Inner automorphisms of Lie algebras related with generic 2 × 2 matrices

Let Fm = Fm(var(sl₂(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl₂(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion Fmˆ of Fm w...

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Опубліковано в: :	Algebra and Discrete Mathematics
Дата:	2012
Автори:	Drensky, V., Fındık, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2012
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Цитувати:	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices / V. Drensky, S. Fındık // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 49-70. — Бібліогр.: 23 назв. — англ.

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spelling	Drensky, V. Fındık, S. 2019-06-09T05:50:12Z 2019-06-09T05:50:12Z 2012 Inner automorphisms of Lie algebras related with generic 2 × 2 matrices / V. Drensky, S. Fındık // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 49-70. — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:17B01, 17B30, 17B40, 16R30. https://nasplib.isofts.kiev.ua/handle/123456789/152228 Let Fm = Fm(var(sl₂(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl₂(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion Fmˆ of Fm with respect to the formal power series topology. Our results are more precise for m = 2 when F₂ is isomorphic to the Lie algebra L generated by two generic traceless 2×2 matrices. We give a complete description of the group of inner automorphisms of Lˆ. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm / Fm c⁺¹ = Fm(var(sl₂(K)) ∩ Nc) The research of the first author was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the second author was partially supported by the Council of Higher Education (YÖK) in Turkey. The second named author is grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences for the creative atmosphere and the warm hospitality during his visit when most of this project was carried out. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Inner automorphisms of Lie algebras related with generic 2 × 2 matrices Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices
spellingShingle	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices Drensky, V. Fındık, S.
title_short	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices
title_full	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices
title_fullStr	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices
title_full_unstemmed	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices
title_sort	inner automorphisms of lie algebras related with generic 2 × 2 matrices
author	Drensky, V. Fındık, S.
author_facet	Drensky, V. Fındık, S.
publishDate	2012
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	Let Fm = Fm(var(sl₂(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl₂(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion Fmˆ of Fm with respect to the formal power series topology. Our results are more precise for m = 2 when F₂ is isomorphic to the Lie algebra L generated by two generic traceless 2×2 matrices. We give a complete description of the group of inner automorphisms of Lˆ. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm / Fm c⁺¹ = Fm(var(sl₂(K)) ∩ Nc)
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/152228
citation_txt	Inner automorphisms of Lie algebras related with generic 2 × 2 matrices / V. Drensky, S. Fındık // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 49-70. — Бібліогр.: 23 назв. — англ.
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first_indexed	2025-11-26T14:36:54Z
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fulltext	Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 14 (2012). Number 1. pp. 49 – 70 c© Journal “Algebra and Discrete Mathematics” Inner automorphisms of Lie algebras related with generic 2 × 2 matrices Vesselin Drensky1 and Şehmus Fındık2 Communicated by I. Ya. Subbotin Dedicated to 100th anniversary of Professor Sergei Nikolaevich Chernikov Abstract. Let Fm = Fm(var(sl2(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl2(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion F̂m of Fm with respect to the formal power series topology. Our results are more precise for m = 2 when F2 is isomorphic to the Lie algebra L generated by two generic traceless 2 × 2 matrices. We give a complete description of the group of inner automorphisms of L̂. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm/F c+1 m = Fm(var(sl2(K)) ∩ Nc) in the subvariety of var(sl2(K)) consisting of all nilpotent algebras of class at most c in var(sl2(K)). Introduction Let Lm be the free Lie algebra of rank m ≥ 2 over a field K of characteristic 0 and let G be an arbitrary Lie algebra. Let I(G) = Im(G) 1The research of the first author was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. 2The research of the second author was partially supported by the Council of Higher Education (YÖK) in Turkey. 