An Unexpected Cyclic Symmetry of Iun\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I{\mathfrak u}_n$$\end{document}

We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra Iun:=un⋉un∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I{\mathfrak u}_n{:}{=}{\mathfrak u}_n < imes {\mathfrak u}_n^*$$\end{document}, where un\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak u}_n$$\end{document} is the Lie algebra of upper triangular n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document} matrices. Our results also extend to gln+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_{n+}^\epsilon $$\end{document}, a “solvable approximation” of gln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_n$$\end{document}, as defined within.

An expected anti-automorphism (left), an unexpected automorphism (middle), and an alternative presentation of the "layers" table (right) Let u n be the Lie algebra of upper triangular n × n matrices over a field in which 2 is invertible. Beyond inner automorphisms, u n and hence I u n has one obvious and expected anti-automorphism corresponding to flipping matrices along their anti-main-diagonal, as shown in the first image of Fig. 1. With x i j denoting the n × n matrix with 1 in position (i j) and zero everywhere else (i ≤ j in u n ), is given by There clearly isn't an automorphism of u n that acts by "sliding down and right parallel to the main diagonal", as in the second image in Fig. 1. Where would the last column go? Yet the sliding map, when restricted to where it is clearly defined (u n with the last column excluded), does extend to an automorphism of I u n as in the theorem below.

Theorem 1 With the basis {x
Note that our choice of bases, using similar symbols x i j / x ji for the non-diagonal matrices and their duals, hides the intricacy of ; e.g., : x n−1,n → x n1 maps an element of u n to an element of u * n (also see Fig. 1, right). It may be tempting to think that has a simple explanation in gl n language: u n is a subset of gl n , gl n has a metric (the Killing form) such that the dual of x i j is x ji as is the case for us, and every permutation of the indices induces an automorphism of gl n . But this explains nothing and too much: nothing because the bracket of I u n simply isn't the bracket of gl n (even away from the diagonal matrices), and too much because every permutation of indices induces an automorphism of gl n , whereas only ψ and its powers induce automorphisms of I u n .

Proof of Theorem 1 Recall that as a vector space
. With that and some case checking and explicit computations, the commutation relations of I u n are given by (1) 2 The awkward factor of 2 in b i , a j is irrelevant for Theorem 1 yet crucial for Theorem 2. Removing this factor removes the factor 1 2 in (1), see also Chapter 4 of [13] Here χ is the indicator function of truth (so χ 5<7 = 1 while χ 7<5 = 0), and λ(x i j ) is the It is easy to verify that the length λ(x i j ) is -invariant, and hence everything in (1) is -equivariant.
I u n is a solvable Lie algebra (as a semi-direct product of solvable with Abelian, and as will be obvious from the table below). It is therefore interesting to look at the structure of its commutator subalgebras. This structure is summarized in the following table (an alternative view is in Fig. 1): layer 0 g = I u n a 1 → a 2 → · · · → a n−2 → a n−1 → a n → layer 1 In this table (all assertions are easy to verify): • Apart for the treatment of the a i 's and the b i 's, layer=length= λ(x i j ).
• The layers indicate a filtration; each layer should be considered to contain all the ones below it. The generators marked at each layer generate it modulo the layers below. • The bracket of an element at layer p with an element of layer q is in layer p + q (and it must vanish if p + q > n). • If p ≥ 2, every generator in layer p is the bracket of a generator in layer 1 with a generator in layer p − 1. • In layer p, the first n − p generators indicated belong to u n and the last p belong to u * n . So as we go down, u * n slowly "overtakes" the table. • The automorphism acts by following the arrows and shifting every generator one step to the right (and pushing the rightmost generator in each layer back to the left). • The anti-automorphism acts by mirroring the u n part of each layer left to right and by doing the same to the u * n part, without mixing the two parts. • Note that I u n can be metrized by pairing the u n summand with the u * n one. The metric only pairs generators indicated in layer p with generators indicated in layer (n − p).
Note also that the brackets of the generators indicated in layer 1 yield the generators indicated in layer 2 as follows: x 12 x 23 x 34 · · · x n−1,n x n1 x 13 x 24 · · · x n−1,1 x n2 (with the diagram continued cyclically). The symmetry group of the above cycle is the dihedral group D n and this strongly suggests that the group of outer automorphisms and anti-automorphisms of I u n (all automorphisms and anti-automorphisms modulo inner automorphisms) is D n , generated by and . We did not endeavor to prove this formally. Extension. The Drinfel'd double / Manin triple construction [12,13], when applied to u n , is a way to reconstruct gl n from its subalgebras of upper triangular matrices u n and lower triangular matrices l n . Precisely, one endows the vector space g = u n ⊕ l n with a nondegenerate symmetric bilinear form by declaring that the subspaces u n and l n are isotropic ( u n , u n = l n , l n = 0) and by setting x kl , x i j = δ li δ jk , b i , a j = 2δ i j , and x ji , a k = b k , x i j = 0 as in Theorem 1 and where a i stands for the diagonal matrix x ii considered as an element of u n and b i stands for the same matrix as an element of l n . There is then a unique bracket on g that extends the brackets on the summands u n and l n and relative to which the inner product of g is invariant 3 With our judicious choice of bilinear form, this bracket on g satisfies the Jacobi identity and turns g into a Lie algebra isomorphic to gl n+ = gl n ⊕ h n , where h n denotes a second copy of the diagonal matrices in gl n . We let gl n+ be the Inonu-Wigner [14] contraction of g along its l n summand, with parameter . 4 All that this means is that the bracket of l n gets multiplied by to give l n , and then the Drinfel'd double / Manin triple construction is repeated starting with u n ⊕ l n , without changing the bilinear form. The result is a Lie algebra gl n+ over the ring of polynomials in which specializes to I u n at = 0 and which is isomorphic to gl n ⊕ h n when is invertible. 5 We care about gl n+ a lot [3-5, 8-10, 18]; when reduced modulo k+1 = 0 for some natural number k it becomes solvable, and hence a "solvable approximation" of gl n with applications to computability of knot invariants.

