I2 are semisimple subalgebras of IO The algebras g1 = [I B id r and g2 = [2 B id r, where (1, 12 are semisimple subalgebras of IO, are isomorphic if and only if I2 = AIlAWl, where A is an automorphism of r such that AKA-1 = I.
Linear Lie Groups
The classification problem focuses on analyzing derivations and automorphisms of the Lie algebra Notably, universal Lie algebras, as examined in the work of Onishchik and Hakimjanov (1975), serve as key examples in this study.
Example (a) If r is abelian, then Aut r = GL(r), and ~0 = sl(t) B id r
By virtue of Theorem 4.4, the classification of Lie algebras with the radical r is reduced to the classification of semisimple subalgebras of sI(r) up to conjugacy
Let \( r = tn(K) \) represent the algebra of upper triangular matrices of order \( n \) In this context, it follows that \( Derr = adr \), resulting in \( \sim 0 = t \) Any Lie algebra containing the radical \( r \ allows for a direct Levi decomposition This principle also applies when \( r \) is identified as the Borel subalgebra, which is the maximal solvable subalgebra of a semisimple Lie algebra, as noted by Tolpygo (1972).
The Heisenberg algebra of dimension 21c + 1, denoted as r = hk(K), can be embedded in sl~~~+~(K) as a subalgebra consisting of operators that annihilate the vector el of the standard basis, as demonstrated by Onishchik and Hakimjanov [1975] Consequently, any faithful Lie algebra with a radical hk is isomorphic to a subalgebra of spakfZ(K) that includes 61~ Additionally, Onishchik and Hakimjanov [1975] explored the scenario where t: represents the nilradical of a parabolic subalgebra within a semisimple Lie algebra.
5.1 Basic Notions A Lie group G (either real or complex) is said to be linear if it is a virtual Lie subgroup of the group GL(V) of linear transfor- mations of a finite-dimensional vector space (over the field K = R or c, respectively)
A Lie group over a field K is considered linearizable if it is isomorphic to a linear Lie group This means that it must have a faithful finite-dimensional linear representation over the field K.
Linear Lie groups are essentially linearizable Lie groups that possess a specific finite-dimensional linear representation For practical purposes, we often equate the concepts of linearity and linearizability, referring to a group as linear if it has a faithful finite-dimensional linear representation.
The study of linear Lie groups is intricately linked to the broader investigation of arbitrary linear groups, which may not be classified as Lie groups and can be defined over various fields For more information on these groups, refer to the works of Merzlyakov (1987) and Suprunenko (1979) Additionally, linear Lie groups are closely associated with algebraic groups over the fields ℝ and ℂ, with each algebraic group being closed in the Zariski topology within GL(V), thus qualifying as a linear algebraic group Among linear groups, algebraic linear groups represent the most extensively researched category.
The group GL(C) can be considered a real algebraic subgroup of GL(R), which means that any complex linear Lie group or algebraic linear group can also be interpreted as a real group.
5.2 Some Examples Example 1 The classical groups described in Vinberg and Onishchik [1988], Chap 1, Sect 1.2 (such as GL,(K), SL,(K), Sp,(K), O,(K), etc., where K = lR or @), are linear algebraic groups
Other linear algebraic groups are T,(K), N,(K), as well as the group D,(K) of all diagonal nonsingular matrices over the field K
Example 2 The group Aff K” of all nonsingular affine transformations of the space K” is linearizable Its embedding in GL,+l(K) is of the form
AffK”=GL,(K).Kn3(A,u)~ E (%,+I (IO, where A E GL,(K), u = (~1, , u,)~ E Kn Clearly, the image of the group Aff Kn under this embedding is an algebraic subgroup of GL,+l (K)
Example 3 Consider the Lie group G = K D(~ Kn, the semidirect product corresponding to some homomorphism ‘p: K + GL,(K) The Lie group G is simply-connected, solvable, and has the following faithful linear representa- tion:
G 3 (t, v) H where t E K, cp(t) E GL,(K), v = (VI, ,u,)~ E K” Therefore the group
A group G is considered linear if it qualifies as a linear algebraic group, which occurs when the one-parameter subgroup q(t) is an algebraic subgroup of GL(K) This condition is met if and only if specific criteria are fulfilled.
