## Kyoto Journal of Mathematics

### Projective unitary representations of infinite-dimensional Lie groups

#### Abstract

For an infinite-dimensional Lie group $G$ modeled on a locally convex Lie algebra ${\mathfrak{g}}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth structure on $G^{\sharp}$.) For infinite-dimensional Lie groups $G$ which are $1$-connected, regular, and modeled on a barreled Lie algebra ${\mathfrak{g}}$, we characterize the unitary ${\mathfrak{g}}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of $G$, smooth linear unitary representations of $G^{\sharp}$, and the appropriate unitary representations of its Lie algebra ${\mathfrak{g}}^{\sharp}$.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 2 (2019), 293-341.

Dates
Revised: 17 January 2017
Accepted: 15 February 2017
First available in Project Euclid: 2 April 2019

https://projecteuclid.org/euclid.kjm/1554170605

Digital Object Identifier
doi:10.1215/21562261-2018-0016

Mathematical Reviews number (MathSciNet)
MR3960295

Zentralblatt MATH identifier
07080106

#### Citation

Janssens, Bas; Neeb, Karl-Hermann. Projective unitary representations of infinite-dimensional Lie groups. Kyoto J. Math. 59 (2019), no. 2, 293--341. doi:10.1215/21562261-2018-0016. https://projecteuclid.org/euclid.kjm/1554170605

#### References

• [1] H. Araki and E. J. Woods, Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. Math. Phys. 4 (1963), no. 5, 637–662.
• [2] V. Bargmann, On unitary ray representations of continuous groups, Ann. of Math. (2) 59 (1954), no. 1, 1–46.
• [3] D. Beltiţă and K.-H. Neeb, A nonsmooth continuous unitary representation of a Banach–Lie group, J. Lie Theory 18 (2008), no. 4, 933–936.
• [4] R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. (2) 23 (1977), nos. 3–4, 209–220.
• [5] N. Bourbaki, Sur certains espaces vectoriels topologiques, Ann. Inst. Fourier (Grenoble) 2 (1950), 5–16.
• [6] L. Gårding, Note on continuous representations of Lie groups, Proc. Natl. Acad. Sci. USA 33 (1947), 331–332.
• [7] H. Glöckner, Lie groups over non-discrete topological fields, preprint, arXiv:math/0408008v1 [math.GR].
• [8] H. Glöckner, Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces, in preparation.
• [9] H. Glöckner and K.-H. Neeb, Infinite-Dimensional Lie Groups, I: Basic Theory and Main Examples, in preparation.
• [10] P. Goddard, A. Kent, and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Comm. Math. Phys. 103 (1986), no. 1, 105–119.
• [11] R. Goodman and N. R. Wallach, Projective unitary positive-energy representations of $\operatorname{Diff}(S^{1})$, J. Funct. Anal. 63 (1985), no. 3, 299–321.
• [12] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.
• [13] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978.
• [14] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland Ser. Stat. Probab. 1, North-Holland, Amsterdam, 1982.
• [15] B. Janssens and K.-H. Neeb, Positive energy representations of gauge groups, in preparation.
• [16] B. Janssens and C. Wockel, Universal central extensions of gauge algebras and groups, J. Reine Angew. Math. 682 (2013), 129–139.
• [17] V. G. Kac, An elucidation of “Infinite-dimensional algebras, Dedekind’s $\eta$-function, classical Möbius function and the very strange formula.” $E^{(1)}_{8}$ and the cube root of the modular invariant $j$, Adv. Math. 35 (1980), no. 3, 264–273.
• [18] V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
• [19] V. G. Kac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, Adv. Ser. Math. Phys. 2, World Scientific, Teaneck, NJ, 1987.
• [20] B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, Ergeb. Math. Grenzgeb. (3) 51, Springer, Berlin, 2009.
• [21] G. Köthe, Topological Vector Spaces, I, Grundlehren Math. Wiss. 159, Springer, New York, 1969.
• [22] R. P. Langlands, “On unitary representations of the Virasoro algebra” in Infinite-Dimensional Lie Algebras and Their Applications (Montreal, PQ, 1986), World Scientific, Teaneck, NJ, 1988, 141–159.
• [23] T. Loke, Operator algebras and conformal field theory of the discrete series representation of $\operatorname{Diff}^{+}({\mathbb{S}}^{1})$, Ph.D. dissertation, University of Cambridge, Cambridge, 1994.
• [24] J. Manuceau and A. Verbeure, Quasi-free states of the C.C.R.—algebra and Bogoliubov transformations, Comm. Math. Phys. 9 (1968), 293–302.
• [25] J. Mickelsson, Two-Cocycle of a Kac-Moody Group, Phys. Rev. Lett. 55 (1985), no. 20, 2099–2101.
• [26] J. Mickelsson, Current Algebras and Groups, Plenum Press, New York, 1989.
• [27] J. Milnor, “Remarks on infinite-dimensional Lie groups” in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, 1007–1057.
• [28] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, De Gruyter Exp. Math. 28, de Gruyter, Berlin, 2000.
• [29] K.-H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1365–1442.
• [30] K.-H. Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2006), no. 2, 291–468.
• [31] K.-H. Neeb, On differentiable vectors for representations of infinite dimensional Lie groups, J. Funct. Anal. 259 (2010), no. 11, 2814–2855.
• [32] K.-H. Neeb, Semibounded representations and invariant cones in infinite dimensional Lie algebras, Confluentes Math. 2 (2010), no. 1, 37–134.
• [33] K.-H. Neeb, Positive energy representations and continuity of projective representations for general topological groups, Glasg. Math. J. 56 (2014), no. 2, 295–316.
• [34] K.-H. Neeb and H. Salmasian, Classification of positive energy representations of the Virasoro group, Int. Math. Res. Not. IMRN 2015, no. 18, 8620–8656.
• [35] K.-H. Neeb and C. Wockel, Central extensions of groups of sections, Ann. Global Anal. Geom. 36 (2009), no. 4, 381–418.
• [36] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615.
• [37] D. Petz, An Invitation to the Algebra of Canonical Commutation Relations, Leuven Notes Math. Theoret. Phys. Ser. A Math. Phys., Leuven Univ. Press, Leuven, 1990.
• [38] D. Pickrell, On the Mickelsson-Faddeev extension and unitary representations, Comm. Math. Phys. 123 (1989), no. 4, 617–625.
• [39] A. Pressley and G. Segal, Loop Groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1986.
• [40] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
• [41] M. A. Rieffel, “Unitary representations of group extensions: An algebraic approach to the theory of Mackey and Blattner” in Studies in Analysis, Adv. Math. Suppl. Stud. 4, Academic Press, New York, 1979, 43–82.
• [42] W. Rudin, Functional Analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991.
• [43] G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981), no. 3, 301–342.
• [44] V. Toledano Laredo, Integrating unitary representations of infinite-dimensional Lie groups, J. Funct. Anal. 161 (1999), no. 2, 478–508.
• [45] A. van Daele, Quasi-equivalence of quasi-free states on the Weyl algebra, Comm. Math. Phys. 21 (1971), 171–191.
• [46] A. van Daele and A. Verbeure, Unitary equivalence of Fock representations on the Weyl algebra, Comm. Math. Phys. 20 (1971), 268–278.
• [47] V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer, New York, 1985.
• [48] J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), no. 1, 570–578.
• [49] A. J. Wassermann, Operator algebras and conformal field theory, III: Fusion and positive energy representations of $\operatorname{LSU}(N)$ using bounded operators, Invent. Math. 133 (1998), no. 3, 467–538.