Abstract
We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra \(I{\mathfrak u}_n{:}{=}{\mathfrak u}_n < imes {\mathfrak u}_n^*\), where \({\mathfrak u}_n\) is the Lie algebra of upper triangular \(n\times n\) matrices. Our results also extend to \(\mathfrak {gl}_{n+}^\epsilon \), a “solvable approximation” of \(\mathfrak {gl}_n\), as defined within.
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Given any Lie algebra \({\mathfrak a}\) one may form its “inhomogeneous version” \(I{\mathfrak a}{:}{=}{\mathfrak a} < imes {\mathfrak a}^*\), its semidirect product with its dual \({\mathfrak a}^*\) where \({\mathfrak a}^*\) is considered as an Abelian Lie^{Footnote 1} algebra and \({\mathfrak a}\) acts on \({\mathfrak a}^*\) via the coadjoint action. (Over \({\mathbb R}\) if \({\mathfrak a}= \mathfrak {so}_3\) then \({\mathfrak a}^*={\mathbb R}^3\) and so \(I{\mathfrak a}= \mathfrak {so}_3 < imes {\mathbb R}^3\) is the Lie algebra of the Euclidean group of rotations and translations, explaining the name).
In general, we care about \(I{\mathfrak a}\). It is a special case of the Drinfel’d double / Manin triple construction [12, 13] when the cobracket is 0. These Lie algebras occur in the study of the KashiwaraVergne problem [1, 7] and they provide the simplest quantum algebra context for the Alexander polynomial [2, 6]. We care especially for the case where \({\mathfrak a}\) is a Borel subalgebra of a semisimple Lie algebra (e.g., upper triangular matrices) as then the algebras \(I{\mathfrak a}\) are the \(\epsilon =0\) “base case” for “solvable approximation” [3,4,5, 8, 9], and their automorphisms are expected to become symmetries of the resulting knot invariants.
Let \({\mathfrak u}_n\) be the Lie algebra of upper triangular \(n\times n\) matrices over a field in which 2 is invertible. Beyond inner automorphisms, \({\mathfrak u}_n\) and hence \(I{\mathfrak u}_n\) has one obvious and expected antiautomorphism \(\Phi \) corresponding to flipping matrices along their antimaindiagonal, as shown in the first image of Fig. . With \(x_{ij}\) denoting the \(n\times n\) matrix with 1 in position (ij) and zero everywhere else (\(i\le j\) in \({\mathfrak u}_n\)), \(\Phi \) is given by \(x_{ij}\mapsto x_{n+1j,n+1i}\).
There clearly isn’t an automorphism of \({\mathfrak u}_n\) 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 (\({\mathfrak u}_n\) with the last column excluded), does extend to an automorphism of \(I{\mathfrak u}_n\) as in the theorem below.
With the basis \(\{x_{ij}\}_{1\le i<j\le n} \cup \{a_i=x_{ii}\}_{1\le i\le n}\) for \({\mathfrak u}_n\) and dual basis \(\{x_{ji}\}_{1\le i<j\le n}\cup \{b_i\}_{1\le i\le n}\) for \({\mathfrak u}_n^*\) (and duality \(\langle x_{kl},x_{ij}\rangle =\delta _{li}\delta _{jk}\), \(\langle b_i,a_j\rangle =2\delta _{ij}\),^{Footnote 2} and \(\langle x_{ji},a_k\rangle = \langle b_k,x_{ij}\rangle = 0\)), the map \(\Psi :I{\mathfrak u}_n\rightarrow I{\mathfrak u}_n\) defined by “incrementing all indices by 1 mod n” (precisely, if \(\psi \) is the singlecycle permutation \(\psi =(123\ldots n)\) then \(\Psi \) is defined by \(\Psi (x_{ij})=x_{\psi (i)\psi (j)}\), \(\Psi (a_i)=a_{\psi (i)}\), and \(\Psi (b_i)=b_{\psi (i)}\)) is a Lie algebra automorphism of \(I{\mathfrak u}_n\).
Note that our choice of bases, using similar symbols \(x_{ij}\) / \(x_{ji}\) for the nondiagonal matrices and their duals, hides the intricacy of \(\Psi \); e.g., \(\Psi :x_{n1,n}\mapsto x_{n1}\) maps an element of \({\mathfrak u}_n\) to an element of \({\mathfrak u}_n^*\) (also see Fig. 1, right).
It may be tempting to think that \(\Psi \) has a simple explanation in \(\mathfrak {gl}_n\) language: \({\mathfrak u}_n\) is a subset of \(\mathfrak {gl}_n\), \(\mathfrak {gl}_n\) has a metric (the Killing form) such that the dual of \(x_{ij}\) is \(x_{ji}\) as is the case for us, and every permutation of the indices induces an automorphism of \(\mathfrak {gl}_n\). But this explains nothing and too much: nothing because the bracket of \(I{\mathfrak u}_n\) simply isn’t the bracket of \(\mathfrak {gl}_n\) (even away from the diagonal matrices), and too much because every permutation of indices induces an automorphism of \(\mathfrak {gl}_n\), whereas only \(\psi \) and its powers induce automorphisms of \(I{\mathfrak u}_n\).
