We give a natural proof of the Ado theorem.Comment: 2 page

Neretin, Yurii A.

English

arXiv

summary:Summary: The Ado theorem is a fundamental fact, which has a reputation of being a `strange theorem'. We give its natural proof

Institute of Mathematics AS CR, v. v. i.

WSGP 22Yurii A. NeretinA construction of finite-dimensional faithful representation of Lie algebraIn: Jarolím Bureš (ed.): Proceedings of the 22nd Winter School "Geometry and Physics". CircoloMatematico di Palermo, Palermo, 2003. Rendiconti del Circolo Matematico di Palermo, Serie II,Supplemento No. 71. pp. [159]--161.Persistent URL: http://dml.cz/dmlcz/701715Terms of use:© Circolo Matematico di Palermo, 2003Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czRENDICONTIDEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 71 (2003), pp. 159-161 A CONSTRUCTION OF FINITE-DIMENSIONAL FAITHFUL REPRESENTATION OF LIE ALGEBRA YURII A. NERETIN ABSTRACT. The Ado theorem is a fundamental fact, which has a reputation to be a 'strange theorem'. We give its natural proof. 1. CONSTRUCTION OF FAITHFUL REPRESENTATION Consider a finite-dimensional Lie algebra g. Assume that g is a semidirect product p tx n of a subalgebra p and a nilpotent ideal n. Assume that the adjoint action of p on n is faithful, i.e., for any z G p, there exists x € n such that [z, x] ^ 0. Consider the minimal k such that all the commutators [...[[xi,x2],x3],...,xfc], Xj en are 0. Denote by U(n) the enveloping algebra of n. The algebra n acts on U(n) by the left multiplications. The algebra p acts on U(n) by the derivations dzx\X2xz... xi = [z, Xi]x2xs ...xi+ xi[z, x2]x$ . . . x\ + . . . , where z E p. This defines the action of the semidirect product p K n = g on U(n). Denote by I the subspace in U(n) spanned by all the products x\x2.. .XJV, where N > k + 2. Obviously, 1. I is the two-side ideal in U(n). 2. Consider the linear span A C U(n) of 1 and all the x G g . Obviously, In .A = 0. 3. I is invariant with respect to the derivations dz. Obviously, the module U(n)/I is a finite-dimensional faithful module over g. 2. THE ADO THEOREM Lemma 1. Any finite-dimensional Lie algebra q admits an embedding to an algebra g such that (a) g is a semidirect product of a reductive subalgebra p and a nilpotent ideal n; (b) the action ofp onn is completely reducible. The paper is in final form and no version of it will be submitted elsewhere. 160 YURII A. NERETIN Obviously, Lemma 1 implies the Ado theorem. Indeed, g admits a decomposition g = p' 0 (p" tx n) where p', p" are reductive subalgebras and the action of p" on n is faithful. After this, it is sufficient to apply the construction of p.L REMARK. The Ado theorem implies Lemma 1 modulo the Chevalley construction of algebraic envelope of a Lie algebra. But Lemma 1 itself can be easily proved directly 3. KILLING LEMMA Let g be a Lie algebra, let d be its derivation. For an eigenvalue A, denote by Q\ its root subspace Qx = Ufcker(d — A)fc; we have g = ®g\. As it was observed by Killing, z € flA, y € fl,, implies [x, y] G Qx+^ Thus the Lie algebra g admits the gradation by the eigenvalues of d. Consider the gradation operator ds : g —> g defined by dsv = \v if v £ Q\. Obviously, ds is a derivation, and dds = dsd. We also consider the derivation dn:= d — dsi this operator is nilpotent (the equality d = dn + ds is called the Jordan-Chevalley decomposition). Clearly (1) kevds D kerd; kerdn D kerd; (2) im ds C im ds\ im dn C im ds. 4. ELEMENTARY EXPANSIONS Let q be a Lie algebra, let I be an ideal of codimension 1. Let x fi I. Denote by d the operator Adx : / —• I. Consider the corresponding pair of derivations dS) dn. Consider the space q' = Cy + Cz + I where y, z are formal vectors. We equip this space with a structure of a Lie algebra by the rule [y, z] = 0, [y, u] = dsu, [z, u] = dnu, for all u G I and the commutator of tx, v G I is the same as it was in I. The subalgebra C(y + z) © I C q' is isomorphic q. We say that q' is an elementary expansion of q. Obviously, [q',q'] = [q,q]. For a general Lie algebra, the required embedding to a semidirect product can be obtained by a sequence of elementary expansions. 5. PROOF OF LEMMA 1 Let q be a Lie algebra. Let f) be its Levi part, and r be the radical. Denote by m the nilradical of q, i.e., m = [q,r]; recall that m is a nilpotent ideal, and [q, q] = f) K m (see [1], 1.4.9). Consider a nilpotent ideal n of q containing the nilradical m. Consider a subalgebra p D J) such that the adjoint action of p on q is completely reducible and p fl n = 0; for instance, the can choice n = m, p = fj. A CONSTRUCTION OF FINITE-DIMENSIONAL FAITHFUL REPRESENTATION,... 161 Obviously, the q-module q/(p K n) is trivial. Consider any subspace I of codimen-sion 1 containing p K n, obviously I is an ideal in q. Since the action of p on q is completely reducible, there exists a p-invariant complementary subspace for I. Let x be an element of this subspace. Since the p-module q/I is trivial, x commutes with p. We apply the elementary expansion to these data. We obtain the new algebra q7 = Cy + Cz + I with the nilpotent ideal n7 = Cz + n and with the reductive subalgebra p7 = Cy © p (by (1), y commutes with p). It remains to notice that dim q7 - dim p7 — dim n7 = dim q — dim p — dim n — 1 and we can repeat the same construction. REFERENCES [1] Dixmier, J., Enveloping Algebras, North-Holland Publ. Co, 1977. MATH.PHYSICS GROUP ITEP B. CHEREMUSHKINSKAYA, 25 Moscow RUSSIA AND INDEPENDENT UNIVERSITY OF MOSCOW AND ESI, VIENNA AUSTRIA (JANUARY, 2002) E-mail address: neretinQmain.mccme.rssi.ru 

A construction of finite-dimensional faithful representation of Lie algebra

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.