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Sortability of representations of Lie algebras
JOURNAL
OF ALGEBRA
27, 486-490 (1973)
Sortability
of Representations
of Lie Algebras
D. W. BARNES Department
of Mathematics,
University
of Sydney, Sydney NSW 2006, Australia
Communicated by N. Jacobson Received
March
8, 1971
A module over a nilpotent Lie algebra decomposes into a direct sum of submodules, each involving only one type of composition factor. We say that a module which decomposes in this way is sortable. Sortability can be regarded as a generalization of complete reducibility. In this we show that the Lie algebras, all of whose representations are sortable are, in the case of nonzero characteristic, the nilpotent algebras, and, in the characteristic zero case, the direct sums of semisimple and nilpotent algebras. All algebras and modules considered are finite-dimensional over a field F on which we place no restrictions. If N is an ideal of the Lie algebra L, we write N 4 L. If N is a subalgebra of L and there exist subalgebras Ni such that N a Nr a ... a N, = L, we say that N is subnormal in L and write N 4~ L. Throughout this note, L is a Lie algebra and N is a given subnormal subalgebra of L. F can be regarded as a one-dimensional L-module with L acting trivially. We shall need to distinguish in notation between F regarded as an L-module and F regarded as an N-module. We denote the latter by FN . For any L-module V, VL = (n E V 1XV = 0 fov aZZ x EL}. For background material on the cohomology of Lie algebras, the reader is referred to Jacobson [2, Chapter 3, Section 101 and to Barnes [l]. DEFINITIONS.
An A-component (i) (ii)
Let V be anl-module and let A be an irreducible of V is an L-submodule A(V) such that
every N-composition factor of A(V) is isomorphic to A, and no N-composition factor of V/A(Y) is isomorphic to A.
A dual A-component (i) (ii)
N-module.
of V is an L-submodule
A’(V) such that
no N-composition factor of A’(V) is isomorphic to A, and every N-composition factor of V/A’(V) is isomorphic to A.
The module V is called N-sortable
if an A-component
of V exists for all A.
1 The author has not been able to find precisely this result in the literature. It is closely related to Theorem 6, p. 42, of Jacobson [2] and is an easy consequence of Barnes [l, Lemma 31.
486 Copyright0 1973by Academic Press,Inc. All rightsof reproduction in anyformreserved.
SORTABILITY
OF REPRESENTATIONS
87
emarks. An A-component of V, if one exists, is &early unique. Sf V is sortable, it is the direct sum of its components. We note that, if P is an irreducible E-module, then, by Zassenhaus [3, Lemma I], all N-composition factors of P are isomorphic. An A-component A(V) exists for all V if and only if: whenever a composition series V = V, > a/, > ..~ > VT = 0 has VSJ Vi N P involving A and Vi/Vi+l Y Q not involving A, there exists a submodule Vi’ between VieI and Vi+l such that Vipl/Vi ‘v Q and Vi’/Vi+l ‘v P, that is, the factors of the composition series can be rearrange bringing P below Q. Thus, the existence of A(V) for aI1 V is equivalent to the splitting of all extensions of such Q by such P. This splitting is equivalent to EP(L, hom,(P, Q)) = 0. Dually, the existence of A’(V) for all V is equivalent to Hl(L, hom,(Q, P)) = 0 for all P, Q as above. t
v = v.
“i
%EMMA
1.
SupposeL’ # L and HI(L, V) = 0. Then VL = 0.
Proof. If dim L = 1, then H1(L, V) E V/LV and the result fooliows. Suppose dimL > 1. There exists M 4 L such that dimL/M = 1. ochschild-Serre exact sequence
0 + H”(L/M,
ii/M) + H”(L, V),
H1(L, V) = 0
implies @(L/M, VM) = 0 and so (%ihf)Ll~M= 0. (Vl”)“P = VL. The next lemma is essentially due to J.H.C. Whitehead. LEMMA 2. BooJ.
Suppose L’ = L. Then H1(L, F) = 0.
