# Loop algebra

A **loop algebra** is an infinite-dimensional [Lie algebra](https://www.edgepedia.org/lie-algebra) formed from a Lie algebra \( \mathfrak{g} \) by extending its scalars to Laurent polynomials, written \( L\mathfrak{g} = \mathfrak{g} \otimes K[t, t^{-1}] \), with the bracket \( [X \otimes t^{m},\, Y \otimes t^{n}] = [X, Y] \otimes t^{m+n} \).<sup>[1](https://en.wiki.beta.math.wmflabs.org/wiki/Loop_algebra)</sup> The name comes from the geometric picture: the algebra arises in the study of Lie algebras of loop groups, that is, groups of maps from a circle into a [Lie group](https://www.edgepedia.org/lie-group).<sup>[2](https://planetmath.org/loopalgebra)</sup> Loop algebras are the building blocks of affine Kac-Moody algebras and appear throughout integrable systems and theoretical physics.<sup>[1](https://en.wiki.beta.math.wmflabs.org/wiki/Loop_algebra)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( L\mathfrak{g} = \mathfrak{g} \otimes K[t, t^{-1}] \) with bracket \( [X \otimes t^{m}, Y \otimes t^{n}] = [X, Y] \otimes t^{m+n} \)<sup>[1](https://en.wiki.beta.math.wmflabs.org/wiki/Loop_algebra)</sup> |
| Geometric meaning | The Lie algebra of maps from the circle \( S^{1} \) into \( \mathfrak{g} \); the polynomial version uses Laurent polynomials \( z \mapsto \sum_{m \in \mathbb{Z}} a_{m} z^{m} \) with finitely many nonzero coefficients<sup>[3](https://stacky.net/files/written/LieGroups/LieGroups.pdf)</sup> |
| Dimension | Infinite-dimensional whenever \( \mathfrak{g} \neq 0 \), since each eigenspace of \( \mathrm{ad}(h) \) in \( L(\mathfrak{sl}_{2}) \) is already infinite-dimensional over \( \mathbb{C} \)<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup> |
| Central extension | A one-dimensional central extension \( \widehat{L\mathfrak{g}} = L\mathfrak{g} \oplus \mathbb{C}c \) with bracket term \( n\,\mathrm{tr}(xy)\,\delta_{m,-n}\, c \)<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup> |
| Affine Kac-Moody algebra | \( L\mathfrak{g} \oplus \mathbb{C}\partial \oplus \mathbb{C}c \), the loop algebra plus an outer derivation and the central element<sup>[3](https://stacky.net/files/written/LieGroups/LieGroups.pdf)</sup> |
| Realization theorem | Any affine Kac-Moody algebra can be realized, that is, constructed explicitly, using loop algebras<sup>[5](https://arxiv.org/abs/0709.0975)</sup> |
| Twisted form | For an involution \( \sigma \) on \( \mathfrak{g} \), the twisted loop algebra \( L(\mathfrak{g}, \sigma) = \{ A \in \mathfrak{g}[\lambda, \lambda^{-1}] \mid A(-\lambda) = \sigma(A(\lambda)) \} \)<sup>[6](https://audin.pages.math.unistra.fr/livres/Audin_Integrability.pdf)</sup> |

## Definition and first example

Given a Lie algebra \( \mathfrak{g} \) over a field \( K \), the loop algebra is the tensor product \( \mathfrak{g} \otimes K[t, t^{-1}] \), where \( K[t, t^{-1}] \) is the space of Laurent polynomials. The bracket on \( \mathfrak{g} \) extends coefficientwise, so that \( [X \otimes t^{m}, Y \otimes t^{n}] = [X, Y] \otimes t^{m+n} \).<sup>[1](https://en.wiki.beta.math.wmflabs.org/wiki/Loop_algebra)</sup> Equivalently, elements are finite [Laurent series](https://www.edgepedia.org/laurent-series) \( \sum_{k} A_{k} z^{k} \) with coefficients in \( \mathfrak{g} \).<sup>[7](https://www.math.utoronto.ca/khesin/papers/Lecture_notes.pdf)</sup>

