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Homeloop algebra

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# loop algebra

Let $\mathfrak{g}$ be a Lie algebra over a field $\mathbb{K}$. The loop algebra based on $\mathfrak{g}$ is defined to be $\mathcal{L}(\mathfrak{g}):=\mathfrak{g}\otimes_{{\mathbb{K}}}\mathbb{K}[t,t^{{% -1}}]$ as a vector space over $\mathbb{K}$. The Lie bracket is determined by

$\left[X\otimes t^{k},Y\otimes t^{l}\right]=\left[X,Y\right]_{{\mathfrak{g}}}% \otimes t^{{k+l}}$ |

where $\left[\,,\,\right]_{{\mathfrak{g}}}$ denotes the Lie bracket from $\mathfrak{g}$.

This clearly determines a Lie bracket. For instance the three term sum in the Jacobi identity (for elements which are homogeneous in $t$) simplifies to the three term sum for the Jacobi identity in $\mathfrak{g}$ tensored with a power of $t$ and thus is zero in $\mathcal{L}(\mathfrak{g})$.

The name “loop algebra” comes from the fact that this Lie algebra arises in the study of Lie algebras of loop groups. For the time being, assume that $\mathbb{K}$ is the real or complex numbers so that the familiar structures of analysis and topology are available. Consider the set of all mappings from the circle $S^{1}$ (we may think of this circle more concretely as the unit circle of the complex plane) to a finite-dimensional Lie group $G$ with Lie algebra is $\mathfrak{g}$. We may make this set into a group by defining multiplication pointwise: given $a,b\colon S^{1}\to G$, we define $(a\cdot b)(x)=a(x)\cdot b(x)$.

# References

- 1
Victor Kac,
*Infinite Dimensional Lie Algebras*, Third edition. Cambridge University Press, Cambridge, 1990.

## Mathematics Subject Classification

22E60*no label found*22E65

*no label found*22E67

*no label found*

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