An element of .P(L, F) is a linear map flXY> = xf( Y> - YfH
f
: L ---f F such that
BARNES
488
for all X, y EL. Since L acts trivially, this condition onf reduces to f(xy) As L’ = L, it follows that Zl(L, F) = 0. As an immediate consequence of Lemmas 1,2, we have LEMMA 3.
= 0.
L’ = L if and only if HI(L, F) = 0.
LEMMA 4. Suppose H1(L, V) = 0 f OY every L-module simple if char F = 0, and L = 0 if char F # 0.
V. Then L is semi-
Boof. The assumptions imply that every L-module V is completely reducible. If char F # 0, this implies L = 0 by Theorem 2, p. 205 of Jacobson [2]. Suppose charF = 0. Since L is completely reducible as an L-module, L is a direct sum of minimal ideals. Any ideal of one of these minimal ideals is itself an ideal of L. The minimal ideals are, therefore, simple. By Lemma 3, L’ = L, so these ideals are non-Abelian. THEOREM. Let L be a jinite-dimensional Lie algebra over the jield F, and let N 4 L. Then the following conditions are equivalent: (a)
Every jinite-dimensional
L-module is N-sortable.
(b)
FN(V) exists fey every finite-dimensional
(c)
FN’( V) exists fey every f&e-dimensional
(d) If V is an irreducible jkite-dimensional as an N-composition factor, then H1(L, V) = 0. (e)
IfV
is a ji ni t-d e imensionall-module
(f) If V is afinite-dimensionalL-module for all Y.
L-module L-module
V. V.
L-module not containing FN
and VN = 0, then H1(L, V) = 0. and VN = 0, then Hr(L,
(g) In the case char F # 0, N is nilpotent. In N = S @ R where S is semisimple and R is nilpotent.
V) = 0
the case char F = 0,
Proof. Trivially, (a) implies (b) and (c). By the remarks above, (b) is equivalent to H-I(L, hom,(F, V)) = 0 f or all irreducible modules V not involving FN , while (c) is equivalent to Hl(L, hom,( V, F)) = 0 for all such V. Since hom,(F, V) ci V, (b) is . eq uivalent to (d). Since hom,( V, F) is irreducible if and only if V is irreducible, and involves FN if and only if V involves FN , (c) also is equivalent to (d). Suppose (d) holds and that VN = 0. By (b), V has no N-composition factor isomorphic to FN . By (d), H1(L, P) = 0 for every L-composition factor P of V. If U is a submodule of V, we have the exact sequence H1(L, U) -+ Hl(L,
V) -+ Hl(L,
V/U).
SORTABILITY
OF REPRESENTATIONS
489
By induction on the length of a composition series, LP(L, U) = Ni(L, V/ LJ) = @ and LP(L, V) = 0 follows. Suppose (e) holds. Let P, ,Q be irreducible L-modules. Then each involves only one type of N-composition factor. Suppose P involves the irreducible N-module A and Q does not. Then hom,(P, ,Q)” = 63 and by (e), LP(L, hom,(P, Q)) = 0. It follows that A(V) exists for all V. We now have the equivalence of the first five conditions. Suppose these hold. We apply (b) to the regular representation of L, and denote &(L> by L, , Since N CI~ L, every composition factor of the N-module L/N is isomorphic to FM . But L/LO does not have FN as an N-composition factor, Therefore, N + L, = L. Ciearly N n L, is the hypercenter of JV. Case 1. Suppose L, = 0. Then N = L. Since L has no principal factors isomorphic toF,L = L’ and Hl(L, F) = 0 by Lemma 2. By (d), P(L, V) = 8 for all irreducible L-modules V not isomorphic to F. Thus we have ,?P(L, V) = 0 for all irreducible L-modules 5’ and so for all L-modules. By Lemma 4, L is semisimple if char F = 0, and L = 0 if char F f 0. Case 2. Suppose L/LO # 0. Every L/L,-module is an L-module and so is N + L,/L,-sortable. Thus, L/L, satisfies (a) and has the structure asserted in Case 1. If char F f 0, this implies L, = L and N R L, = N. Thus, N is nilpotent. Suppose char F = 0. Let R be the radical of N and let S be a Levi factor, Since N/N n L, ‘v L/LO