Theorem 2 With the same conventions as in Theorem 1 the map
is also a Lie algebra automorphism of gl n+ .
Proof By some case checking and explicit computations, the commutation relations of gl n+ are given by where χ True = 1 and χ False = . These relations are clearly -equivariant.
Note 3 There is of course an "sl" version of everything, in which linear combinations α i a i and β i b i are allowed only if α i = β i = 0, with obvious modifications throughout.

Note 4
At n = 2 and = 0, the algebra sl 0 2+ is the "diamond Lie algebra" of [15,Chapter 4.3], which is sometimes called "the Nappi-Witten algebra" [17]. With a = (a 1 − a 2 )/2, Here : (a, x, y, b) → (−a, x, y, −b) and : (a, x, y, b) → (−a, y, x, −b). 3 Indeed we only need to determine [u, l] for u ∈ u n and l ∈ l n . Writing [u, l] = u + l with u ∈ u n and l ∈ l n , we determine u using the non-degeneracy of the inner product from the relation u , l = [u, l], l = u, [l, l ] which holds for every l ∈ l n due to the invariance of , . Similarly l is determined from l , u = [u, l], u = l, [u , u] . 4 Alternatively, make u n into a Lie bialgebra with cobracket δ using its given duality with l n , and double it as in [12,13] but using the cobracket δ. 5 Hence gl n+ → I u n is a counter-example to the feel-true statement "a contraction of a direct sum is a direct sum". Indeed with notation as in Theorem 2, as → 0 the decomposition gl n+ = gl n ⊕ h n = x i j , b i + a i ⊕ b i − a i collapses.
Note 5 Upon circulating this paper as an eprint we received a note from A. Knutson informing us of [16, esp. sec. 2.3], where the algebra I u n (except reduced modulo b i and considered globally rather than infinitesimally) is considered from a different perspective. It is shown to be a subquotient of the affine algebra gl n in a manner preserved by its automorphisms corresponding to its Dynkin diagram, which is a cycle. Similar comments apply to the other algebras considered here.
Note 6 A day later we received a note [11] from M. Bulois and N. Ressayre reporting on an explanation of Theorem 1 in terms of affine Kac-Moody Lie algebras, similarly to Note 5.