Kashiwara-Vergne Lie Algebra

• Kashiwara-Vergne Lie Algebra is a specialized Lie algebra defined by derivations of free Lie algebras that satisfy strict trace and symmetry conditions.
• It connects deeply with the double shuffle and Grothendieck–Teichmüller Lie algebras, clarifying the arithmetic of multiple zeta values.
• Generalizations to higher genus, cyclotomic, and elliptic variants expand its applications in deformation quantization, topology, and quantum invariants.

The Kashiwara-Vergne (KV) Lie algebra is a highly structured Lie algebra intimately related to deformation quantization, the theory of associators, the arithmetic of multiple zeta values, and the topological paper of mapping class groups of surfaces. Its construction encodes specialized derivations of free Lie algebras with explicit symmetry and trace conditions, and it admits deep connections with the Grothendieck–Teichmüller Lie algebra, the double shuffle Lie algebra, and generalizations to higher genus and cyclotomic settings.

1. Foundational Definition and Structural Properties

The Kashiwara–Vergne Lie algebra, often denoted $\mathfrak{krv}$ or more specifically $\mathfrak{krv}_2$ in the two-generator/free Lie algebra case, consists of a subalgebra of derivations of the free Lie algebra $\operatorname{Lie}[x,y]$ defined by strict algebraic and combinatorial constraints. Its most canonical realization is:

• Derivations of the form

$$D_{F,G} : \begin{cases} x \mapsto [x, G] \ y \mapsto [y, F] \end{cases}$$

where $F,G \in \operatorname{Lie}[x,y]$.

• The "specialness" or balancing condition:

$[x, G] + [y, F] = 0$

which ensures the derivation annihilates $x + y$.

• The trace (or divergence) condition:

$$\mathrm{tr}(F_{yy} + G_{xx}) = A \cdot \mathrm{tr}((x+y)^n - x^n - y^n)$$

for some constant $A$ depending on degree, where the trace map projects to cyclically invariant classes.

A key reformulation of the specialness condition relates to combinatorial symmetries:

• $F_y$ (the $y$-initial monomial component of $F$) is antipalindromic,
• $F$ is push-invariant (invariant under cyclic permutations of $x$-powers in its monomials).

These symmetries, together with the trace condition, completely characterize elements of the KV Lie algebra (Schneps, 2012).

Grading is respected: for example, the degree 1 part is one-dimensional (generated by $D_{y,x}$), and degree 2 is trivial.

2. Deep Relationship with the Double Shuffle Lie Algebra

The double shuffle Lie algebra $\mathfrak{ds}$ (“os” in some literature) is defined combinatorially in terms of “stuffle” (and regularized shuffle) conditions on Lie polynomials and encodes the structure of formal multiple zeta value relations. The KV Lie algebra receives a remarkable injection:

• For $f \in \mathfrak{ds}$, define $F(x,y) = f(-x - y, y)$.
• There exists a unique $G$ such that $[y, F] + [x, G] = 0$ (by symmetry/antipalindromicity).
• The corresponding special derivation $D_{F,G}$ lies in $\mathfrak{krv}$.

The map $f \mapsto D_{F,G}$ is injective and Lie-algebra morphic, yielding a conceptual bridge — $\mathfrak{ds}$ (dual to formal multiple zeta values) is a subalgebra in $\mathfrak{krv}$ (Schneps, 2012). This injection reflects and clarifies the algebraic structure previously seen only in the combinatorics of multiple zeta value relations, and, importantly, the embedding is compatible with the Poisson bracket structure on $\mathfrak{ds}$.

3. Generalizations: Higher Genus, Cyclotomic, Elliptic, and Bigraded Variants

Higher Genus and Associated Formality

For a surface $\Sigma_{g,n+1}$, the higher-genus Kashiwara–Vergne problem ($\mathrm{KV}^{(g,n)}$) replaces the standard free Lie algebra with a completed Lie algebra generated by $x_i,y_i$ ($i=1,\ldots,g$) and $z_j$ ($j=1,\ldots,n$) (corresponding to commutators and boundary elements). Tangential derivations must preserve boundary conditions and satisfy specially adapted divergence criteria. The set of solutions corresponds directly to group-like expansions inducing isomorphisms between the Goldman–Turaev Lie bialgebra of surface loops and its associated graded (Alekseev et al., 2016, Alekseev et al., 2018, Taniguchi, 10 Feb 2025).