which is semisimple, R < N n L, DTherefore R is the hypercenter of N. N is represented on R by nilpotent linear transformations. Since S is semisimple, it follows by Engel’s theorem that § is in the kernel of this representation. Thus, SR = 8 and S is an ideal off _Ri. Clearly (f) implies (e). To complete the proof of the theorem, we suppose (g) holds, and we prove (f). Suppose V is irreducible and VN = 0. Let K be the kernel of the representation of L on V. Then K n N # N. If K > 8, then, since V is completely reducible as an S-module and only involves one type of S-composition factor, V=O. By the Whitehead lemma (see Jacobson [2, p. 96, Theorem 14:), Hr(S, V) = 0 for all Y. If K 3 S, then VR = 0, and by Barnes [l, Lemma 31, H’(R, V) = 0 for all Y. Thus, in either case, we have a subnormal subalgebra M ofL such that LP(M, V> = 0 for all Y. Let M = lMO 4 M1 CJ ... 4 ni7, = L be a subnormal chain linking M to L. If LP’(ll& , V) = 0 for all Y, then M’(M+, , V) = 0 for all Y by the Hochschild-Serre spectral sequence. Thus, Nr(L, V) = 0 for all Y. In particular, (d) holds. Let V be any L-module such that VN = 0. Since V is N-sortable, P” = 0 for every L-composition factor P of V, and, therefore, H’(L, P) = 0 for all T. By induction on the length of a composition series, it follows that LP(L, V) = for all T.
490
BARNES REFERENCES
1. D. W. BARNES,
On the cohomology
of soluble Lie algebras, Math.
2. 101 (1967),
343-349.
2. N. JACOBSON, “Lie Algebras,” Interscience, New York/London, 1962. 3. H. ZASSENHAUS, On trace bilinear forms on Lie-algebras, Proc. Glasgow Math. Sot. 14 (1959), 62-72.
OF ALGEBRA
27, 486-490 (1973)
Sortability
of Representations
of Lie Algebras
D. W. BARNES Department
of Mathematics,
University
of Sydney, Sydney NSW 2006, Australia
Communicated by N. Jacobson Received
March
8, 1971
A module over a nilpotent Lie algebra decomposes into a direct sum of submodules, each involving only one type of composition factor. We say that a module which decomposes in this way is sortable. Sortability can be regarded as a generalization of complete reducibility. In this we show that the Lie algebras, all of whose representations are sortable are, in the case of nonzero characteristic, the nilpotent algebras, and, in the characteristic zero case, the direct sums of semisimple and nilpotent algebras. All algebras and modules considered are finite-dimensional over a field F on which we place no restrictions. If N is an ideal of the Lie algebra L, we write N 4 L. If N is a subalgebra of L and there exist subalgebras Ni such that N a Nr a ... a N, = L, we say that N is subnormal in L and write N 4~ L. Throughout this note, L is a Lie algebra and N is a given subnormal subalgebra of L. F can be regarded as a one-dimensional L-module with L acting trivially. We shall need to distinguish in notation between F regarded as an L-module and F regarded as an N-module. We denote the latter by FN . For any L-module V, VL = (n E V 1XV = 0 fov aZZ x EL}. For background material on the cohomology of Lie algebras, the reader is referred to Jacobson [2, Chapter 3, Section 101 and to Barnes [l]. DEFINITIONS.
An A-component (i) (ii)
Let V be anl-module and let A be an irreducible of V is an L-submodule A(V) such that
every N-composition factor of A(V) is isomorphic to A, and no N-composition factor of V/A(Y) is isomorphic to A.
A dual A-component (i) (ii)
N-module.
of V is an L-submodule
A’(V) such that
no N-composition factor of A’(V) is isomorphic to A, and every N-composition factor of V/A’(V) is isomorphic to A.
The module V is called N-sortable
if an A-component
of V exists for all A.