2010 MSC: 17B01, 17B30, 17B40, 16R30. Key words and phrases: free Lie algebras, generic matrices, inner automor- phisms, Baker-Campbell-Hausdorff formula. Jo ur na l A lg eb ra D is cr et e M at h. 50 Inner Automorphisms be the ideal of Lm consisting of all Lie polynomial identities in m variables for the algebra G. The factor algebra Fm(G) = Fm(var G) = Lm/I(G) is the relatively free Lie algebra of rank m in the variety of Lie algebras generated by G. Typical examples of relatively free algebras are free solvable of class k Lie algebras when I(G) = L (k) m (e.g., free metabelian Lie algebras with I(G) = L′′ m), free nilpotent of class c Lie algebras when I(G) = Lc+1, relatively free algebras in a variety generated by a finite dimensional simple Lie algebra G, etc. See the books by Bahturin [1] and Mikhalev, Shpilrain and Yu [21] for a background on relatively free Lie algebras and their automorphisms, respectively. Let Fm(G) be a relatively free Lie algebra freely generated by x1, . . . , xm. An automorphism ϕ of Fm(G) is called linear if it is of the form ϕ(xj) = m∑ i=1 αijxi, αij ∈ K, j = 1, . . . , m. It is triangular if ϕ(xj) = αjxj + fj(xj+1, . . . , xm), 0 6= αj ∈ K, fj ∈ Fm(G), j = 1, . . . , m. (Here the polynomials fj = fj(xj+1, . . . , xm) do not depend on the vari- ables x1, . . . , xj .) The automorphism ϕ is tame if it can be presented as a product of linear and triangular automorphisms. Otherwise ϕ is wild. Cohn [5] showed that every automorphism of the free Lie algebra Lm is tame. In particular, the group of automorphisms Aut(L2) is isomorphic to the general linear group GL2(K). Quite often relatively free algebras Fm(G) possess wild automorphisms and for better understanding of the group Aut(Fm(G)) one studies its important subgroups. When we consider a finite dimensional simple Lie algebra G over C, the general theory gives that the series exp(adg) = ∑ n≥0 adng n! which defines an inner automorphism converges for all g ∈ G. Consid- ering inner automorphisms of a relatively free Lie algebra Fm(G), the first problem arising is that the formal power series defining the inner automorphism has to be well defined. This means that the operator adw, w ∈ Fm(G), has to be locally nilpotent. In many important cases adw is not locally nilpotent for some w ∈ Fm(G). Hence we have two possibilities to study inner automorphisms: Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 51 (1) to restrict the consideration to locally nilpotent derivations adw, or (2) to consider nilpotent relatively free algebras Fm(G)/F c+1 m (G) = Lm/(I(G) + Lc+1 m ) when exp(adw) is well defined for all w ∈ Fm(G)/F c+1 m (G). The well known formula exey = ex+y from Calculus is not more true if we consider non-commuting variables x and y. If x and y are the gen- erators of the free associative algebra A = K〈x, y〉, we may consider the completion Â with respect to the formal power series topology. Then the classical Baker-Campbell-Hausdorff formula gives the solution z of the equation ez = exey in Â. It is a formal power series in Â and its homogeneous components are Lie elements, i.e., z belongs to the comple- tion L̂2 of the free Lie algebra L2 generated by x and y. Similarly, if we consider the relatively free algebra Fm = Fm(G), it has a completion F̂m with respect to the formal power series topology. The composition of two inner automorphisms of F̂m is an inner automorphism obtained by the Baker-Campbell-Hausdorff formula. In the former case (1), when we consider only locally nilpotent deriva- tions of the algebra Fm = Fm(G), let adu and adv be locally nilpotent for some u, v ∈ Fm(G). It is not clear a priori whether the solution w ∈ F̂m of the equation exp(adu) exp(adv) = exp(adw) belongs to Fm. If this hap- pens for all u, v ∈ Fm with adu, adv locally nilpotent, then the product of two inner automorphisms of Fm is inner again and the group of inner automorphisms Inn(Fm) is well defined. For example, if Fm = Lm/L′′ m is the free metabelian Lie algebra, then adu is locally nilpotent for u ∈ Fm if and only if u belongs to the commutator ideal F ′ m and exp(adu) exp(adv) = exp(ad(u + v)), u, v ∈ F ′ m. It is easy to construct an example of a graded associative algebra R and two elements u, v ∈ R such that uk = vn = 0 for some positive integers k and n (which guarantees that eu, ev ∈ R) and the equation euev = ew has a solution w in R̂ which does not belong to R, see Example 1. This gives also an example of a Lie algebra H which is graded as an algebra, with well defined inner automorphisms exp(adu) and exp(adv), u, v ∈ H, and such that the solution w of the equation exp(adu) exp(adv) = exp(adw) belongs to Ĥ and not to H, see Example 2. But we do not know any example of a relatively free algebra Fm(G) with the property that the product of inner automorphisms is not an inner automorphism of the Jo ur na l A lg eb ra D is cr et e M at h. 52 Inner Automorphisms algebra. (By the Baker-Campbell-Hausdorff formula the product of inner automorphisms is inner in the