---
title: Covering Extended Affine Lie Algebras
url: https://www.emergentmind.com/topics/covering-extended-affine-lie-algebras
type: topic
---

# Covering Extended Affine Lie Algebras

Covering extended affine Lie algebras refers, in the standard structural language of EALA theory, to several closely related extension phenomena: the passage from the **centerless core** to the **core** by central extension, the reconstruction of the full EALA by adjoining derivations and cocycle data, and, in more specialized settings, universal central extensions of Lie tori, multiloop and torsor realizations of centerless cores, local affine coverings by root subsystems, and Steinberg-type coverings on the group side [1003.2352]. The subject is therefore not governed by a single universal notion of covering; rather, it is organized by the canonical hierarchy
\[
E_{cc}\;\longleftarrow\;E_c\;\longrightarrow\;E,
\]
together with the realization of \(E_{cc}\) as a centerless Lie torus and the analysis of how automorphisms, integral structures, modules, and group constructions lift across these successive extensions [2302.00446].

## 1. Core, centerless core, and the canonical extension ladder

An extended affine Lie algebra is a triple \((E,(\cdot,\cdot),H)\) satisfying the usual axioms, with root system \(R=R^\times\cup R^0\), where \(R^\times\) is the set of nonisotropic roots and \(R^0\) the isotropic roots. Its **core** is the subalgebra generated by the nonisotropic root spaces,
\[
E_c=\langle E_\alpha\mid \alpha\in R^\times\rangle,
\]
and the **centerless core** is
\[
E_{cc}:=E_c/Z(E_c).
\]
A central structural fact is that \(E_{cc}\) is precisely a centerless Lie torus; conversely, starting from a centerless Lie torus \(\mathcal L\), one reconstructs an EALA by adjoining a central part and a derivation part [2302.00446].

In Neher’s realization, one starts with a centerless Lie torus \(\mathcal L\) carrying an invariant nondegenerate graded form \((\cdot,\cdot)_\mathcal L\), chooses a permissible graded subalgebra
\[
D\subseteq \operatorname{SCDer}_K(\mathcal L)
\]
of skew centroidal derivations, and an affine cocycle
\[
\kappa:D\times D\to D^{\mathrm{gr}*}.
\]
The resulting EALA is built on
\[
E=E(\mathcal L,D,\kappa):=\mathcal L\oplus D^{\mathrm{gr}*}\oplus D,
\]
with bracket
\[
[x_1+c_1+d_1,\ x_2+c_2+d_2]
=
\big([x_1,x_2]_{\mathcal L}+d_1(x_2)-d_2(x_1)\big)
+
\big(\sigma_D(x_1,x_2)+d_1\cdot c_2-d_2\cdot c_1+\kappa(d_1,d_2)\big)
+
[d_1,d_2],
\]
where
\[
\sigma_D(x,y)(d)=(d(x)\mid y)_\mathcal L.
\]
Its invariant form is
\[
(x_1+c_1+d_1,\ x_2+c_2+d_2)
=
(x_1,x_2)_\mathcal L + c_1(d_2)+c_2(d_1),
\tag{2.2}
\]
and its Cartan subalgebra is
\[
H:=\mathfrak h\oplus (D^0)^*\oplus D^0,\qquad \mathfrak h:=\mathcal L_0.
\tag{2.3}
\]
Crucially,
\[
E_c=\mathcal L\oplus D^{\mathrm{gr}*},
\]
so the core is literally a central extension of the centerless core \(\mathcal L\), while the full EALA is obtained from the core by adjoining \(D\) [2302.00446].

Neher’s lectures formulate the same picture in the language of coverings and central extensions. An **extension** of a Lie algebra \(L\) is a surjective homomorphism \(f:K\to L\); it is a **central extension** if \(\ker f\subset Z(K)\); it is a **covering** if it is a central extension with \(K\) perfect; and a **universal central extension** exists precisely for perfect Lie algebras [1003.2352]. In EALA theory, the most immediate covering object is therefore not the full algebra \(E\), but the central extension \(E_c\to E_{cc}\).