1 The author has not been able to find precisely this result in the literature. It is closely related to Theorem 6, p. 42, of Jacobson [2] and is an easy consequence of Barnes [l, Lemma 31.
486 Copyright0 1973by Academic Press,Inc. All rightsof reproduction in anyformreserved.
SORTABILITY
OF REPRESENTATIONS
87
emarks. An A-component of V, if one exists, is &early unique. Sf V is sortable, it is the direct sum of its components. We note that, if P is an irreducible E-module, then, by Zassenhaus [3, Lemma I], all N-composition factors of P are isomorphic. An A-component A(V) exists for all V if and only if: whenever a composition series V = V, > a/, > ..~ > VT = 0 has VSJ Vi N P involving A and Vi/Vi+l Y Q not involving A, there exists a submodule Vi’ between VieI and Vi+l such that Vipl/Vi ‘v Q and Vi’/Vi+l ‘v P, that is, the factors of the composition series can be rearrange bringing P below Q. Thus, the existence of A(V) for aI1 V is equivalent to the splitting of all extensions of such Q by such P. This splitting is equivalent to EP(L, hom,(P, Q)) = 0. Dually, the existence of A’(V) for all V is equivalent to Hl(L, hom,(Q, P)) = 0 for all P, Q as above. t
v = v.
“i
%EMMA
1.
SupposeL’ # L and HI(L, V) = 0. Then VL = 0.
Proof. If dim L = 1, then H1(L, V) E V/LV and the result fooliows. Suppose dimL > 1. There exists M 4 L such that dimL/M = 1. ochschild-Serre exact sequence
0 + H”(L/M,
ii/M) + H”(L, V),
H1(L, V) = 0
implies @(L/M, VM) = 0 and so (%ihf)Ll~M= 0. (Vl”)“P = VL. The next lemma is essentially due to J.H.C. Whitehead. LEMMA 2. BooJ.
Suppose L’ = L. Then H1(L, F) = 0.
An element of .P(L, F) is a linear map flXY> = xf( Y> - YfH
f
: L ---f F such that
BARNES
488
for all X, y EL. Since L acts trivially, this condition onf reduces to f(xy) As L’ = L, it follows that Zl(L, F) = 0. As an immediate consequence of Lemmas 1,2, we have LEMMA 3.
= 0.
L’ = L if and only if HI(L, F) = 0.
LEMMA 4. Suppose H1(L, V) = 0 f OY every L-module simple if char F = 0, and L = 0 if char F # 0.
V. Then L is semi-
Boof. The assumptions imply that every L-module V is completely reducible. If char F # 0, this implies L = 0 by Theorem 2, p. 205 of Jacobson [2]. Suppose charF = 0. Since L is completely reducible as an L-module, L is a direct sum of minimal ideals. Any ideal of one of these minimal ideals is itself an ideal of L. The minimal ideals are, therefore, simple. By Lemma 3, L’ = L, so these ideals are non-Abelian. THEOREM. Let L be a jinite-dimensional Lie algebra over the jield F, and let N 4 L. Then the following conditions are equivalent: (a)
Every jinite-dimensional
L-module is N-sortable.
(b)
FN(V) exists fey every finite-dimensional
(c)
FN’( V) exists fey every f&e-dimensional
(d) If V is an irreducible jkite-dimensional as an N-composition factor, then H1(L, V) = 0. (e)
IfV
is a ji ni t-d e imensionall-module
(f) If V is afinite-dimensionalL-module for all Y.
L-module L-module
V. V.
L-module not containing FN
and VN = 0, then H1(L, V) = 0. and VN = 0, then Hr(L,
(g) In the case char F # 0, N is nilpotent. In N = S @ R where S is semisimple and R is nilpotent.