In the closed surface case, the relevant automorphism group is described as those pro-unipotent automorphisms of the associated graded algebra $\hat T(H)_\omega$ whose divergence (computed via a noncommutative connection) vanishes modulo exact forms on the boundary (Taniguchi, 10 Feb 2025):

$$\mathfrak{krv}_{(g,0)} = \left\{g \in \mathrm{Der}^+(\hat{L}(H)_\omega) : \mathrm{sdiv}^{\nabla'_{\bullet,H}}(g) \in \ker(|\overline{\Delta}_\omega|)\right\}$$

Bigraded and Cyclotomic Generalization

In the presence of a finite abelian group $T$, the “cyclotomic” and bigraded KV Lie algebra is defined using a free Lie algebra on $x$ and $\{y_t\}$ with an additional grading, and leading term conditions encapsulate both the classical and cyclotomic cases. The corresponding mould-theoretic perspective uses specific symmetry classes (alternality, senary relations, push-invariance, pus-neutrality) and embeds Goncharov’s dihedral Lie algebra as a subalgebra (Furusho et al., 2020).

Linearized and Elliptic Versions

The linearized (or depth-graded) KV Lie algebra $\ellkv$ is defined by restricting to the lowest-depth part of the conditions. It consists of push-invariant and circ-neutral elements in the algebra generated by the iterated commutators $C_i = \operatorname{ad}_x^{i-1}(y)$, the bracket being corrected by uniquely associated partners (to maintain the defining symmetries). The elliptic variant $\mathfrak{krv}_{ell}$ is constructed by twisting via denominators in the mould-theoretic formulation, leading to push- and circ-invariant objects only after suitable modification (Raphael et al., 2017, Raphael et al., 2018).

There are injective morphisms from the double shuffle and Grothendieck–Teichmüller elliptic Lie algebras to the elliptic Kashiwara–Vergne Lie algebra, positioning $\mathfrak{krv}_{ell}$ as a key nodal point in the web of Lie algebras interrelating multiple zeta values, associators, and modular forms (Raphael et al., 2017).

4. Representation Theory, Splittings, and Symmetry Groups

Splitting and Filtration Results

It has been conjectured and proven in various settings (notably in depth $2$) that

$$\mathfrak{krv} \cong \mathbb{K}t \oplus \mathfrak{grt}_1$$

where $t$ is a one-dimensional central element and $\mathfrak{grt}_1$ is the Grothendieck–Teichmüller Lie algebra (Alekseev et al., 2014). Divergence, super-divergence, and $q$-divergence cocycles furnish tools for distinguishing elements, with the q-divergence cocycle showing injectivity properties that the standard divergence lacks.

Further, the filtration obtained from graph complexes (internally connected graphs, ICG) yields a nested sequence of Lie subalgebras interpolating between $\mathfrak{krv}$ and $\mathfrak{grt}_1$; their intersection reproduces the latter exactly, formalizing the conceptual role of $\mathfrak{grt}_1$ as the "tree-level core" of the broader $\mathfrak{krv}$ (Felder, 2016).

Symmetry Groups and Topological Characterization

KV groups ($\mathsf{KV}$, $\mathsf{KRV}$) and their graded/associated versions are realized as automorphism groups of completed circuit algebras arising in the paper of welded foams (and thus 4-dimensional knotted surfaces). The relationship to Drinfeld's graded Grothendieck–Teichmüller group ($\mathsf{GRT}_1$) is made concrete: $\mathsf{GRT}_1$ acts as automorphisms of associated graded circuit algebras of arrow diagrams, embedding as a subgroup of $\mathsf{KRV}$ (Dancso et al., 2021). This identification unifies Lie-theoretic and topological approaches to the KV problem via homomorphic expansions of welded foams.

5. Mould-theoretic Interpretation and the Role of Symmetry

A powerful description of $\mathfrak{krv}$ and related Lie algebras is achieved via mould theory (initiated by Écalle):

• The key symmetries are encoded as alternality, push-invariance, and the "senary" relation (a six-term symmetry), all of which control which polynomial-valued moulds correspond to admissible elements (Furusho et al., 2022).
• The KV Lie algebra is isomorphic (under explicit maps) to the Lie algebra of push-invariant, senary-related, pus-neutral moulds; similarly, the double shuffle Lie algebra corresponds to bialternality under swap.
• The inclusion (and sometimes isomorphism) $\mathfrak{ds} \hookrightarrow \mathfrak{krv}$ in the mould setting hinges on the validity of Écalle's senary relation, which has been verified in small depths. These mould operations translate the combinatorial symmetries of Lie polynomials into explicit algebraic conditions, providing an efficient computational and conceptual apparatus (Furusho et al., 2022).

