A177821 a(n) gives the number of nonisomorphic connected compact Lie groups of dimension n which are simple products.

1, 1, 1, 3, 3, 3, 6, 6, 8, 12, 14, 18, 23, 27, 34, 43, 52, 62, 79, 93, 109, 138, 159, 187, 236, 270, 316, 385, 443, 518, 620, 719, 836, 983, 1138, 1314, 1541, 1770, 2041, 2388, 2726, 3122, 3628, 4124, 4720, 5459, 6204, 7063, 8116, 9203, 10440, 11940, 13525, 15306, 17436, 19690, 22231, 25208, 28388, 32013, 36217, 40673, 45729, 51575, 57808, 64817

Offset

0,4

Comments

By the structure theorem for compact Lie groups, every compact connected Lie group is a finite central quotient of a product of copies of the circle group U(1) and compact simple Lie groups which are all known due to Cartan's classification. This sequence counts only those which are direct products of such groups.

Links

Table of n, a(n) for n=0..65.

Formula

G.f.: 1/((1-x)*(1-x^3)^2*(1-x^8)^2*(1-x^10)^2*(1-x^14)*...) = (1/(1-x)) * Product_{k>=0} (1-x^k)^A178176(k) with (1-x^k)^0 taken to be 1.

Example

For n=0, the trivial group is the only such group.
For n=8, the 8 Lie groups are U(1)^8, U(1)^5 x SU(2), U(1)^5 x SO(3), U(1)^2 x SU(2)^2, U(1)^2 x SU(2) x SO(3), U(1)^2 x SO(3)^2, SU(3) and SU(3)/3.

Crossrefs

See also A178176 for enumeration of the simple factors giving these counts.

Sequence in context: A013322 A211534 A219816 * A166273 A244482 A219299

Adjacent sequences: A177818 A177819 A177820 * A177822 A177823 A177824

Keyword

nonn

Author

Andrew Rupinski, Dec 18 2010

Status

approved