completion F̂m of Fm = Fm(G).) In the latter case (2) of nilpotent relatively free algebras Fm(G)/F c+1 m (G), it is more convenient to work in the completion F̂m of Fm = Fm(G) which we prefer to do. Then the results on Inn(Fm/F c+1 m ) can be obtained im- mediately from the corresponding results on Inn(F̂m) taking into account only the first c summands in the formal power series describing inner automorphisms. Kofinas and Papistas [15] found a description of the automorphism group of relatively free nilpotent Lie algebras Fm over Q in terms of the Baker-Campbell-Hausdorff formula. This allowed them to construct generating sets for Aut(Fm) for several varieties of nilpotent Lie algebras. In the completion F̂2 of a concrete relatively free Lie algebra F2 = F2(G) the Baker-Campbell-Hausdorff series may have a simpler form than in Â. For example, Gerritzen [14] and later Kurlin [18, 19] found an expression of the series in the completion of the free metabelian Lie algebra L2/L′′ 2. In our recent paper [9], see also [8], we have used their results to describe the inner and outer automorphisms of the free metabelian nilpotent Lie algebras (Lm/L′′ m)/(Lm/L′′ m)c+1 for any c and m. Going back to the classics, Baker [2] evaluated the Baker-Campbell- Hausdorff series on several finite dimensional Lie algebras given in their adjoint representations, including the three-dimensional simple Lie algebra. In the present paper we study inner automorphisms of the completion of relatively free algebras of the variety of algebras generated by the three- dimensional simple Lie algebra G3. If the field K is algebraically closed, then G3 is isomorphic to the Lie algebra sl2(K) of traceless 2×2 matrices. This implies that I(G3) = I(sl2(K)) for any field K of characteristic 0, see the comments in the end of Section 2. In the theory of algebras with polynomial identities there is a standard generic construction which realizes Fm(G), dim(G) < ∞. Fixing a basis of G, the algebra Fm(G) is isomorphic to a subalgebra of the algebra with elements which are obtained by replacing the coordinates of the elements of G with polynomials in m · dim(G) variables. Our first result translates the result of Baker [2] in terms of this generic construction. We present a multiplication rule for the inner automorphisms of the completion F̂m of Fm = Fm(G3) with respect to the formal power series topology. The most important generic realization of the relatively free algebra Fm(G3) = Fm(sl2(K)) is as the Lie algebra generated by m generic traceless 2 × 2 matrices. This allows to apply the approach of Baker [2] and to obtain a formula for the composition of inner automorphisms of Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 53 Fm(sl2(K)). Although the structure of Fm(sl2(K)) is known for all m ≥ 2, we consider the case m = 2 only. The first reason for this restriction is that the structure of the algebra Fm(sl2(K)) in the case m > 2 is more complicated than for m = 2. The second reason is that the Lie algebra F2(sl2(K)) is naturally embedded into an associative algebra which is a free module over the polynomial algebra in three variables and the commutator ideal of F2(sl2(K)) is a free submodule which allows to use methods of linear algebra. We work in the completion Ŵ of the associative algebra W generated by two generic traceless 2 × 2 matrices x = (xij) and y = (yij), where xij , yij , (i, j) = (1, 1), (1, 2), (2, 1), are algebraically independent commuting variables, x22 = −x11, y22 = −y11. Let L be the Lie subalgebra of W gen- erated by x and y. Then L ∼= F2(sl2(K)). We have obtained the complete description of the group of inner automorphisms of the associative algebra Ŵ . We give a lemma to recognize whether an element in W belongs to L and if the answer is affirmative, an algorithm to write the element as a linear combination of commutators. This leads to the description of the group of inner automorphisms of the Lie algebra L̂: the multiplication rule for Inn(L̂) and the associated matrices of inner automorphisms of L̂. Finally, factorizing the algebra Fm(sl2(K)) modulo F c+1 m (sl2(K)) we derive the corresponding results for the relatively free algebras in the variety var(sl2(K)) ∩ Nc of the nilpotent of class ≤ c Lie algebras in var(sl2(K)). It would be interesting to have a similar description for Inn(F̂m) and Inn(Fm/F c+1 m ) also for m > 2, where Fm = Fm(sl2(K)), but it seems that the answer will be quite technical. Recently the second named author of the present paper [13] has described the group Out(L̂) = Aut(L̂)/Inn(L̂) of outer automorphisms of L̂, where Aut(L̂) is the group of continuous automorphisms of L̂. This gives immediately the description of the group Out(L/Lc+1) of outer automorphisms of L/Lc+1. The results of the present paper have been announced without proofs in [8]. Jo ur na l A lg eb ra D is cr et e M at h. 54 Inner Automorphisms 1. Preliminaries We fix a field K of characteristic 0 and the associative