Recall that as a vector space \(I{\mathfrak u}_n={\mathfrak u}_n\oplus {\mathfrak u}_n^*\), yet with bracket \([(x,f),(y,g)]=\left( [x,y],x\cdot gy\cdot f\right) \) where \(\cdot \) denotes the coadjoint action, \((x\cdot f)(v)=f([v,x])\). With that and some case checking and explicit computations, the commutation relations of \(I{\mathfrak u}_n\) are given by
Here \(\chi \) is the indicator function of truth (so \(\chi _{5<7}=1\) while \(\chi _{7<5}=0\)), and \(\lambda (x_{ij})\) is the “length” of \(x_{ij}\), defined by \(\lambda (x_{ij}){:}{=}{\left\{ \begin{array}{ll} ji &{} i<j \\ n(ij) &{} i>j \end{array}\right. }\).
It is easy to verify that the length \(\lambda (x_{ij})\) is \(\Psi \)invariant, and hence everything in (1) is \(\Psi \)equivariant. \(\square \)
\(I{\mathfrak u}_n\) is a solvable Lie algebra (as a semidirect 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):
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\(=\lambda (x_{ij})\).

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 \(p\ge 2\), every generator in layer p is the bracket of a generator in layer 1 with a generator in layer \(p1\).

In layer p, the first \(np\) generators indicated belong to \({\mathfrak u}_n\) and the last p belong to \({\mathfrak u}_n^*\). So as we go down, \({\mathfrak u}_n^*\) slowly “overtakes” the table.

The automorphism \(\Psi \) 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 antiautomorphism \(\Phi \) acts by mirroring the \({\mathfrak u}_n\) part of each layer left to right and by doing the same to the \({\mathfrak u}_n^*\) part, without mixing the two parts.

Note that \(I{\mathfrak u}_n\) can be metrized by pairing the \({\mathfrak u}_n\) summand with the \({\mathfrak u}_n^*\) one. The metric only pairs generators indicated in layer p with generators indicated in layer \((np)\).
Note also that the brackets of the generators indicated in layer 1 yield the generators indicated in layer 2 as follows:
(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 antiautomorphisms of \(I{\mathfrak u}_n\) (all automorphisms and antiautomorphisms modulo inner automorphisms) is \(D_n\), generated by \(\Phi \) and \(\Psi \). We did not endeavor to prove this formally.
Extension. The Drinfel’d double / Manin triple construction [12, 13], when applied to \({\mathfrak u}_n\), is a way to reconstruct \(\mathfrak {gl}_n\) from its subalgebras of upper triangular matrices \({\mathfrak u}_n\) and lower triangular matrices \({\mathfrak l}_n\). Precisely, one endows the vector space \({\mathfrak g}={\mathfrak u}_n\oplus {\mathfrak l}_n\) with a nondegenerate symmetric bilinear form by declaring that the subspaces \({\mathfrak u}_n\) and \({\mathfrak l}_n\) are isotropic (\(\langle {\mathfrak u}_n,{\mathfrak u}_n\rangle =\langle {\mathfrak l}_n,{\mathfrak l}_n\rangle =0\)) and by setting \(\langle x_{kl},x_{ij}\rangle =\delta _{li}\delta _{jk}\), \(\langle b_i,a_j\rangle =2\delta _{ij}\), and \(\langle x_{ji},a_k\rangle = \langle b_k,x_{ij}\rangle = 0\) as in Theorem 1 and where \(a_i\) stands for the diagonal matrix \(x_{ii}\) considered as an element of \({\mathfrak u}_n\) and \(b_i\) stands for the same matrix as an element of \({\mathfrak l}_n\). There is then a unique bracket on \({\mathfrak g}\) that extends the brackets on the summands \({\mathfrak u}_n\) and \({\mathfrak l}_n\) and relative to which the inner product of \({\mathfrak g}\) is invariant^{Footnote 3} With our judicious choice of bilinear form, this bracket on \({\mathfrak g}\) satisfies the Jacobi identity and turns \({\mathfrak g}\) into a Lie algebra isomorphic to \(\mathfrak {gl}_{n+}=\mathfrak {gl}_n\oplus {\mathfrak h}'_n\), where \({\mathfrak h}'_n\) denotes a second copy of the diagonal matrices in \(\mathfrak {gl}_n\).
