nLab conformal symplectic group

Definition

Let $k$ denote the ground field and $k^\times$ its group of units.

Definition

Given a symplectic vector space $(C,\omega)$, its conformal symplectic group $CSp(V,\omega)$ is the group of linear isomorphisms $f \colon V \to V$ which preserve the symplectic form up to a scalar $\lambda(f) \in k^\times$:

$$\omega\big( f(-) ,\, f(-) \big) \;=\; \lambda(f) \cdot \omega(-,-) \,.$$

(e.g. Guillemin & Sternberg 1977 p 115, Malle & Testerman 2012 p. 7, Taylor 2021 §1).

Remark

Remembering just this conformal scale $\lambda$ evidently constitutes a group homomorphism from $CSp(V,\omega)$ to $k^\times$, whose kernel is the ordinary symplectic group:

$$0 \to Sp(V,\omega) \longrightarrow OSp(V,\omega) \xrightarrow{\phantom{-} \lambda \phantom{-}} k^\times \to 0 \,.$$

Remark

In contrast to symmetric bilinear forms, where the conformal group is typically assumed to rescale only by positive numbers, for symplectic forms it more generally makes sense to rescale by any non-vanishing number. But beware that some authors constrain elements of a conformal symplectic group to have positive multiplier (e.g. Jensen & Kruglikov 2020 §7).

Related concepts

• conformal group, symplectic group

• conformal invariance and scale invariance

References

• Victor Guillemin, Shlomo Sternberg, p. 115 of: Geometric asymptotics, Mathematical Surveys and Monographs 14, AMS (1977) [ams:surv-14]

• Rudolf Scharlau, Pham Huu Tiep: Symplectic group lattices, Trans. Amer. Math. Soc. 351 (1999) 2101-2139 [doi:10.1090/S0002-9947-99-02469-1]

• Gunter Malle, Donna Testerman, p. 7 of: Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge University Press (2012) [doi:10.1017/CBO9780511994777]

• D. E. Taylor: Conjugacy Classes in Finite Conformal Symplectic Groups (2021) [pdf]

See also:

• Jørn Olav Jensen, Boris Kruglikov: Differential Invariants of Linear Symplectic Actions [arXiv:2010.08024]