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We ﬁnd and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra I u :=u u ,where u is the Lie algebra of upper triangular n × n matrices. Our n n n results also extend to gl , a “solvable approximation” of gl , as deﬁned within. n+ n Keywords Lie algebras · Lie bialgebras · Lie algebra automorphism · Solvable approximation · Triangular matrices Mathematics Subject Classiﬁcation 17B40 · 57M25 Given any Lie algebra a one may form its “inhomogeneous version” I a:=aa , its semidirect ∗ ∗ 1 ∗ product with its dual a where a is considered as an Abelian Lie algebra and a acts on a ∗ 3 3 via the coadjoint action. (Over R if a = so then a = R and so I a = so R is the Lie 3 3 algebra of the Euclidean group of rotations and translations, explaining the name). In general, we care about I 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 Kashiwara-Vergne problem [1, 7] and they provide the simplest quantum algebra context for the Alexander polynomial [2, 6]. We care especially for the case where a is a Borel subalgebra of a semi-simple Lie algebra (e.g., upper triangular matrices) as then the algebras I a are the = 0 “base case” for “solvable approximation” [3–5, 8, 9], and their automorphisms are expected to become symmetries of the resulting knot invariants. Two Norwegians! Communicated by Managing Editors. This work was partially supported by grant RGPIN-2018-04350 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. B Dror Bar-Natan drorbn@math.toronto.edu http://www.math.toronto.edu/drorbn B Roland van der Veen roland.mathematics@gmail.com http://www.rolandvdv.nl/ Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada University of Groningen, Bernoulli Institute, P.O. Box 407, 9700 AK Groningen, The Netherlands 123 D. Bar-Natan, R.V.D. Veen Fig. 1 An expected anti-automorphism (left), an unexpected automorphism (middle), and an alternative presentation of the “layers” table (right) Let u be the Lie algebra of upper triangular n × n matrices over a ﬁeld in which 2 is invertible. Beyond inner automorphisms, u and hence I u has one obvious and expected n n anti-automorphism corresponding to ﬂipping matrices along their anti-main-diagonal, as showninthe ﬁrst imageofFig. 1. With x denoting the n × n matrix with 1 in position (ij ) ij and zero everywhere else (i ≤ j in u ), is given by x → x . n ij n+1− j ,n+1−i There clearly isn’t an automorphism of u 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 with the last column excluded), does extend to an automorphism of I u as in the theorem below. Theorem 1 With the basis {x } ∪{a = x } for u and dual basis ij 1≤i < j ≤n i ii 1≤i ≤n n ∗ 2 {x } ∪{b } for u (and duality x , x = δ δ , b , a = 2δ , and ji 1≤i < j ≤n i 1≤i ≤n kl ij li jk i j ij x , a =b , x = 0), the map : I u → I u deﬁned by “incrementing all indices by ji k k ij n n 1 mod n” (precisely, if ψ is the single-cycle permutation ψ = (123 ... n) then is deﬁned by (x ) = x , (a ) = a , and (b ) = b ) is a Lie algebra automorphism ij ψ(i )ψ ( j ) i ψ(i ) i ψ(i ) of I u . Note that our choice of bases, using similar symbols x / x for the non-diagonal matrices ij ji and their duals, hides the intricacy of ; e.g., : x → x maps an element of u to n−1,n n1 n an element of u (also see Fig. 1, right). It may be tempting to think that has a simple explanation in gl language: u is a subset of gl , gl has a metric (the Killing form) such that the dual of x is x as is the case for n n ij ji us, and every permutation of the indices induces an automorphism of gl . But this explains nothing and too much: nothing because the bracket of I u simply isn’t the bracket of gl (even away from the diagonal matrices), and too much because every permutation of indices induces an automorphism of gl , whereas only ψ and its powers induce automorphisms of I u . Proof of Theorem 1 Recall that as a vector space I u = u ⊕ u , yet with bracket n n [(x , f ), (y, g)]= ([x , y], x · g − y · f ) where · denotes the coadjoint action, (x · f )(v) = f ([v, x ]). With that and some case checking and explicit computations, the commutation relations of I u are given by [x , x ]= χ (δ x − δ x ) unless both j = k and l = i , ij kl λ(x )+λ(x )<n jk il li kj ij kl (1) [x , x ]= (b − b ), [a , x ]= (δ − δ )x , [b , x ]= 0, ij ji i j i jk ij ik jk i jk [a , a ]=[b , b ]=[a , b ]= 0. i j i j i j The awkward factor of 2 in b , a is irrelevant for Theorem 1 yet crucial for Theorem 2. Removing this i j factor removes the factor in (1), see also Chapter 4 of [13] 123 An Unexpected Cyclic Symmetry... Here χ is the indicator function of truth (so χ = 1 while χ = 0), and λ(x ) is the 5<7 7<5 ij j −ii < j “length” of x ,deﬁnedby λ(x ):= . ij ij n − (i − j ) i > j It is easy to verify that the length λ(x ) is -invariant, and hence everything in (1)is ij -equivariant. I u 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 a → a → ··· → a → a → a → n 1 2 n−2 n−1 n layer 1 g = g =[g, g] x → x → ··· → x → x → x → 12 23 n−2,n−1 n−1,n n1 layer 2 g =[g , g ] x → x → ··· → x → x → x → 13 24 n−2,n n−1,1 n2 2 1 layer 3 g =[g , g ] x → x → ··· → x → x → x → 14 25 n−2,1 n−1,2 n3 3 2 . . . . . . . . . . . . . . . . . . . . . layer (n − 1) g =[g , g ] x → x → ··· → x → x → x → 1n 21 n−2,n−1 n−1,n−2 n,n−1 n−1 n−2 layer n g =[g , g ] b → b → ··· → b → b → b → 1 2 n−2 n−1 n n n−1 In this table (all assertions are easy to verify): • Apart for the treatment of the a ’s and the b ’s, layer=length= λ(x ). i i 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 ≥ 