## 2. Lie tori, universal central extensions, and reconstruction

Lie tori are the structural base objects for EALAs. Neher’s lectures emphasize that the centerless core of every EALA is an **invariant centreless Lie torus**, and that every EALA is reconstructed from such a Lie torus by taking a suitable central extension and adjoining derivations with an affine cocycle [1003.2352]. In this framework, covering theory is concentrated at the level of the Lie torus and its central extensions.

A basic fact is that Lie tori are perfect, hence admit universal central extensions. Moreover, if
\[
u:\operatorname{uce}(L)\to L
\]
is the universal central extension of a Lie torus \(L\), then \(\operatorname{uce}(L)\) is again a Lie torus of the same type [1003.2352]. This is the abstract form of the statement that the class of centerless-core models relevant to EALAs is stable under universal central covering.

The affine and toroidal examples provide the standard prototypes. For affine Kac–Moody theory, the loop algebra \(L(g,\sigma)\) is the centerless core, its canonical central extension \(\widehat L=L\oplus Fc\) is the core, and the affine algebra \(\widetilde L=\widehat L\rtimes Fd\) is obtained by adjoining the degree derivation [1003.2352]. For untwisted multiloop algebras
\[
L(g)=g\otimes F[t_1^{\pm1},\dots,t_n^{\pm1}],
\]
the universal central extension is larger than the one-variable affine analogue; Neher writes it using a graded center
\[
C=\bigoplus_{\gamma\in\Gamma} C^\gamma,\qquad C^\gamma=F^n/F\gamma,
\]
and a universal cocycle
\[
\psi_u(u\otimes t^\lambda,v\otimes t^\mu)_\gamma
=
\kappa(u,v)\delta_{\lambda+\mu,-\gamma}\lambda.
\tag{1.17}
\]
The higher-nullity EALA then has the form
\[
E=(L(g)\oplus C)\rtimes D,
\]
with \(D\) spanned by degree derivations [1003.2352].

This central-extension picture can also be formulated torsorially. Gille and Pianzola treat multiloop algebras as \(R_n\)-forms of split simple Lie algebras over Laurent polynomial rings,
\[
R_n=k[t_1^{\pm1},\dots,t_n^{\pm1}],
\]
split by finite étale extensions \(R_{n,m}/R_n\). They are classified by torsors under \(\operatorname{Aut}(A)\), and a torsor is a **loop torsor** if its cohomology class lies in the image of
\[
H^1(\pi_1(\mathfrak X,a),G(k_s))\to H^1(\mathfrak X,G).
\tag{3.2}
\]
For reductive groups over \(R_n\), loop torsors coincide with toral torsors:
\[
H^1_{\mathrm{toral}}(R_n,G)=H^1_{\mathrm{loop}}(R_n,G),
\]
and a reductive \(R_n\)-group is loop reductive iff it admits a maximal torus [1109.3405]. This identifies multiloop centerless cores as the Lie algebras of loop reductive group schemes, thereby attaching a geometric classification to the covering layer beneath EALAs.

## 3. Multiloop algebras, nullity \(2\), and the classification of centerless cores

The paper on multiloop algebras and nullity \(2\) organizes the subject around three classes:
\[
\mathbb M_n,\qquad \mathbb I_n,\qquad \mathbb E_n,
\]
where \(\mathbb M_n\) denotes multiloop algebras of finite-dimensional simple Lie algebras, \(\mathbb I_n\) iterated loop algebras, and \(\mathbb E_n\) Lie algebras isomorphic to centerless cores of EALAs of nullity \(n\) [1002.2674].

| Class | Meaning | Nullity \(2\) role |
|---|---|---|
| \(\mathbb M_n\) | multiloop algebras | first-kind affine loop algebras |
| \(\mathbb I_n\) | iterated loop algebras | larger than \(\mathbb M_2\) |
| \(\mathbb E_n\) | centerless cores of EALAs | centerless-core class |

For \(n=0\) and \(n=1\), the classes coincide:
\[
\mathbb M_0=\mathbb I_0=\mathbb E_0,\qquad
\mathbb M_1=\mathbb I_1=\mathbb E_1.
\]
For \(n=2\), they separate, and the paper gives a complete description. The main theorem states that for a Lie algebra \(\mathcal L\), the following are equivalent: \(\mathcal L\in\mathbb M_2\); \(\mathcal L\simeq L(\bar{\mathfrak g},\sigma)\) with \(\mathfrak g\) untwisted affine and \(\sigma\) a diagram automorphism; \(\mathcal L\in\mathbb I_2\) with centroid
\[
C(\mathcal L)\simeq R_2=k[t_1^{\pm1},t_2^{\pm1}].
\]
Thus, in nullity \(2\), multiloop algebras are exactly the iterated loop algebras of affine type with Laurent polynomial centroid [1002.2674].

