loop algebra
Let $\mathrm{\pi \x9d\x94\u20ac}$ be a Lie algebra^{} over a field $\mathrm{\pi \x9d\x95\x82}$. The loop algebra based on $\mathrm{\pi \x9d\x94\u20ac}$ is defined to be $\mathrm{\beta \x84\x92}\beta \x81\u2019(\mathrm{\pi \x9d\x94\u20ac}):=\mathrm{\pi \x9d\x94\u20ac}{\beta \x8a\x97}_{\mathrm{\pi \x9d\x95\x82}}\mathrm{\pi \x9d\x95\x82}\beta \x81\u2019[t,{t}^{-1}]$ as a vector space^{} over $\mathrm{\pi \x9d\x95\x82}$. The Lie bracket is determined by
$$[X\beta \x8a\x97{t}^{k},Y\beta \x8a\x97{t}^{l}]={[X,Y]}_{\mathrm{\pi \x9d\x94\u20ac}}\beta \x8a\x97{t}^{k+l}$$ |
where ${[,]}_{\mathrm{\pi \x9d\x94\u20ac}}$ denotes the Lie bracket from $\mathrm{\pi \x9d\x94\u20ac}$.
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 $\mathrm{\pi \x9d\x94\u20ac}$ tensored with a power of $t$ and thus is zero in $\mathrm{\beta \x84\x92}\beta \x81\u2019(\mathrm{\pi \x9d\x94\u20ac})$.
The name βloop algebraβ comes from the fact that this Lie algebra arises in the study of Lie algebras of loop groups. For the time being, assume that $\mathrm{\pi \x9d\x95\x82}$ 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 $\mathrm{\pi \x9d\x94\u20ac}$. We may make this set into a group by defining multiplication^{} pointwise: given $a,b:{S}^{1}\beta \x86\x92G$, we define $(a\beta \x8b\x85b)\beta \x81\u2019(x)=a\beta \x81\u2019(x)\beta \x8b\x85b\beta \x81\u2019(x)$.
References
- 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
Title | loop algebra |
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Canonical name | LoopAlgebra |
Date of creation | 2013-03-22 15:30:07 |
Last modified on | 2013-03-22 15:30:07 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 12 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 22E60 |
Classification | msc 22E65 |
Classification | msc 22E67 |
Defines | loop algebra |