We let \(\mathfrak {gl}_{n+}^\epsilon \) be the InonuWigner [14] contraction of \({\mathfrak g}\) along its \({\mathfrak l}_n\) summand, with parameter \(\epsilon \).^{Footnote 4} All that this means is that the bracket of \({\mathfrak l}_n\) gets multiplied by \(\epsilon \) to give \({\mathfrak l}_n^\epsilon \), and then the Drinfel’d double / Manin triple construction is repeated starting with \({\mathfrak u}_n\oplus {\mathfrak l}_n^\epsilon \), without changing the bilinear form. The result is a Lie algebra \(\mathfrak {gl}_{n+}^\epsilon \) over the ring of polynomials in \(\epsilon \) which specializes to \(I{\mathfrak u}_n\) at \(\epsilon =0\) and which is isomorphic to \(\mathfrak {gl}_n\oplus {\mathfrak h}'_n\) when \(\epsilon \) is invertible.^{Footnote 5} We care about \(\mathfrak {gl}_{n+}^\epsilon \) a lot [3,4,5, 8,9,10, 18]; when reduced modulo \(\epsilon ^{k+1}=0\) for some natural number k it becomes solvable, and hence a “solvable approximation” of \(\mathfrak {gl}_n\) with applications to computability of knot invariants.
With the same conventions as in Theorem 1 the map \(\Psi \) is also a Lie algebra automorphism of \(\mathfrak {gl}_{n+}^\epsilon \).
FormalPara ProofBy some case checking and explicit computations, the commutation relations of \(\mathfrak {gl}_{n+}^\epsilon \) are given by
where \(\chi ^{\epsilon }_{\texttt {True}}=1\) and \(\chi ^{\epsilon }_{\texttt {False}}=\epsilon \). These relations are clearly \(\Psi \)equivariant. \(\square \)
FormalPara Note 3There is of course an “\(\mathfrak {sl}\)” version of everything, in which linear combinations \(\sum \alpha _ia_i\) and \(\sum \beta _ib_i\) are allowed only if \(\sum \alpha _i=\sum \beta _i=0\), with obvious modifications throughout.
FormalPara Note 4At \(n=2\) and \(\epsilon =0\), the algebra \(\mathfrak {sl}_{2+}^0\) is the “diamond Lie algebra” of [15, Chapter 4.3], which is sometimes called “the NappiWitten algebra” [17]. With \(a=(a_1a_2)/2\), \(x=x_{12}\), \(y=x_{21}\), and \(b=(b_1b_2)/2\), it is
Here \(\Phi :(a,x,y,b)\mapsto (a,x,y,b)\) and \(\Psi :(a,x,y,b)\mapsto (a,y,x,b)\).
FormalPara Note 5Upon circulating this paper as an eprint we received a note from A. Knutson informing us of [16, esp. sec. 2.3], where the algebra \(I{\mathfrak u}_n\) (except reduced modulo \(\langle b_i\rangle \) and considered globally rather than infinitesimally) is considered from a different perspective. It is shown to be a subquotient of the affine algebra \(\widehat{\mathfrak {gl}_n}\) 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.
FormalPara Note 6A day later we received a note [11] from M. Bulois and N. Ressayre reporting on an explanation of Theorem 1 in terms of affine KacMoody Lie algebras, similarly to Note 5.
Notes
Two Norwegians!
Indeed we only need to determine [u, l] for \(u\in {\mathfrak u}_n\) and \(l\in {\mathfrak l}_n\). Writing \([u,l]=u'+l'\) with \(u'\in {\mathfrak u}_n\) and \(l'\in {\mathfrak l}_n\), we determine \(u'\) using the nondegeneracy of the inner product from the relation \(\langle u',l''\rangle = \langle [u,l],l''\rangle = \langle u,[l,l'']\rangle \) which holds for every \(l''\in {\mathfrak l}_n\) due to the invariance of \(\langle ,\,\rangle \). Similarly \(l'\) is determined from \(\langle l',u''\rangle = \langle [u,l],u''\rangle = \langle l,[u'',u]\rangle \).
Hence \(\mathfrak {gl}_{n+}^\epsilon \rightarrow I{\mathfrak u}_n\) is a counterexample to the feeltrue statement “a contraction of a direct sum is a direct sum”. Indeed with notation as in Theorem 2, as \(\epsilon \rightarrow 0\) the decomposition \(\mathfrak {gl}_{n+}^\epsilon = \mathfrak {gl}_n\oplus {\mathfrak h}'_n = \langle x_{ij},b_i+\epsilon a_i\rangle \oplus \langle b_i\epsilon a_i\rangle \) collapses.
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Acknowledgements
We wish to thank M. Bulois, A. Knutson, A. Referees, N. Ressayre, and N. Williams for comments and suggestions.
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This work was partially supported by grant RGPIN201804350 of the Natural Sciences and Engineering Research Council of Canada. It is available in electronic form, along with source files and a verification Mathematica notebook at http://drorbn.net/UnexpectedCyclic and at arXiv:2002.00697.
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BarNatan, D., Veen, R.v.d. An Unexpected Cyclic Symmetry of \(I{\mathfrak u}_n\). Abh. Math. Semin. Univ. Hambg. 93, 71–76 (2023). https://doi.org/10.1007/s1218802300266w
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DOI: https://doi.org/10.1007/s1218802300266w