The intersection with the EALA-core class is sharply characterized:
\[
\mathbb M_2\cap \mathbb E_2
=
\{\text{fgc centreless cores of nullity-2 EALAs}\}
=
\{\text{isotropic algebras in }\mathbb M_2\}.
\]
Equivalently, an algebra in \(\mathbb M_2\) is the centerless core of a nullity-\(2\) EALA iff it is isotropic, or equivalently iff it comes from a nontransitive affine diagram automorphism [1002.2674]. The anisotropic part
\[
\mathbb M_2\setminus \mathbb E_2
\]
consists of algebras such as
\[
\mathfrak{sl}_1(Q(\theta)),\qquad \theta\neq 1 \text{ a root of unity},
\]
while
\[
\mathbb I_2\setminus \mathbb M_2
\]
consists of loop algebras defined by automorphisms of second kind [1002.2674].

The same paper gives an explicit affinization that recovers the full EALA above its centerless core. For an affine Lie algebra \(\mathfrak g\) and a nontransitive diagram automorphism \(\sigma\),
\[
\operatorname{Aff}_m(\mathfrak g,\sigma)
=
L_m(\mathfrak g,\sigma)\oplus k\check c\oplus k\check d,
\]
with bracket
\[
[x\otimes z^i+r_1\check c+r_2\check d,\ y\otimes z^j+s_1\check c+s_2\check d]
=
[x,y]\otimes z^{i+j}
+jr_2y\otimes z^j
-is_2x\otimes z^i
+i\delta_{i+j,0}(x|y)\check c.
\]
If \(\sigma\) is nontransitive, this is a nullity-\(2\) EALA whose centerless core is
\[
L_m(\bar{\mathfrak g},\sigma).
\]
This is the most explicit nullity-\(2\) realization of the extension
\[
\text{multiloop/iterated loop algebra}
\longleftrightarrow
\text{centreless core of an EALA}
\]
in the literature summarized here [1002.2674].

## 4. Lifting automorphisms and involutions through the extension hierarchy

A central structural problem is whether automorphisms or involutions of the centerless core can be lifted through the central extension \(E_c\to E_{cc}\) and then through the derivation extension \(E_c\subset E\). This is treated explicitly for Chevalley involutions by the paper on Lie tori and EALAs [2302.00446].

For a \(\Lambda\)-graded algebra \(A=\bigoplus_{\lambda\in\Lambda}A^\lambda\), a **pre-Chevalley involution** is an involution \(T\) such that
\[
T(A^\lambda)=A^{-\lambda}.
\]
For a centerless Lie torus \(\mathcal L\), it is a **Chevalley involution** if it also satisfies
\[
T(h)=-h\qquad (h\in \mathcal L^0).
\]
The main existence theorem states that every centerless Lie torus of reduced type admits a Chevalley involution, with the additional assumption in type \(A_\ell\), \(\ell>2\), that its coordinate algebra is equipped with a \(\Lambda\)-grading anti-involution [2302.00446].

The lifting mechanism begins by conjugation on endomorphisms,
\[
\widetilde T:\operatorname{End}(\mathcal L)\to \operatorname{End}(\mathcal L),\qquad
\phi\mapsto T\phi T^{-1},
\]
and dually on \(\operatorname{End}(\mathcal L)^*\). If \(d_\theta\) is the degree derivation attached to \(\theta\in\operatorname{Hom}_\mathbb Z(\Lambda,K)\), then
\[
Td_\theta=-\,d_\theta T,\qquad \widetilde T(d_\theta)=-d_\theta.
\]
More generally,
\[
\widetilde T(x^\mu d_\theta)=x^{-\mu}d_{-\theta},
\qquad
\widetilde T(c^{(\mu)}_\lambda)=c^{(-\mu)}_{-\lambda}.
\]
These formulas determine how the derivation and central pieces transform under the involution [2302.00446].

