## Journal of the Mathematical Society of Japan

### Extensions of current groups on $S^3$ and the adjoint representations

Tosiaki KORI

#### Abstract

Let $\Omega^3(SU(n))$ be the Lie group of based mappings from $S^3$ to $SU(n)$. We construct a Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$ by the abelian group $\exp 2\pi i {\cal A}_3^{\ast}$, where ${\cal A}_3^{\ast}$ is the affine dual of the space of $SU(n)$-connections on $S^3$. J. Mickelsson in 1987 constructed a similar Lie group extension. In this article we give several improvement of his results, especially we give a precise description of the extension of those components that are not the identity component. We also correct several argument about the extension of $\Omega^3(SU(2))$ which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of $\Omega^3(SU(2))$ reduces to the extension by Z$_2$ is correct. Then we shall investigate the adjoint representation of the Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 819-838.

Dates
First available in Project Euclid: 24 July 2014

https://projecteuclid.org/euclid.jmsj/1406206974

Digital Object Identifier
doi:10.2969/jmsj/06630819

Mathematical Reviews number (MathSciNet)
MR3238319

Zentralblatt MATH identifier
1298.81107

#### Citation

KORI, Tosiaki. Extensions of current groups on $S^3$ and the adjoint representations. J. Math. Soc. Japan 66 (2014), no. 3, 819--838. doi:10.2969/jmsj/06630819. https://projecteuclid.org/euclid.jmsj/1406206974

#### References

• J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progr. Math., 107, Birkhäuser Boston, Basel, Berlin, 1993.
• B. A. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, Ergeb. Math. Grenzgeb. (3), 51, Springer-Verlag, Berlin, Heidelberg, 2009.
• T. Kori, Four-dimensional Wess-Zumino-Witten actions, J. Geom. Phys., 47 (2003), 235–258.
• T. Kori, Chern-Simons pre-quantizations over four-manifolds, Differential Geom. Appl., 29 (2011), 670–684.
• J. Mickelsson, Current Algebras and Groups, Plenum Monogr. Nonlinear Phys., Plenum Press, New York, 1989.
• J. Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Comm. Math. Phys., 110 (1987), 173–183.
• A. M. Polyakov and P. B. Wiegmann, Goldstone fields in two dimensions with multivalued actions, Phys. Lett. B, 141 (1984), 223–228.
• A. Pressley and G. Segal, Loop Groups, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, New York, 1986.
• G. Segal, Unitary representations of some infinite dimensional groups, Comm. Math. Phys., 80 (1981), 301–342.
• E. Witten, Current algebra, baryons, and quark confinement, Nuclear Phys. B, 223 (1983), 433–444.
• B. Zumino, Chiral anomalies and differential geometry, In: Relativité, Groupes Et Topologie. II, Les Houches, 1983, (ed. B. S. DeWitt and R. Stora), North-Holland Publishing Co., Amsterdam, 1984, pp.,1291–1332.