The simplest example starts from \( \mathfrak{sl}_{2}(\mathbb{C}) \), the algebra of 2 by 2 traceless complex matrices with generators \( h, e, f \) satisfying \( [h, e] = 2e \), \( [h, f] = -2f \), and \( [e, f] = h \). Its loop algebra is

\[ L(\mathfrak{sl}_{2}) = \mathfrak{sl}_{2} \otimes_{\mathbb{C}} \mathbb{C}[z, z^{-1}], \]

with bracket \( [x \otimes f, y \otimes g] = [x, y] \otimes fg \).<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup> As a vector space, \( L(\mathfrak{sl}_{2}) \) is the space of 2 by 2 traceless matrices over Laurent polynomials, and it decomposes into eigenspaces of \( \mathrm{ad}(h) \) in which each eigenspace is infinite-dimensional over \( \mathbb{C} \).<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup>

## Geometric interpretation

The loop algebra can be described as the Lie algebra of maps from the circle \( S^{1} \) into \( \mathfrak{g} \). This description admits several versions, including continuous maps, smooth maps, and polynomial maps; the polynomial version, the most tractable, is the base change of \( \mathfrak{g} \) by the ring of Laurent polynomials \( R[z, z^{-1}] \).<sup>[8](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Loop.html)</sup> In the polynomial model one considers maps \( z \mapsto \sum_{m \in \mathbb{Z}} a_{m} z^{m} \) with \( a_{m} \in \mathfrak{g} \) and all but finitely many coefficients zero, and the bracket is the pointwise bracket \( [f, g]_{L\mathfrak{g}}(z) = [f(z), g(z)]_{\mathfrak{g}} \).<sup>[3](https://stacky.net/files/written/LieGroups/LieGroups.pdf)</sup>

## Central extensions and affine Kac-Moody algebras

**From loop algebra to affine algebra.** Loop Lie algebras admit central extensions attached to invariant inner products on the base Lie algebra.<sup>[8](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Loop.html)</sup> Concretely, when \( \mathfrak{g} \subseteq \mathfrak{gl}_{n} \), the induced trace gives a nondegenerate inner product on \( L\mathfrak{g} \) by a contour integral around the unit circle, and the associated 2-cocycle is

\[ c(f, g) = \frac{1}{2\pi i} \oint_{|z|=1} \mathrm{tr}\, f(z)\, g'(z) \, \frac{dz}{z} = \operatorname{Res}_{z=0} \bigl( \mathrm{tr}\, f(z)\, g'(z) \bigr), \]

the coefficient of \( z^{-1} \) in the Laurent series \( \mathrm{tr}\, f(z) g'(z) \).<sup>[3](https://stacky.net/files/written/LieGroups/LieGroups.pdf)</sup> For \( \mathfrak{sl}_{2} \) this gives the one-dimensional central extension \( \widehat{\mathfrak{sl}_{2}} = L(\mathfrak{sl}_{2}) \oplus \mathbb{C}c \) with bracket

\[ [x \otimes z^{n} + \lambda c,\; y \otimes z^{m} + \mu c] = [x, y] \otimes z^{m+n} + n\, \mathrm{tr}(xy)\, \delta_{m,-n}\, c, \]

in which \( c \) commutes with everything, hence the name central extension.<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup>

Adding an outer derivation \( \partial \), which acts on Laurent polynomial loops by differentiation with respect to \( z \), produces the full affine Kac-Moody algebra \( L\mathfrak{g} \oplus \mathbb{C}\partial \oplus \mathbb{C}c \).<sup>[3](https://stacky.net/files/written/LieGroups/LieGroups.pdf)</sup><sup> • </sup><sup>[17](https://sporadic.stanford.edu/reference/algebras/sage/algebras/lie_algebras/affine_lie_algebra.html)</sup> For \( \mathfrak{sl}_{2} \) the result is the Kac-Moody algebra of type \( A_{1}^{(1)} \), with vector space \( \mathfrak{sl}_{2} \otimes_{\mathbb{C}} \mathbb{C}[z, z^{-1}] \oplus \mathbb{C}c \oplus \mathbb{C}d \) and a bracket that combines the loop bracket, the derivation action, and the central term \( \delta_{n,-m}\, n\, \mathrm{tr}(xy)\, c \).<sup>[4](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)</sup> When the base Lie algebra is finite-dimensional and simple, the extended algebra with its derivation admits an infinite root system with an affine [Weyl group](https://www.edgepedia.org/weyl-group), and these extended Lie algebras are called untwisted affine Kac-Moody Lie algebras.<sup>[8](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Loop.html)</sup>