6. Interplay with Multiple Zeta Values and Associators

The duality between the double shuffle Lie algebra and the space of new formal multiple zeta values allows the explicit embedding of arithmetic structures into $\mathfrak{krv}$. Through this embedding, $\mathfrak{krv}$ can be seen as encoding the algebraic and combinatorial nature of multiple zeta value relations, as well as the deformation-theoretic structures encoded in Drinfeld associators and the Grothendieck–Teichmüller group (Schneps, 2012).

In particular, in the genus zero and genus one (elliptic) settings, the algebra governs the existence of homomorphic expansions intertwining the Goldman–Turaev Lie bialgebra of loops on surfaces with necklace Lie bialgebras (gr$\mathfrak{g}(\Sigma)$). The solution of the higher-genus KV problem corresponds to the existence of a formality isomorphism between the Goldman–Turaev Lie bialgebra and its associated graded (Alekseev et al., 2016, Alekseev et al., 2018, Alekseev et al., 2018, Taniguchi, 10 Feb 2025).

7. Explicit Morphisms, Conjectures, and Arithmetic Applications

Explicit Lie Algebra Morphisms

There exists an explicit injective Lie algebra morphism from $\mathfrak{ds}$ to $\mathfrak{krv}$, constructed via the assignment

$f \mapsto f + D_{g,h}$

with $g = f(-x-y,-y)$ and $h$ determined by $[x, h] + [y, g] = 0$. This morphism fits compatibly into a commutative triangle with the known injections from $\mathfrak{grt}_1$ (Schneps, 19 Apr 2025).

Conjectures and Numerical Evidence

Chains of inclusions (and isomorphisms in low degrees) are numerically supported:

$$\text{free} \subset \mathfrak{grt}_1 \subset \mathfrak{ds} \subset \mathfrak{krv}_2$$

with all having coincident graded dimensions up to substantial weight/depth. This supports the Deligne–Drinfeld, Broadhurst–Kreimer, and Alekseev–Torossian conjectures on the structure of these algebras and the number of multiple zeta values in each graded piece (Naef et al., 11 Aug 2025).

The numerical computations are performed by analyzing special tangential derivations, computing images under divergence-like operators, and bruteforce enumeration of Lie expressions in high weights and depths. All data so far matches the predictions from conjectured generating functions and dimension formulas.

Summary Table: Core Properties and Relations

Aspect	$\mathfrak{krv}$	Related structure
Defining properties	Special derivations; trace/divergence conditions	Lie[x,y], cyclic
Symmetries	Antipalindromicity, push-invariance, circ-neutrality	Moulds, symmetry
Poisson structure	Yes, via [Alekseev–Torossian–Schneps bracket]	sder, tder
Subalgebra inclusions	$\mathfrak{grt}_1$, $\mathfrak{ds}$ (injective map)	$\mathfrak{krv}$
Elliptic/cyclotomic	Elliptic: $\mathfrak{krv}_{ell}$; Cyclotomic: $T$-variant bigraded	Moulds, ARI
Topological actions	Automorphisms of circuit algebras for welded foams	4D topology
Surface generality	Full Kashiwara–Vergne problem for all $(g,n)$ surfaces	Mapping class Gp

Concluding Remarks

The Kashiwara–Vergne Lie algebra serves as a central node in the web of Lie-type, combinatorial, and topological structures at the heart of modern deformation quantization, multiple zeta values, and low-dimensional topology. Its combinatorial definition, symmetry constraints, and diverse morphisms give rise to a model for the interlocking algebraic structure underlying associators, period relations, and even quantum invariants of knots and 4-manifolds. Ongoing research continues to elucidate its connections with the Grothendieck–Teichmüller Lie algebra, the double shuffle Lie algebra, and new cyclotomic and elliptic generalizations, as well as applications in the topological interpretation of formality and deformation problems across geometry and mathematical physics.