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 and the last p belong to u . So as we go down, u 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 part of each layer left to right and by doing the same to the u part, without mixing the two parts. • Note that I u can be metrized by pairing the u summand with the u one. The metric n n 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 x x ··· x x 12 23 34 n−1,n n1 x x ··· x x 13 24 n−1,1 n2 (with the diagram continued cyclically). The symmetry group of the above cycle is the dihedral group D and this strongly suggests that the group of outer automorphisms and anti-automorphisms of I u (all automorphisms and anti-automorphisms modulo inner automorphisms) is D , 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 , is a way to reconstruct gl from its subalgebras of upper triangular matrices u and lower 123 D. Bar-Natan, R.V.D. Veen triangular matrices l . Precisely, one endows the vector space g = u ⊕ l with a non- n n n degenerate symmetric bilinear form by declaring that the subspaces u and l are isotropic n n (u , u =l , l = 0) and by setting x , x = δ δ , b , a = 2δ ,and x , a = n n n n kl ij li jk i j ij ji k b , x = 0 as in Theorem 1 and where a stands for the diagonal matrix x considered as k ij i ii an element of u and b stands for the same matrix as an element of l . There is then a unique n i n bracket on g that extends the brackets on the summands u and l and relative to which the n n inner product of g is invariant 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 = gl ⊕ h , n+ n n where h denotes a second copy of the diagonal matrices in gl . n n We let gl be the Inonu-Wigner [14] contraction of g along its l summand, with param- n+ eter . All that this means is that the bracket of l gets multiplied by to give l ,and then the Drinfel’d double / Manin triple construction is repeated starting with u ⊕ l , without changing the bilinear form. The result is a Lie algebra gl over the ring of polynomials in n+ which specializes to I u at = 0 and which is isomorphic to gl ⊕ h when is invertible. n n k+1 We care about gl alot [3–5, 8–10, 18]; when reduced modulo = 0 for some natural n+ number k it becomes solvable, and hence a “solvable approximation” of gl 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 [x , x ]= χ (δ x − δ x ) unless both j = k and l = i , ij kl jk il li kj λ(x )+λ(x )<n ij kl [x , x ]= (b − b ) + (a − a ), [a , x ]= (δ − δ )x , ij ji i j i j i jk ij ik jk 2 2 [b , x ]= (δ − δ )x , i jk ij ik jk [a , a ]=[b , b ]=[a , b ]= 0, i j i j i j where χ = 1and χ = . These relations are clearly -equivariant. True False Note 3 There is of course an “sl” version of everything, in which linear combinations α a i i and β b are allowed only if α = β = 0, with obvious modiﬁcations throughout. i i i i Note 4 At n = 2and = 0, the algebra sl is the “diamond Lie algebra” of [15, Chap- 2+ ter 4.3], which is sometimes called “the Nappi-Witten algebra” [17]. With a = (a − a )/2, 1 2 x = x , y = x ,and b = (b − b )/2, it is 12 21 1 2 a, x , y, b/ ([a, x]= x , [a, y]=−y, [x , y]= b, [b, −] = 0) . Here : (a, x , y, b) → (−a, x , y, −b) and : (a, x , y, b) → (−a, y, x , −b). Indeed we only need to determine [u, l] for u ∈ u and l ∈ l . Writing [u, l]= u + l with u ∈ u n n n and l ∈ l , 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 due to the invariance of , . Similarly l is determined from l , u =[u, l], u =l, [u , u]. Alternatively, make u into a Lie bialgebra with cobracket δ using its given duality with l , and double it as n n in [12, 13] but using the cobracket δ. Hence gl → I u is a counter-example to the feel-true statement “a contraction of a direct sum is a n+ direct sum”. Indeed with notation as in Theorem 2,as → 0 the decomposition gl = gl ⊕ h = n+ n x , b + a ⊕b − a collapses. ij i i i i 123 An Unexpected Cyclic Symmetry... 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 thealgebra I u (except reduced modulo b and considered n i globally rather than inﬁnitesimally) is considered from a different perspective. It is shown to be a subquotient of the afﬁne algebra gl 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. Acknowledgements We wish to thank M. Bulois, A. Knutson, A. Referees, N. Ressayre, and N. Williams for comments and suggestions. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Bar-Natan, D.: Convolutions on Lie Groups and Lie Algebras and Ribbon 2-Knots, talk at Chern-Simons Gauge Theory: 20 years after, Bonn (2009). Handout and video at http://www.math.toronto.edu/~drorbn/ Talks/Bonn-0908 2. Bar-Natan, D.: From the ax + b Lie Algebra to the Alexander Polynomial and Beyond, talk at Knots in Chicago (2010). Handout and video at http://www.math.toronto.edu/~drorbn/Talks/Chicago-1009 3. Bar-Natan, D.: What Else Can You Do with Solvable Approximations? Talk at the McGill University HEP Seminar (2017). Handout and video at http://www.math.toronto.edu/~drorbn/Talks/McGill-1702 4. Bar-Natan, D.: The Dogma is Wrong, talk at Lie Groups in Mathematics and Physics, Les Diablerets (2017). Handout and video at http://www.math.toronto.edu/~drorbn/Talks/LesDiablerets-1708 5. Bar-Natan, D.: Everything Around sl is DoPeGDO. So What? Talk at Quantum Topology and Hyper- 2+ bolic Geometry, Da Nang (2019). 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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Mar 16, 2023
Keywords: Lie algebras; Lie bialgebras; Lie algebra automorphism; Solvable approximation; Triangular matrices; 17B40; 57M25
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