The connection runs in both directions. A theorem of infinite-dimensional Lie theory states that any affine Kac-Moody algebra can be realized explicitly using loop algebras, and the multiloop construction, which uses sequences of commuting finite-order automorphisms of a simple Lie algebra, extends this realization to the extended affine Lie algebras that generalize the affine ones.<sup>[5](https://arxiv.org/abs/0709.0975)</sup> Kac-Moody algebras themselves were introduced independently in the mid-1960s by Robert Moody at the [University of Toronto](https://www.edgepedia.org/university-of-toronto), working from a generalization of the [Cartan matrix](https://www.edgepedia.org/cartan-matrix), and [Victor Kac](https://www.edgepedia.org/victor-kac) at Moscow State University, studying simple graded Lie algebras of finite growth.<sup>[9](https://math.berkeley.edu/~barrett/resources/km.pdf)</sup>

## Twisted and multiloop algebras

If \( \mathfrak{g} \) carries an involution \( \sigma \), the fixed-point construction produces a subalgebra of \( \mathfrak{g}[\lambda, \lambda^{-1}] \), the twisted loop algebra

\[ L(\mathfrak{g}, \sigma) = \{ A \in \mathfrak{g}[\lambda, \lambda^{-1}] \mid A(-\lambda) = \sigma(A(\lambda)) \}. \]

These twisted algebras arise naturally in the theory of Lax equations with spectral parameter.<sup>[6](https://audin.pages.math.unistra.fr/livres/Audin_Integrability.pdf)</sup> More generally, from a finite-dimensional simple Lie algebra \( \mathfrak{g} \), a sequence of mutually commuting finite-order automorphisms \( \boldsymbol{\sigma} = (\sigma_{1}, \dots, \sigma_{n}) \), and positive integers \( \boldsymbol{m} = (m_{1}, \dots, m_{n}) \) with \( \sigma_{i}^{m_{i}} = \mathrm{id} \), one constructs a \( \mathbb{Z}^{n} \)-graded multiloop algebra \( M_{\boldsymbol{m}}(\mathfrak{g}, \boldsymbol{\sigma}) \).<sup>[10](https://arxiv.org/abs/0807.2019)</sup> When \( n = 1 \) and the automorphism is a diagram automorphism, the construction recovers the classical realization theorem for affine Kac-Moody Lie algebras.<sup>[5](https://arxiv.org/abs/0709.0975)</sup>

Central extensions of these algebras are well studied. C. Kassel constructed the universal central extensions of untwisted multiloop Lie algebras using Kähler differentials in 1984, and [Jie Sun](https://www.edgepedia.org/jie-sun) gave sufficient conditions under which a descent construction yields the universal central extension of a twisted form of a split simple Lie algebra over a ring, obtaining in particular the universal central extensions of twisted multiloop Lie tori.<sup>[11](https://arxiv.org/abs/1011.4086)</sup>

## Applications to integrable systems

Loop algebras provide a standard framework for integrable Hamiltonian systems. John Harnad, a mathematician at [Concordia University](https://www.edgepedia.org/concordia-university), showed that standard examples including the Neumann oscillator, the cubically nonlinear Schrödinger systems, and the sine-Gordon equation can be cast as isospectral flows in loop algebras; each system has an associated invariant spectral curve and is integrated by the Liouville-Arnold technique, with the isospectral property following from the Adler-Kostant-Symes theorem.<sup>[12](https://www.researchgate.net/publication/226497977_Isospectral_flow_and_Liouville-Arnold_integration_in_loop_algebras)</sup> Many finite-dimensional integrable systems, as well as finite-gap reductions of integrable partial differential equations, can be regarded as flows on finite-dimensional [coadjoint orbits](https://www.edgepedia.org/coadjoint-orbit) of the loop algebra \( \widetilde{\mathfrak{gl}}(r) \), described by \( r \times r \) Lax equations \( \dot{L}(\lambda) = [L(\lambda), M(\lambda)] \) with a spectral parameter \( \lambda \in \mathbb{C} \).<sup>[13](https://upcommons.upc.edu/server/api/core/bitstreams/3a25bcc4-8212-4077-ba15-ff7c2819ef98/content)</sup> In the Lax framework the coefficients of the matrices are polynomials in \( \lambda \) and \( \lambda^{-1} \), which is precisely the loop-algebra setting.<sup>[6](https://audin.pages.math.unistra.fr/livres/Audin_Integrability.pdf)</sup>