(i) the subgroup p(t) is unipotent (i.e the matrices p(t) are unipotent for all t E K);
(ii) the subgroup q(t) consists of semisimple elements and is conjugate to the subgroup of the form expX t, where X = diag (Xl, , X,) and all Xi are integers
Example 4 The subgroup G, = {diag (et,eat)lt e K} is linear for any (u E K, but it is an algebraic linear group if and only if Q E Q
For irrational values of the parameter cy E K the group G, is isomorphic to an algebraic linear subgroup of GLz(K), namely, the subgroup N2( K) c
GLz( K) The next example shows that there exist linear Lie groups noniso- morphic to any linear algebraic group
Example 5 Consider the Lie group G = R D( PR2, where the homomorphism cp is of the form
The Lie group G is linearizable but not isomorphic to any real linear algebraic group due to the matrix p(t) being neither unipotent nor semisimple for t ≠ 0 Consequently, the one-parameter subgroup p(t) does not qualify as an algebraic subgroup of GL2(R) If G were isomorphic to a linear algebraic group, it would possess a Chevalley decomposition G = H.U, where H represents a reductive subgroup (algebraic torus) and U denotes a unipotent radical Given that U is greater than the derived subgroup (G, G) and the dimension of (G, G) is 2, it follows that the dimension of U must also be 2.
The image of the torus H, when subjected to conjugations on U, should align (up to similarity) with the subgroup p(t) of GLz(lR) However, this implies that the one-parameter subgroup p(t) is algebraic, which we have established is not the case.
Consider now some examples of nonlinearizable Lie groups Many such examples owe their origin to the following statement
Proposition 5.1 Let G be a connected semisimple linear Lie group (either real or complex) Then its centre Z(G) is finite
Proof This is a consequence of the complete reducibility of linear semisim- ple Lie groups (see Sect 3.4), the Schur lemma, and the discreteness of the group Z(G) q
Example 6 Consider the Lie group A = s?;,(R), the simply-connected uni- versal covering of the group SL2(Jl%) The group A is nonlinearizable because
Nonlinearizable Lie groups S%(p, 2) (for p ≥ 1) and %(p,q) (with p > q ≥ 1) serve as simply-connected coverings for the groups WP, 2) and Sub, cd, respectively However, there exist nonlinearizable semisimple Lie groups that have finite centers For instance, none of the Lie groups covering SLz(lK) (where the multiplicity n > 1) is linearizable, while Z(d,) P zan is considered (refer to Chapter 4, Section 3.6).
Example 7 Consider the group N = Ns(lR), and its subgroup Z = Z(N)(Z), which is the group of integer points of the centre Z(Nz(lR)) 2 KY.:
Let G be defined as N/Z, and we aim to demonstrate that the nilpotent Lie group G is nonlinearizable In this context, if T represents a maximal compact subgroup of G, then T is identified as a one-dimensional torus, specifically T = (G, G) Should G be isomorphic to a linear group, it would imply that T n (G, G) equals the identity element {e}, as outlined in Section 5.4 This leads to a contradiction, thereby establishing that G cannot be nonlinearized.
5.3 Ado’s Theorem As we have just seen, not every connected Lie group is linearizable However, it turns out that a Lie group is always locally iso- morphic to a linear Lie group This follows from Ado’s theorem, one of the main results of the Lie algebra theory
Theorem 5.1 (Ado, see Bourbaki [1975], Jacobson [1962], Serre [1987]) Let g be a finite-dimensional Lie algebra over a field K of characteristic 0 Then the Lie algebra g has a faithful finite-dimensional linear representation over K
If the center b(g) of the Lie algebra g is trivial, the adjoint representation ad: g → gI(g) is faithful Conversely, if a(g) is non-trivial, a faithful representation π of the Lie algebra g must be identified that acts faithfully on a(g) Consequently, p = plead serves as a faithful representation of the Lie algebra g.
