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Table 3 System algebras of quasifree fermionic systems with d modes satisfying translation-invariance

From: A dynamic systems approach to fermions and their relation to spins

d General case (see Theorem 34) (Twisted) Reflection symmetry (see Eq. (45) and Corollary 35)
2 u(1)+u(1) u(1)+u(1)
3 u(2)+u(1) su(2)+u(1)
4 u(2)+u(1)+u(1) su(2)+u(1)+u(1)
5 u(2)+u(2)+u(1) su(2)+su(2)+u(1)
6 u(2)+u(2)+u(1)+u(1) su(2)+su(2)+u(1)+u(1)
2n − 1 i = 1 n 1 u(2)+u(1) i = 1 n 1 su(2)+u(1)
2n i = 1 n 1 u(2)+u(1)+u(1) i = 1 n 1 su(2)+u(1)+u(1)