algebra W generated by two generic traceless 2 × 2 matrices x = ( x11 x12 x21 −x11 ) , y = ( y11 y12 y21 −y11 ) , where xij , yij , (i, j) = (1, 1), (1, 2), (2, 1), are algebraically independent commuting variables. We assume that W is a subalgebra of the 2 × 2 matrix algebra M2(K[xij , yij ]) and identify the polynomial f ∈ K[xij , yij ] with the scalar matrix with entries f on the diagonal. In particular, for any matrix z ∈ W we assume that the trace tr(z) belongs to the centre of M2(K[xij , yij ]). Let L be the Lie subalgebra of W generated by x and y. This is the smallest subspace of the vector space W containing x and y and closed with respect to the Lie multiplication [z1, z2] = z1adz2 = z1z2 − z2z1, z1, z2 ∈ L. Similarly we define the associative algebra Wm generated by m ≥ 2 generic traceless 2 × 2 matrices. We assume that all commutators are left normed, i.e., [z1, . . . , zn−1, zn] = [[z1, . . . , zn−1], zn], n = 3, 4, . . . . The following results give the description of the algebras Wm, W = W2 and L and some equalities in W . Theorem 1. Let Wm, W and L be as above. Then: (i) (Razmyslov [22]) The algebra of generic traceless matrices Wm is isomorphic to the factor-algebra K〈x1, . . . , xm〉/I(M2(K), sl2(K)) of the free associative algebra K〈x1, . . . , xm〉, where the ideal I(M2(K), sl2(K)) of the weak polynomial identities in m variables for the pair (M2(K), sl2(K)) consists of all polynomials from K〈x1, . . . , xm〉 which vanish on sl2(K) considered as a subset of M2(K). As a weak T -ideal I(M2(K), sl2(K)) is generated by the weak polynomial identity [x2 1, x2] = 0. The Lie subalgebra of Wm generated by the m generic traceless matrices is isomorphic to the relatively free algebra Fm(sl2(K)) in the variety of Lie algebras generated by sl2(K). (ii) (Drensky, Koshlukov [11], see also the comments in [7] and Koshlukov [16, 17] for the case of positive characteristic) The algebra Wm has the presentation Wm ∼= K〈x1, . . . , xm \| [x2 i , xj ] = [xixj+xjxi, xk] = s4(xi1 , xi2 , xi3 , xi4 ) = 0〉, Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 55 where i, j, k, ik = 1, . . . , m, i 6= j 6 k 6 i, i1 < i2 < i3 < i4, and s4(x1, x2, x3, x4) = ∑ σ∈S4 sign(σ)xσ(1)xσ(2)xσ(3)xσ(4) is the standard polynomial of degree 4. In particular, W ∼= K〈x1, x2 \| [x2 1, x2] = [x2 2, x1] = 0〉. (iii) (see e.g. Le Bruyn [20]) The centre of W is generated by t = tr(x2), u = tr(y2), v = tr(xy). The elements t, u, v are algebraically independent and W is a free K[t, u, v]- module with free generators 1, x, y, [x, y]. (iv) (see, e.g., Drensky and Gupta [10]) For k ≥ 1 the following equalities hold in W : x2 = t 2 ; y2 = u 2 ; xy + yx = v; [x, y]2 = v2 − tu; yad2kx = 2ktk−1(−vx + ty); yad2k+1x = 2ktk[y, x]; xad2ky = 2kuk−1(ux − vy); xad2k+1y = 2kuk−1[x, y]. Theorem 1 (iii) and (iv) gives immediately that L is embedded into the free K[t, u, v]-module with free generators x, y, [x, y]. The next lemma gives the precise description of the Lie elements in W . It also provides an algorithm how to express in Lie form the elements of L given as elements of the free K[t, u, v]-module with basis x, y, [x, y]. Lemma 1. (i) The commutator ideal L′ of L ∼= F2(sl2(K)) is a free K[t, u, v]-module of rank 3, with free generators xv − yt, xu − yv, [x, y]. (ii) The elements of L′ = (xv − yt)K[t, u, v] ⊕ (xu − yv)K[t, u, v] ⊕ [x, y]K[t, u, v] can be expressed in Lie form using the identities 2a+b+c+1(xv −yt)taubvc = [x, y, y](ady)2b−1(adx)2a+1(adyadx)c, b > 0, 2a+c+1(xv − yt)tavc = [x, y, x](adx)2a(adyadx)c, 2a+b+c+1(xu−yv)taubvc = [x, y, x](adx)2a−1(ady)2b+1(adxady)c, a > 0, 2b+c+1(xu − yv)ubvc = [x, y, y](ady)2b(adxady)c, 2a+b+c[x, y]taubvc = [x, y](adx)2a(ady)2b(adxady)c. Jo ur na l A lg eb ra D is cr et e M at h. 56 Inner Automorphisms Proof. (i) We make induction on the degree of the commutators using that every element in a Lie algebra is a linear combination of left normed commutators. The following equalities [x, y, x] = −[y, x, x] = 2(xv − yt), [x, y, y] = −[y, x, y] = 2(xu − yv), [x, y, x, x] = 2[x, y]t, [x, y, x, y] = [x, y, y, x] = 2[x, y]v, [x, y, y, y] = 2[x, y]u [xv − yt, x, x] = 2(xv − yt)t, [xv − yt, x, y] = 2(xu − yv)t, [xv − yt, y, x] = 2(xv − yt)v, [xv − yt, y, y] = 2(xu − yv)v, [xu − yv, x, x] = 2(xv − yt)v, [xu − yv, x, y] = 2(xu − yv)v, [xu − yv, y, x] = 2(xv − yt)u, [xu − yv, y, y] = 2(xu − yv)u show that L′ coincides with the set of elements of W in the form (xv − yt)f + (xu − yv)g + [x, y]h, f, g, h ∈ K[t, u, v]. This means that L′ is the K[t, u, v]-module generated by xv − yt, xu − yv, [x, y]. If such an element is equal to 0 then we have x(vf + ug) + y(−tf − vg) + [x, y]h = 0 in the free K[t, u, v]-module xK[t, u, v]+yK[t, u, v]+[x, y]K[t, u, v]. Hence vf + ug = 0, tf + vg = 0, h = 0. Since the determinant v2 − tu of the system of the first two equations (with unknowns f, g) is different from 0 we have f = g = 0. (ii) The proof follows from by easy induction using the equations in the proof of (i). The relatively