Complexification and Real Forms
Theorem 6.8 (see Onishchik and Vinberg [1990], Chap 6) Let H be a reductive Levi subgroup of an algebraic group G, and let S be a reductive algebraic subgroup of G Then there exists g E Rad,G such that gSg-l c H
Corollary 1 Any two reductive Levi subgroups of an algebraic group are conjugate
Corollary 2 An algebraic subgroup is a reductive Levi subgroup if and only if it is a maximal reductive subgroup § 7 Complexification and Real Forms
7.1 Complexifkation and Real Forms of Lie Algebras Let g be a real Lie algebra The vector space g @ c is naturally equipped with the structure of a Lie algebra The resulting complex Lie algebra is denoted by g(cC) and is called the complexification of g For example, if g = nn(Jw) is the Lie algebra of all real nilpotent upper triangular matrices of order n, then g(C) = nn(C) is the Lie algebra of all complex nilpotent upper triangular matrices of order n
An arbitrary element z E g(C) is uniquely represented in the form z = TC + iy, where x,y E g Let F = x - iy; the correspondence z H 2 is an antilinear automorphism of the Lie algebra g(@), i.e z’ + 2” = 2 + p, where z’, z” E g(C), x2=x.2?, XEC, ,zEQ(C)
The process that serves as the inverse to complexification is known as the realification of a complex Lie algebra For a Lie algebra \( g \) defined over the field \( \mathbb{C} \), its realification is represented as \( g_{\mathbb{R}} \), which considers the same algebra over the field \( \mathbb{R} \) Thus, \( g_{\mathbb{R}} \) is recognized as the realification of the complex Lie algebra \( g \).
In the Lie algebra g the operator of the complex structure is defined, i.e the operator of multiplication by i:
The operator I is linear over the field W and satisfies the relations
I(IX? Yl) = [xc, IYl> where x,y E gn
A real Lie algebra \( g \) can be endowed with a complex structure through a linear operator \( I \) that satisfies specific relations By defining \( (cz + i, 0)x = cyx + 0.I(x) \), we establish a complex Lie algebra structure on \( g \), which, upon realification, aligns perfectly with the original Lie algebra \( g \).
If I is a complex structure on a real Lie algebra, then -1 is evidently also a complex structure The complex structures I and -I are said to be conjugate For example, for an arbitrary complex Lie algebra g, by reversing the sign of the operator of the complex structure we obtain another structure of a complex Lie algebra We denote this algebra by 3, and it is said to be conjugate to g If X = a + ip E c, and z E g, then the operation of multiplication by the number X in the Lie algebra 3 is of the form X z (o - ip) 5, where the right-hand side features the scalar multiplication in g Clearly, gR ru & The Lie algebras g and g are isomorphic (over @) if and only if 0 admits an antilinear automorphism
Let g be an arbitrary complex Lie algebra Then, as one can easily verify, (gR)(@) z g @ zij In particular, if g N g, then &J(C) P g @ g
A real subalgebra \( h \) of a complex Lie algebra \( g \) is defined as a real form for \( g \) if the natural embedding \( l_j \subset g \) can be extended to an isomorphism \( h(\mathbb{C}) \cong g \) of complex Lie algebras For instance, \( \mathfrak{n}(\mathbb{K}) \) serves as a real form of \( \mathfrak{n}(\mathbb{Q}_1) \), while \( \mathfrak{l} = \mathfrak{su}(2) \) is a real form for \( g = \mathfrak{sl}(\mathbb{C}) \), and \( \mathfrak{g}' = \mathfrak{so}(2, \mathbb{R}) \) is another real form for \( g \) It is evident that two real forms \( h' \) and \( h'' \) for \( g \) are isomorphic over \( \mathbb{R} \) if there exists an automorphism \( \varphi \in \text{Aut}(g) \) such that \( \varphi(h') = h'' \).
Let \( b \) represent a real form of a complex Lie algebra \( g \) The operation of complex conjugation on \( h(@) \) produces an involutory antilinear automorphism \( c \) of \( g \), with \( Q \) aligning with the fixed points set \( gg \) Conversely, any involutory antilinear automorphism \( \tilde{7} \) of a complex Lie algebra \( g \) results in \( g'' \) being a real form of \( g \) Furthermore, isomorphic real forms are linked to involutory antilinear homomorphisms that are conjugate by elements from \( \text{Aut} \, \mathcal{G} \).
By a real structure in g we will mean an arbitrary involutory antilinear au- tomorphism of this algebra
A complex Lie algebra \( g \) that possesses a real form is isomorphic to itself, indicating a relationship between \( g \) and its real counterpart Specifically, this implies that \( (g_w)(\mathbb{C}) \) is a subset of \( g \) within the context of \( g \) and its structures Notably, every complex semisimple Lie algebra has at least one real form, as illustrated in Chapter 4, Section 1.1, Example 6, and some may even have multiple real forms However, it is important to recognize that certain complex Lie algebras do not have any real forms at all.