**Lie-theoretic refinements.** Quinn and Singer used the structure theory of affine Kac-Moody algebras, rather than that of loop algebras, to construct the natural analog of the nonperiodic [Toda lattice](https://www.edgepedia.org/toda-lattice). The result is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space, prescribed purely in terms of Lie-theoretic data, whose commuting functions are the gauge-invariant functions obtained by viewing elements of the loop algebra as connections on a bundle over \( S^{1} \).<sup>[14](https://arxiv.org/abs/solv-int/9606008)</sup>

In the quantum theory, evaluation representations play a central role: from a representation \( \rho \) of a finite-dimensional Lie algebra \( \mathfrak{g} \) on a space \( V \) one constructs a one-parameter family \( \rho_{u} \) of representations of a half loop algebra by \( \rho_{u}(J_{a}^{n}) := u^{n} \rho(J_{a}) \), where the constant \( u \in \mathbb{C} \) is called the evaluation parameter.<sup>[15](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> In applications to quantum integrable systems one considers families of representations of a quantum loop algebra parametrized by a complex spectral parameter.<sup>[16](https://pure.mpg.de/rest/items/item_3176510_3/component/file_3176512/content)</sup>

## References

1. [Loop algebra, testwiki](https://en.wiki.beta.math.wmflabs.org/wiki/Loop_algebra)
2. [Loop algebra, PlanetMath](https://planetmath.org/loopalgebra)
3. [Lecture notes on Lie groups, Lecture 10: Affine Kac-Moody algebras](https://stacky.net/files/written/LieGroups/LieGroups.pdf)
4. [Structure and Representation Theory of Kac-Moody Algebras, thesis (University of Pittsburgh)](https://sites.pitt.edu/~bion/pdf/mehrle-thesis.pdf)
5. [Multiloop realization of extended affine Lie algebras and Lie tori, arXiv:0709.0975](https://arxiv.org/abs/0709.0975)
6. [Audin, An Introduction to Lax Equations (Integrability lecture text)](https://audin.pages.math.unistra.fr/livres/Audin_Integrability.pdf)
7. [Khesin, Lecture notes (periodic matrices and polynomial loop algebras)](https://www.math.utoronto.ca/khesin/papers/Lecture_notes.pdf)
8. [Mathlib.Algebra.Lie.Loop, Lean mathematical library documentation](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Loop.html)
9. [Kac-Moody Algebras and Applications (Berkeley lecture notes)](https://math.berkeley.edu/~barrett/resources/km.pdf)
10. [Multiloop Lie algebras and the construction of extended affine Lie algebras, arXiv:0807.2019](https://arxiv.org/abs/0807.2019)
11. [Jie Sun, Universal central extensions of twisted forms of split simple Lie algebras over rings, arXiv:1011.4086](https://arxiv.org/abs/1011.4086)
12. [John Harnad, Isospectral flow and Liouville-Arnold integration in loop algebras, Lecture Notes in Physics 424 (Springer, 1993)](https://www.researchgate.net/publication/226497977_Isospectral_flow_and_Liouville-Arnold_integration_in_loop_algebras)
13. [Yuri Fedorov, Bäcklund Transformations on Coadjoint Orbits of the Loop Algebra gl(r), Journal of Nonlinear Mathematical Physics 9 (2002)](https://upcommons.upc.edu/server/api/core/bitstreams/3a25bcc4-8212-4077-ba15-ff7c2819ef98/content)
14. [M. Quinn and S. F. Singer, Loop algebras, gauge invariants and a new completely integrable system, arXiv:solv-int/9606008](https://arxiv.org/abs/solv-int/9606008)
15. [Beisert, Introduction to Integrability, Lecture Notes (ETH Zürich)](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)
16. [Vertex Models and Spin Chains in Formulas and Pictures (SIMGA)](https://pure.mpg.de/rest/items/item_3176510_3/component/file_3176512/content)
17. [Affine Lie Algebras, Sage 9.3.beta9 Reference Manual: Algebras](https://sporadic.stanford.edu/reference/algebras/sage/algebras/lie_algebras/affine_lie_algebra.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Kac–Moody and affine Lie algebras › Affine Kac–Moody algebras, loop algebras, and central extensions*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 11, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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