free Lie algebra Fm(sl2(K)), m ≥ 2, has a nice de- scription in terms of representation theory of the general linear group GLm(K), see [6, Exercise 12.6.10, p. 245]. But for m > 2 the center of the algebra Wm has a lot of defining relations, see [7] and there is no analogue of Lemma 1. This makes the explicit computations in Fm(sl2(K)) more complicated. Let R be a (not necessarily associative) graded K-algebra, R = ⊕ n≥0 R(n) = R(0) ⊕ R(1) ⊕ R(2) ⊕ · · · , Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 57 where R(n) is the homogeneous component of degree n in R, and R(0) = 0 or R(0) = K. We consider the formal power series topology on R induced by the filtration ω0(R) ⊇ ω1(R) ⊇ ω2(R) ⊇ · · · , ωn(R) = ⊕ k≥n R(k), n = 0, 1, 2, . . . , where ω(R) = R if R(0) = 0, and ω(R) is the augmentation ideal of R when R(0) = K. This is the topology in which the sets r + ωn(R), r ∈ R, n ≥ 0, form a basis for the open sets. We shall denote by R̂ the completion of R with respect to the formal power series topology and shall identify it with the Cartesian sum ⊕̂ n≥0R(n). The elements f ∈ R̂ are formal power series f = f0 + f1 + f2 + · · · , fn ∈ R(n), n = 0, 1, 2, . . . , A sequence f (k) = fk0 + fk1 + fk2 + · · · , k = 1, 2, . . . , where fkn ∈ R(n), converges to f = f0 + f1 + f2 + · · · , where fn ∈ R(n), if for every n0 there exists a k0 such that fkn = fn for all n < n0 and all k ≥ k0, i.e., for all sufficiently large k the first n0 terms of the formal power series f (k) are the same as the first n0 terms of f . Let Fm = Fm(G) be a relatively free algebra freely generated by x1, . . . , xm. Then Fm is graded and the nth homogeneous component is spanned by all commutators [xi1 , . . . , xin ] of length n. Hence the elements of F̂m are formal series of commutators. Since [F n m, u] = F n madu ⊂ F n+1 m for any u ∈ Fm, we derive that the inner automorphisms exp(adu) of F̂m are continuous automorphisms. Let W(n) be the subspace of W spanned by all monomials of total degree n in x, y. The elements f ∈ Ŵ are formal power series f = f0 + f1 + f2 + · · · , fn ∈ W(n), n = 0, 1, 2, . . . , and Ŵ is a free K[[t, u, v]]-module with free generators 1, x, y, [x, y], where K[[t, u, v]] is the algebra of formal power series in the variables t, u, v. Since L̂ is embedded canonically into Ŵ , Lemma 1 gives that (L̂)′ is a free K[[t, u, v]]-module with free generators xv − yt, xu − yv, [x, y] and L̂ = {αx+βy+a(xv−yt)+b(xu−yv)+c[x, y] \| α, β ∈ K, a, b, c ∈ K[[t, u, v]]}. Jo ur na l A lg eb ra D is cr et e M at h. 58 Inner Automorphisms The Baker-Campbell-Hausdorff formula gives the solution z of the equation ez = exey for non-commuting x and y, see e.g. [4] and [23]. If x, y are the generators of the free associative algebra A = K〈x, y〉, then z = x + y + [x, y] 2 − [x, y, x] 12 + [x, y, y] 12 − [x, y, x, y] 24 + · · · is a formal power series in the completion Â. The homogeneous compo- nents of z are Lie elements and z ∈ L̂2 where L2 is canonically embedded into A. The composition of two inner automorphisms in Inn(Ŵ ) is also an inner automorphism which can be obtained by the Baker-Campbell- Hausdorff formula. Hence, studying the inner automorphisms of L̂, it is convenient to work in Ŵ and to study the group of its inner automor- phisms. If δ is an endomorphism of the free K[[t, u, v]]-submodule of Ŵ with basis {x, y, [x, y]}, then we denote by M(δ) the associated matrix of δ with respect to this basis. If δ(x) = σ11x + σ21y + σ31[x, y], δ(y) = σ12x + σ22y + σ32[x, y], δ([x, y]) = σ13x + σ23y + σ33[x, y], σij ∈ K[[t, u, v]], then M(δ) =   σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33   . Clearly M(δ) behaves as a matrix of a usual linear operator. In particular, M(δ1δ2) = M(δ1)M(δ2). Since the derivation adX, X ∈ Ŵ , acts trivially on the centre of Ŵ , it is an endomorphism of Ŵ as a K[[t, u, v]]-module. Its restriction on the submodule generated by x, y, [x, y] satisfies the above conditions. Hence the matrix M(adX) is well defined, and similarly for the matrix M(exp(adX)). Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 59 2. The work of Baker and its consequences In the sequel we shall use calculations of Baker [2]. Let G3 be the three-dimensional complex simple Lie algebra with the basis {p1, p2, p3} and multiplication [p1, p2] = p1, [p1, p3] = 2p2, [p2, p3] = p3, and let X = x1p1 + x2p2 + x3p3 ∈ G3, x1, x2, x3 ∈ C. We denote respec- tively by P (X) and Q(exp(adX)) the matrices of the linear operators adX and exp(adX) with respect to the basis {p1, p2, p3}. Clearly, P (X) =   x2 −x1 0 2x3 0 −2x1 0 x3 −x2   . Theorem 2. (Baker [2]) (i) In the above notation, the matrices P (X) and Q(exp(adX)) satisfy P 3(X) = g(X)P (X), g(X) = x2 2 − 4x1x3, Q(exp(adX)) = I3 + P (X) ( 1 + g(X) 3! + g2(X) 5! + g3(X) 7! + · · · ) + + P 2(X) ( 1 2 + g(X) 4! + g2(X) 6! + · · · ) = I3 + A(X)P (X) + B(X)P 2(X), where I3 is the 3 × 3 identity matrix and A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) . (ii) Let X, Y ∈ G3 and let Z = z1p1 + z2p2 + z3p3 ∈ G3, z1, z2, z3 ∈ C, be such that exp(adZ) = exp(adX) exp(adY ). If Q(exp(adZ)) = (σij), then z1 = M1 2A , z2 = M2 2A , z3 = M3 2A , Jo ur na l A lg eb ra D is cr et e M at h. 60 Inner Automorphisms where M1 = −σ12 − 1 2 σ23, M2 = σ11 − σ33, M3 = 1 2 σ21 + σ32, A = sinh( √ g(Z))√ g(Z) , B = cosh( √ g(Z)) − 1 g(Z) , g(Z) = z2 2 − 4z1z3 = ( log 1 − M + √ 1 − 2M M )2 , M = 2σ11 + 2σ33 − 4 M2 2 − 2M1M3 = 2σ13 M2 1 = 2σ31 M2 3 = σ21 − 2σ32 M2M3 = σ23 − 2σ12 M1M2 = 1 − 2σ22 M1M3 = B A2 . Now we consider the algebra G3 from the consideration of Baker over an arbitrary field K of characteristic 0. The following generic construction is well known. Let m ≥ 2 and let K[xij ] = K[xij \| i = 1, 2, 3, j = 1, . . . , m] be the polynomial algebra in 3m variables. We consider the tensor product K[xij ] ⊗K G3 which is a Lie K-algebra. It is a free K[xij ]-module with basis {p1, p2, p3}. We shall omit the symbol ⊗ for the tensor product in the elements of K[xij ] ⊗K G3. We fix the elements Xj = x1jp1 + x2jp2 + x3jp3, j = 1, . . . , m. The Lie subalgebra generated by X1, . . . , Xm in K[xij ]⊗KG3 is isomorphic to the relatively free algebra Fm(G3) = Fm(varG3) and we shall identify both algebras. Clearly, the completion of Fm(G3) with respect to the formal power series topology is canonically embedded into K[[xij ]] ⊗K G3. For an arbitrary V = v1p1 + v2p2 + v3p3, vi ∈ K[[xij ]], the operators adV and exp(adV ) of K[[xij ]]⊗K G3 act as endomorphisms of the free K[[xij ]]-module K[[xij ]] ⊗K G3. We denote respectively by P (V ) and Q(exp(adV )) the matrices of these endomorphisms with respect to the basis {p1, p2, p3}. Our first result is the following corollary which translates the result of Baker given in Theorem 2 in terms of the above generic construction. Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 61 Corollary 1. If X, Y, Z ∈ K[[xij ]] ⊗K G3, X = x1p1+x2p2+x3p3, Y = y1p1+y2p2+y3p3, Z = z1p1+z2p2+z3p3, xi, yi, zi ∈ K[[xij ]], are such that exp(adZ) = exp(adX) exp(adY ), then the related matrices P (V ) and Q(exp(adV )), V = X, Y, Z, satisfy the relations of Theorem 2. Proof. Obviously, Z ∈ K[[xij ]] ⊗K G3 is uniquely determined from the condition exp(adZ) = exp(adX) exp(adY ). The computations of Baker in Theorem 2 are performed for power series which converge evaluated in C. They hold also over K[[xij ]] if the corresponding expressions have sense there. Working in K[[w]], we have that sinh( √ w)√ w = 1+ w 3! + w2 5! + w3 7! +· · · , cosh( √ w) − 1 w = 1 2! + w 4! + w2 6! +· · · . These expressions are well defined in K[[xij ]] for w = v2 2 − 4v1v3 when v1, v2, v3 ∈ K[[xij ]] are formal power series without constant term. Since the constant term of M = B/A2 in Theorem 2 is equal to 1/2, we verify directly that the logarithm in the expression of g(Z) has no constant term, and this implies that all formal expressions have sense in K[[xij ]]. The three-dimensional Lie algebra G3 is simple. Extending the field K to its algebraic closure K, we obtain an algebra isomorphic to sl2(K). It is well known that if two K-algebras G1 and G2 become isomorphic over an extension of the infinite field K, then G1 and G2 have the same polynomial identities. Hence the relatively free algebras Fm(G3) and Fm(sl2(K)) are isomorphic. Corollary 1 provides the multiplication formula for arbitrary inner automorphisms of the completion F̂m of Fm = Fm(G3) in terms of the generic elements X1, . . . , Xm. But it does not give the expression of exp(adZ)) as a formal power series of Lie commutators. In the next section we shall mimic the computations of Baker for the Lie algebra generated by two generic traceless 2 × 2 matrices and shall show how to find the Lie expression of the Baker-Campbell-Hausdorff formula modulo the polynomial identities of sl2(K). Jo ur na l A lg eb ra D is cr et e M at h. 62 Inner Automorphisms 3. Inner automorphisms of the Lie algebra of two generic matrices In this section we find the explicit form of the associated matrix of the inner automorphisms of Ŵ and a multiplication rule for Inn(Ŵ ). Then we transfer the obtained results to the algebra W/ω(W )c+1 and obtain the description of Inn(W/ω(W )c+1). Applying Lemma 1 we obtain immediately the corresponding results for the Lie algebras L̂ and L/Lc+1. Let Inn(Ŵ ) denote the set of all inner automorphisms of Ŵ which are of the form exp(adX), X ∈ Ŵ . As we already discussed, since Ŵ is a K[[t, u, v]]-module with the generators 1, x, y, [x, y] and adX acts trivially on 1 it is sufficient to know the action of inner automorphisms on only x, y, [x, y]. Theorem 3. Let X = ax+by +c[x, y], a, b, c ∈ K[[t, u, v]], be an element in Ŵ and let M(adX) =   −2cv −2cu 2(av + bu) 2ct 2cv −2(at + bv) b −a 0   be the associated matrix of adX. Then the associated matrix of exp(adX) is of the form M(exp(adX)) = I3 + A(X)M(adX) + B(X)M2(adX), where A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) , g(X) = 2(a2t + 2abv + b2u + 2c2(v2 − tu)). Proof. From Theorem 1 (iv) we know that [x, [x, y]] = 2(−xv + yt); [y, [x, y]] = 2(−xu + yv). By easy calculations we obtain that xadX = −2cvx + 2cty + b[x, y], yadX = −2cux + 2cvy − a[x, y], [x, y]adX = 2(av + bu)x − 2(at + bv)y, Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 63 M(adX) =   −2cv −2cu 2(av + bu) 2ct 2cv −2(at + bv) b −a 0   , M2(adX) = ( 4c2w + 2b(av + bu) −2a(av + bu) −4acw −2b(at + bv) 4c2w + 2a(at + bv) −4bcw −2c(at + bv) −2c(av + bu) 2a(at + bv) + 2b(av + bu) ) , w = v2 − tu. Calculating M3(adX) we obtain that M3(adX) = g(X)M(adX), g(X) = 2(a2t + 2abv + b2u + 2c2(v2 − tu)). Following the steps in Theorem 2 we obtain that M(exp(adX)) = I3 + A(X)M(adX) + B(X)M2(adX), A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) . Theorem 4. If Q = M(exp(adX)) is the associated matrix of an inner automorphism for an element X in Ŵ , then the associated matrix M(adX) of the derivation adX is M(adX) = A(X) B(X) (Q − I3) − 1 2A(X) (Q2 − I3), where cosh( √ g(X)) = 1 2 (tr(Q) − 1) , A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) . Proof. Since M3(adX) = g(X)M(adX), the Cayley-Hamilton theorem gives that M(adX) ∼   0 0 0 0 √ g(X) 0 0 0 − √ g(X)   , Q = M(exp(adX)) ∼   1 0 0 0 exp( √ g(X)) 0 0 0 exp(− √ g(X))   . Jo ur na l A lg eb ra D is cr et e M at h. 64 Inner Automorphisms Hence the trace of Q is equal to 1 + exp( √ g(X)) + exp(− √ g(X)) which gives the expression for cosh( √ g(X)) = 1 2 ( exp( √ g(X)) + exp(− √ g(X)) ) = 1 2 (tr(Q) − 1) . So we have also A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) . We consider the expressions for Q and Q2 Q = I3 + A(X)M(adX) + B(X)M2(adX), Q2 = I3 + 2A(X)(1 + B(X)g(X))M(exp(adX)) +(A2(X) + B2(X)g(X) + 2B(X))M2(exp(adX)) as a linear system with unknowns M(adX) and M2(adX). Using the equality A2(X) = B2(X)g(X) + 2B(X), the solution of the system gives M(adX) = = −(A2(X) + B2(X)g(X) + 2B(X))(Q − I3) + B(X)(Q2 − I3) −A(X)(A2(X) + B2(X)g(X) + 2B(X)) + 2A(X)B(X)(1 + B(X)g(X)) = −2A2(X)(Q − I3) + B(X)(Q2 − I3) −2A(X)B(X) . (1) Remark 1. Theorem 4 shows how to find X ∈ Ŵ if we know the matrix Q = M(exp(adX)). Since cosh( √ g(X)) = (tr(Q) − 1)/2 is of the form 1 + h for some h ∈ ω(K[[t, u, v]]), we use the formula √ g(X) = arccosh(1 + h) = log ( 1 + h + √ h2 + 2h ) which can be easily verified, and we obtain expressions for g(X), A(X) and B(X). Combined with Theorem 3 we obtain also the multiplication rule for the group of inner automorphisms of Ŵ : Given adX and adY , we calculate consecutively Q1 = M(exp(adX)), Q2 = M(exp(adY )), their product Q = M(exp(adZ)) = Q1Q2 and finally Z. Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 65 Let us denote by ω the augmentation ideal of the polynomial algebra K[t, u, v] consisting of the polynomials without constant terms and let us denote its completion ω̂ ⊂ K[[t, u, v]] with respect to the formal power formal series. Since the elements t = 2x2, u = 2y2, v = xy + yx are of even degree in W , the associated matrices of the automorphisms of Ŵ modulo ω̂(W ) c+1 , c ≥ 3, contain the entries in the factor algebra K[t, u, v]/ω[(c+1)/2]. As a consequence of our Theorems 3 and 4 for Inn(Ŵ ) we imme- diately obtain the description of the group of inner automorphisms of W/ω(W )c+1. We shall give the results for the associated matrices only. The multiplication rule for the group Inn(W/ω(W )c+1) can be stated similarly. Corollary 2. Let X = ax + by + c[x, y], a, b, c ∈ K[[t, u, v]], be an ele- ment in Ŵ and let M(adX) be the associated matrix of adX. Then the associated matrix M(exp(adX)) of the inner automorphism exp(adX) of W/ω(W )c+1 ∼= Ŵ/ω(Ŵ )c+1 is of the form M(exp(adX)) = I3+A(X)M(adX)+B(X)M2(adX) (mod M3(ω[(c+1)/2])), A(X) = sinh( √ g(X))√ g(X) , B(X) = cosh( √ g(X)) − 1 g(X) , where M(adX) =   −2cv −2cu 2(av + bu) 2ct 2cv −2(at + bv) b −a 0   , g(X) = 2(a2t + 2abv + b2u + 2c2(v2 − tu)) and M3(ω[(c+1)/2]) is the 3 × 3 matrix algebra with entries from the [(c + 1)/2]-th power of the augmentation ideal of K[t, u, v]. We shall complete the paper with two examples which are in the spirit of our considerations. They are based on the simplest relations which guarantee that the corresponding exponentials are well defined. We believe that the existence of such examples is folklorely known although we were not able to find any references. Example 1. Let R be the two-generated associative algebra with pre- sentation R = K〈u, v \| u2 = v2 = 0〉. Jo ur na l A lg eb ra D is cr et e M at h. 66 Inner Automorphisms Then eu = 1 + u, ev = 1 + v are elements of R but the equation euev = ew has no solution w in R. Proof. Since R is a monomial algebra it has a basis as a K-vector space consisting of all non-commutative monomials which do not contain as subwords u2 and v2. The completion R̂ of R with respect to the formal power series topology consists of formal power series of such monomials. Consider the 2 × 2 matrices U = ( 0 a 0 0 ) , V = ( 0 0 b 0 ) , where a, b are algebraically independent commuting