The semidirect sum of Lie algebras is represented by g = @ B V@2, corresponding to a homomorphism ‘p: @ -+ 812(c) with q(l) = diag (1,2i) If g possesses a real form IJ, it can be expressed as IJ N IR B JIIW~, involving a homomorphism $X @ + gI2(lR) The eigenvalues of the operator $(l) are denoted as Xi and X2.
Xi X2 # 0 The isomorphism between h @ @ and g implies that Xi = X 1, X2 = X 2i for some X E @, X # 0 Since Xi, X2 cannot be both real (otherwise
X E IR, and therefore 2i E R), they must be complex conjugates, in particular ]Xi] = 1x21 But then IX] = ]2iX], w ic is impossible h h Thus, the Lie algebra g has no real form
The process of complexification in Lie algebras maintains fundamental algebraic structures, as evidenced by the relationship [g(c), g(c)] = [g, g](c) This indicates that a real Lie algebra g is solvable (or nilpotent) if and only if its complex counterpart g(c) exhibits the same properties Furthermore, the radical of the Lie algebra, denoted as rad(g), corresponds to rad(g(c)), suggesting that the Lie algebras g and g(c) can only be semisimple simultaneously Lastly, in the context of a Levi decomposition, if g is expressed as g = s + t, it highlights the structural interplay between the components of real Lie algebras.
5 7 Complexification and Real Forms 35 g (r is the radical, 5 a Levi subalgebra), then g(C) = s(C) + r(C) is a Levi decomposition of the Lie algebra g(C) over C
7.2 ComplexiEcation and Real Forms of Lie Groups Let G be a real Lie group By analogy with the case of Lie algebras, we can consider the question of constructing the complexification G(C) of G If G is algebraic (to be more precise, G is isomorphic to the group of R-points of some algebraic group defined over R), then it is natural to mean by its complexification the group of its G-points G(C) However, if G is an arbitrary Lie group, then there is no candidate naturally entitled to the position of the complexification of G This is not a mere accident, because if we understand the complexification as an embedding in a complex Lie group with some additional properties, such a complexification does not necessarily exist For example, it is not difficult to show that the Lie group A = SL2(lR) is not isomorphic to any virtual Lie subgroup of a complex Lie group (this is related to the fact that the group
A linear Lie group can be complexified by considering a virtual Lie subgroup G of GL(V), where V is a finite-dimensional vector space over R The complexification G(C) is defined as the intersection of all complex virtual Lie subgroups of GL(V(@)) that contain G, making it the smallest complex virtual Lie subgroup of GL(V(C)) containing G This can be constructed using a tangent subalgebra g within gK(V) and its complexification g(C), which corresponds to a connected virtual Lie subgroup G within GL(V(@)) Notably, G(C)’ = G and G(C) = G.G, with complex conjugation inducing an involution in G(C) that has G as its fixed points If G is a real algebraic group, G(C) corresponds to all C-points of G Additionally, for a complex Lie group G, its realification is denoted as GR A real form of G is defined by a real Lie subgroup H, where the tangent algebra h serves as a real form of the Lie algebra g, and H intersects every connected component of G, satisfying G = H.Go.
Gi is a real form of a linear Lie group, then it is a real form of Gi(@) if Gi is closed in Gi(C) (which is not necessarily the case)
A real structure in a complex Lie group \( G \) is defined as an involutory smooth automorphism \( s: G \to G \) for which the differential \( ds \) acts as a real structure on the tangent Lie algebra \( g \) of the group For instance, in the case of an algebraic group defined over \( \mathbb{R} \), complex conjugation creates a real structure on the set \( G(\mathbb{C}) \) comprising all its \( \mathbb{C} \)-points.
If s represents a real structure on the Lie group G, the set G” of fixed points under the involution s forms a real form of G Additionally, if G is algebraic, it follows that G” is also algebraic.
The complex conjugation operation A H ‘21 within the group GL(c) establishes a real structure on GL(c), with the associated real form being GL(R) Additional examples of this concept in the context of complex semisimple Lie groups are detailed in Chapter 4, Section 1.1.
7.3 Universal Complexifkation of a Lie Group As noted above, for an arbitrary Lie group a complexification considered as an embedding in a com- plex Lie group does not necessarily exist Following Hochschild [1966], we now make the notion of the complexification more precise