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Table 1 System algebras of translation-invariant fermionic systems with d modes for (a) nearest-neighbor interactions only and (b) arbitrary translation-invariant interactions

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

d System algebra for (a) System algebra for (b)
2 i = 1 2 u(1) i = 1 2 u(1)
3 i = 1 2 su(2)+ i = 1 3 u(1) i = 1 2 su(2)+ i = 1 4 u(1)
4 i = 1 5 su(2)+ i = 1 4 u(1) i = 1 8 su(2)+ i = 1 6 u(1)
5 i = 1 2 su(4)+ i = 1 8 su(3)+ i = 1 3 u(1) i = 1 2 su(4)+ i = 1 8 su(3)+ i = 1 8 u(1)
6 i = 1 4 su(6)+ i = 1 8 su(5)+ i = 1 3 u(1) i = 1 4 su(6)+ i = 1 8 su(5)+ i = 1 10 u(1)