Given permissible \(D\subseteq \operatorname{SCDer}(\mathcal L)\), the transformed derivation algebra and cocycle are
\[
D_T:=\widetilde T(D),\qquad
\kappa_T(\widetilde T(d),\widetilde T(d')):=\widetilde T(\kappa(d,d')).
\]
Then
\[
\widetilde T:E(\mathcal L,D,\kappa)\xrightarrow{\sim}E(\mathcal L,D_T,\kappa_T)
\]
is an isomorphism. If \(D\) is \(T\)-invariant, then \(\widetilde T\) restricts to a pre-Chevalley involution of the core
\[
E_c=\mathcal L\oplus D^{\mathrm{gr}*}.
\]
If \((D,\kappa)\) is \(T\)-invariant, then \(\widetilde T\) becomes a pre-Chevalley involution of the full EALA. If \(T\) is already a Chevalley involution of \(\mathcal L\), the lifted involution on \(E_c\) or \(E\) is again a Chevalley involution [2302.00446].

The criterion for \(D\)-invariance is concrete. Writing
\[
U^\mu:=\{\theta\in\operatorname{Hom}_{\mathbb Z}(\Lambda,K)\mid x^\mu d_\theta\in D^\mu\},
\tag{3.6}
\]
one has
\[
D \text{ is } T\text{-invariant } \Longleftrightarrow U^\mu = U^{-\mu}\quad \forall \mu\in\Gamma.
\]
In particular, the standard choices
\[
D=D^0
\qquad\text{or}\qquad
D=\operatorname{SCDer}(\mathcal L)
\]
are always \(T\)-invariant, so the core inherits a Chevalley involution in these common realizations [2302.00446].

This lifting viewpoint persists in recent work on integral structures. The paper on integral structures for reduced tame EALAs states that the essential structural features are encoded in the centerless core \(g_{cc}:=g_c/Z(g_c)\), and one then promotes the results from \(g_{cc}\) to \(g_c\) and then to \(g\) [2106.09414]. The rank-one elliptic paper makes the same hierarchy completely explicit for type \(A_1\), nullity \(2\): if \(g=\operatorname{TKK}(J)\), then
\[
E(g,D,\kappa)=g\oplus D^{\mathrm{gr}*}\oplus D,\qquad
E_c=g\oplus D^{\mathrm{gr}*},\qquad
Z(E_c)=D^{\mathrm{gr}*},
\]
so the centerless Lie torus, the core, and the full EALA are visibly separated [2507.10081].

## 5. Local, presentation-theoretic, group-theoretic, and module-theoretic variants

A different covering-like mechanism is **localization by root subsystems**. For a closed subsystem \(R'\subseteq R\), a subspace \(H'\subseteq H\) is called a cover if it contains \(t_\alpha\) for all \(\alpha\in {R'}^\times\) and the form restricted to \(H'\) is nondegenerate. The associated Lie cover is
\[
E_{R',H'}:=H'\oplus \sum_{\alpha\in R'\setminus\{0\}}E_\alpha.
\]
It is again an EALA with Cartan \(H'\) and root system \(R'\). If \(R'\) is affine, then the refined localization
\[
\hat E_{R',H_{R'}}
\]
is an affine Lie algebra. The same paper develops a filtration
\[
E_0\subseteq E_1\subseteq \cdots \subseteq E_\nu=E
\]
by EALAs of increasing nullity, with \(E_1\) affine, so that an EALA is approximated by finite and affine local pieces rather than by a single global cover [2510.08243].

Another covering-like construction is presentation-theoretic. For a connected non-negative unit form \(q\), the paper on \(E(q)\) constructs an EALA with root system
\[
R(q):=q^{-1}(0)\cup q^{-1}(1),
\]
and shows that \(E(q)\) is obtained from the generalized-Serre algebra \(G(q)\) by quotienting out the unique maximal ideal intersecting the Cartan trivially:
\[
G(q)/I \cong E(q).
\]
This is not formulated as a universal central covering, but it is a canonical presentation-to-EALA quotient closely analogous to the Serre presentation of affine Kac–Moody algebras [1209.4380].