V) = 0
the case char F = 0,
Proof. Trivially, (a) implies (b) and (c). By the remarks above, (b) is equivalent to H-I(L, hom,(F, V)) = 0 f or all irreducible modules V not involving FN , while (c) is equivalent to Hl(L, hom,( V, F)) = 0 for all such V. Since hom,(F, V) ci V, (b) is . eq uivalent to (d). Since hom,( V, F) is irreducible if and only if V is irreducible, and involves FN if and only if V involves FN , (c) also is equivalent to (d). Suppose (d) holds and that VN = 0. By (b), V has no N-composition factor isomorphic to FN . By (d), H1(L, P) = 0 for every L-composition factor P of V. If U is a submodule of V, we have the exact sequence H1(L, U) -+ Hl(L,
V) -+ Hl(L,
V/U).
SORTABILITY
OF REPRESENTATIONS
489
By induction on the length of a composition series, LP(L, U) = Ni(L, V/ LJ) = @ and LP(L, V) = 0 follows. Suppose (e) holds. Let P, ,Q be irreducible L-modules. Then each involves only one type of N-composition factor. Suppose P involves the irreducible N-module A and Q does not. Then hom,(P, ,Q)” = 63 and by (e), LP(L, hom,(P, Q)) = 0. It follows that A(V) exists for all V. We now have the equivalence of the first five conditions. Suppose these hold. We apply (b) to the regular representation of L, and denote &(L> by L, , Since N CI~ L, every composition factor of the N-module L/N is isomorphic to FM . But L/LO does not have FN as an N-composition factor, Therefore, N + L, = L. Ciearly N n L, is the hypercenter of JV. Case 1. Suppose L, = 0. Then N = L. Since L has no principal factors isomorphic toF,L = L’ and Hl(L, F) = 0 by Lemma 2. By (d), P(L, V) = 8 for all irreducible L-modules V not isomorphic to F. Thus we have ,?P(L, V) = 0 for all irreducible L-modules 5’ and so for all L-modules. By Lemma 4, L is semisimple if char F = 0, and L = 0 if char F f 0. Case 2. Suppose L/LO # 0. Every L/L,-module is an L-module and so is N + L,/L,-sortable. Thus, L/L, satisfies (a) and has the structure asserted in Case 1. If char F f 0, this implies L, = L and N R L, = N. Thus, N is nilpotent. Suppose char F = 0. Let R be the radical of N and let S be a Levi factor, Since N/N n L, ‘v L/LO which is semisimple, R < N n L, DTherefore R is the hypercenter of N. N is represented on R by nilpotent linear transformations. Since S is semisimple, it follows by Engel’s theorem that § is in the kernel of this representation. Thus, SR = 8 and S is an ideal off _Ri. Clearly (f) implies (e). To complete the proof of the theorem, we suppose (g) holds, and we prove (f). Suppose V is irreducible and VN = 0. Let K be the kernel of the representation of L on V. Then K n N # N. If K > 8, then, since V is completely reducible as an S-module and only involves one type of S-composition factor, V=O. By the Whitehead lemma (see Jacobson [2, p. 96, Theorem 14:), Hr(S, V) = 0 for all Y. If K 3 S, then VR = 0, and by Barnes [l, Lemma 31, H’(R, V) = 0 for all Y. Thus, in either case, we have a subnormal subalgebra M ofL such that LP(M, V> = 0 for all Y. Let M = lMO 4 M1 CJ ... 4 ni7, = L be a subnormal chain linking M to L. If LP’(ll& , V) = 0 for all Y, then M’(M+, , V) = 0 for all Y by the Hochschild-Serre spectral sequence. Thus, Nr(L, V) = 0 for all Y. In particular, (d) holds. Let V be any L-module such that VN = 0. Since V is N-sortable, P” = 0 for every L-composition factor P of V, and, therefore, H’(L, P) = 0 for all T. By induction on the length of a composition series, it follows that LP(L, V) = for all T.
490
BARNES REFERENCES
1. D. W. BARNES,
On the cohomology
of soluble Lie algebras, Math.
2. 101 (1967),
343-349.
2. N. JACOBSON, “Lie Algebras,” Interscience, New York/London, 1962. 3. H. ZASSENHAUS, On trace bilinear forms on Lie-algebras, Proc. Glasgow Math. Sot. 14 (1959), 62-72.