variables. Since U2 = V 2 = 0, the K-subalgebra R1 of M2(K[a, b]) generated by U and V is a homomorphic image of the algebra R. Algebras like R1 appear in the paper by Belov [3]. Tracing his considerations, it follows that the algebras R and R1 are isomorphic. See also [12] for a similar matrix realization of algebras generated by two quadratic elements and direct proof of the isomorphism of R and R1. We shall work in the algebra R1 and its completion R̂1 and show that the solution W of the equation eU eV = eW in R̂1 does not belong to R1. Direct computations give that eU = ( 1 a 0 1 ) , eV = ( 1 0 b 1 ) , eU eV = ( 1 + ab a b 1 ) = I + T = eW , I = ( 1 0 0 1 ) , T = ( ab a b 0 ) , W = log(I + T ) = ∑ n≥1 (−1)n−1T n n . Let c = ab and let ξ1,2 = c ± √ c(c + 4) 2 be the solutions of the quadratic equation ξ2 = c(ξ + 1). Since T 2 = ab(T + I) = c(T + I), by easy induction and the Viète formulas (or using linear recurrence relations arguments) we obtain T n = 1√ c(c + 4) ( (ξn 1 − ξn 2 )T + c(ξn−1 1 − ξn−1 2 )I ) . Jo ur na l A lg eb ra D is cr et e M at h. V. Drensky, Ş. Fındık 67 Hence W = 1√ c(c + 4) ∑ n≥1 (−1)n−1 n ( (ξn 1 − ξn 2 )T + c(ξn−1 1 − ξn−1 2 )I ) = 1√ c(c + 4) ( (log(1 + ξ1) − log(1 + ξ2))T + c ( log(1 + ξ1) ξ1 − log(1 + ξ2) ξ2 ) I ) . The equation (1 + ξ1)(1 + ξ2) = 1 implies log(1 + ξ1) = − log(1 + ξ2), W = log(1 + ξ1)√ c(c + 4) ( 2T + c ( 1 ξ1 + 1 ξ2 ) I ) = log(1 + ξ1)√ c(c + 4) (2T − cI). If W ∈ R1, we would have that log(1 + ξ1)√ c(c + 4) is a polynomial in K[a, b] and hence in K[c]. In this way log(1 + ξ1) = f(c) √ c(c + 4) for some f(c) ∈ K[c]. Solving the equation ξ2 1 = c(ξ1 + 1) with respect to c we obtain c = ξ2 1 1 + ξ1 = ∑ n≥2 (−1)nξn 1 , √ c(c + 4) = ξ1(2 + ξ1) 1 + ξ1 . Since c belongs to the augmentation ideal of the algebra K[[ξ1]] of formal power series in ξ1 we derive that K[[c]] ⊂ K[[ξ1]]. Then log(1 + ξ1)√ c(c + 4) = 1 + ξ1 ξ1(2 + ξ1) log(1 + ξ1) = f ( ξ2 1 1 + ξ1 ) , log(1 + ξ1) = ξ1(2 + ξ1) 1 + ξ1 f ( ξ2 1 1 + ξ1 ) . This is a contradiction because log(1 + ξ1) is not a rational function in ξ1. Example 2. Let L2 be the free Lie algebra freely generated by u, v and let H be its factor algebra modulo the ideal generated by all commu- tators [z, u, u], [z, v, v], z ∈ L2. Then exp(adu) and exp(adv) are well defined inner automorphisms of H but the solution w of the equation exp(adu) exp(adv) = exp(adw) belongs to Ĥ and not to H. Proof. Consider the algebra M(L2) of the multiplications of the Lie algebra L2. It is the associative subalgebra of the algebra of linear operators of the vector space L2 generated by the operators adz, z ∈ L2. Since the Jo ur na l A lg eb ra D is cr et e M at h. 68 Inner Automorphisms elements of L2 are linear combinations of left normed commutators of u and v, the algebra M(L2) is generated by adu and adv. It is well known that M(L2) is isomorphic to the two-generated free associative algebra. From the definition of H as the factor Lie algebra H = L2/([z, u, u], [z, v, v] \| z ∈ L2) we derive that the algebra M(H) of the multiplications of H satisfies the relations ad2u = ad2v = 0. The freedom of M(L2) implies that the above two equalities are the defining relations of M(H) and it has the presentation M = M(H) = K〈adu, adv \| ad2u = ad2v = 0〉. Let M̂ be the completion of M with respect to the formal power series topology. The exponentials exp(adu) = 1 + adu, exp(adv) = 1 + adv are inner automorphisms of H. Now Example 1 gives that the solution adw, w ∈ Ĥ, of the equation exp(adu) exp(adv) = exp(adw) which is in M̂ does not belong to M, i.e., w does not belong to H. Acknowledgements The second named author is grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences for the creative atmosphere and the warm hospitality during his visit when most of this project was carried out. References [1] Yu.A. Bahturin, Identical Relations in Lie Algebras (Russian), “Nauka”, Moscow, 1985. Translation: VNU Science Press, Utrecht, 1987. [2] H.F. 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Free Groups, Polyno- mials, Free Algebras, CMS Books in Mathematics, Springer, New York, 2004. [22] Yu.P. Razmyslov, Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (Russian), Algebra i Logika N.12, 1973, pp. 83-113. Translation: Algebra and Logic N.12, 1973, pp. 47-63. [23] J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin Inc., New York- Amsterdam, 1965. Jo ur na l A lg eb ra D is cr et e M at h. 70 Inner Automorphisms Contact information V. Drensky Institute of Mathematics and Informatics, Bul- garian Academy of Sciences, 1113 Sofia, Bulgaria E-Mail: drensky@math.bas.bg URL: http://www.math.bas.bg/ Ş. Fındık Department of Mathematics, Çukurova Univer- sity, 01330 Balcalı, Adana, Turkey E-Mail: sfindik@cu.edu.tr URL: http://math.cu.edu.tr/ Received by the editors: 30.04.2012 and in final form 23.05.2012. V. Drensky, Ş. Fındık

Inner automorphisms of Lie algebras related with generic 2 × 2 matrices

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