On the group side, the paper on groups of extended affine Lie type introduces an **\(A\)-covering** of a reduced extended affine root system \(R=R(\bar R,S,L)\):
\[
\widetilde R:=R(\bar R,(S),(L)).
\]
The Steinberg group associated to \(\widetilde R\) is called the **universal Steinberg group** of \(R\), and there is a right-split exact sequence
\[
K_\chi \hookrightarrow \operatorname{St}_{\bar R}(C_\sigma)\xrightarrow{\chi}\operatorname{St}_{\bar R,R,\sigma}(\mathbb C),
\tag{4.11}
\]
hence
\[
\operatorname{St}_{\bar R}(C_\sigma)\cong K_\chi\rtimes \operatorname{St}_{\bar R,R,\sigma}(\mathbb C).
\tag{4.12}
\]
The same paper shows that the extended affine Weyl group is recovered as
\[
W\cong \operatorname{Ad}(N)/\operatorname{Ad}(T).
\tag{4.39}
\]
These are covering constructions for groups attached to EALAs rather than for the Lie algebras themselves [1908.07809].

A fourth usage of covering appears in representation theory. In the paper on irreducible modules for EALAs, a **thin covering** of a module \(U\) over a graded Lie algebra \(\mathcal L=\bigoplus_{g\in G}\mathcal L_g\) is a family \(\{U_g\}_{g\in G}\) such that
\[
\sum_{g\in G}U_g=U,\qquad \mathcal L_gU_h\subset U_{g+h},
\]
minimal under termwise inclusion. Thin coverings are then used to construct irreducible modules for twisted toroidal Lie algebras and the corresponding EALAs. This usage is module-theoretic, not Lie-algebraic: it concerns coverings of modules, not coverings of EALAs by central extensions [1002.2262].

## 6. Intrinsic invariants, conjugacy, and the limits of the covering viewpoint

A foundational issue for any covering theory of EALAs is whether the core, the centerless core, and the associated root data are intrinsic to the Lie algebra or depend on the chosen EALA structure. For fgc EALAs, the conjugacy theorem resolves this strongly. If \((E,H)\) and \((E,H')\) are two EALA structures on the same Lie algebra and the centerless core \(E_{cc}\) is fgc, then there exists
\[
f\in \operatorname{Aut}_k(E)
\]
such that
\[
f(H)=H'.
\]
The paper first proves that the core is independent of the EALA structure: if \(E_c\) and \(E_c'\) are the cores attached to two EALA structures on \(E\), then
\[
E_c=E_c'.
\]
Moreover, the core is characteristic:
\[
f(E_c)=E_c\qquad \text{for all }f\in\operatorname{Aut}_k(E).
\]
Thus, for fgc EALAs, the root system and quotient root system are intrinsic invariants, and the covering ladder
\[
E_{cc}\longleftarrow E_c\longrightarrow E
\]
is attached to the Lie algebra itself, not merely to an auxiliary Cartan choice [1602.07759].

This intrinsic viewpoint also clarifies the limits of the terminology. Multiloop algebras are generally not EALAs themselves; in nullity \(2\), isotropic algebras in \(\mathbb M_2\) are centerless cores of EALAs, while the full EALA sits above them as a central extension with derivations [1002.2674]. Likewise, Neher’s lectures stress that one should not identify an EALA with the universal central cover of its centerless core: the general construction allows many suitable central extensions arising from graded subalgebras of skew-centroidal derivations, not only the universal one [1003.2352].

The literature also leaves clear open ends. The conjugacy theorem is proved only for **fgc** EALAs, and the non-fgc case remains unresolved [1602.07759]. On the group side, integral-structure methods currently yield the **adjoint form** over arbitrary fields, while the universal-type group is explicitly left open [2106.09414]. This suggests that covering questions in EALA theory are best understood as a family of related extension problems—central, derivational, local, and group-theoretic—organized around the intrinsic ideal \(E_c\) and its quotient \(E_{cc}\), rather than as a single universal-covering theory valid uniformly across all extended affine Lie algebras.

Source: https://www.emergentmind.com/topics/covering-extended-affine-lie-algebras