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User: Andrew Rupinski

Andrew Rupinski has authored 3 sequences.

A178930 Number of semisimple Lie algebras of dimension n.

0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 5, 3, 4, 8, 4, 5, 8, 7, 8, 11, 10, 11, 12, 13, 15, 19, 16, 21, 24, 21, 24, 32, 27, 34, 43, 37, 39, 53, 47, 54, 65, 65, 68, 79, 80, 90, 98, 102, 114, 129, 122, 138, 160, 157, 172, 207, 193, 211, 247, 244, 262, 306, 305, 329, 363, 378, 399, 448, 460

Offset: 0

Andrew Rupinski, Jan 11 2011

nonn

a(n) is also the number of simply-connected semisimple Lie groups.

Is a(n) eventually monotonically increasing, and if so, beyond what index?

			a(3) = 1 since A_1 is the only semisimple Lie algebra of dimension 3.
For n=21, the a(21) = 5 such Lie algebras are A_1+A_1+A_1+A_1+A_1+A_1+A_1, A_1+A_1+A_3, A_1+A_2+B_2, B_3, and C_3

A178176 a(n) is the number of central quotients of simple compact Lie groups of dimension n.

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Andrew Rupinski, Dec 18 2010

nonn

			a(3) = 2 since the 3-dimensional SU(2) has two central quotients: SU(2) and SU(2)/2 = SO(3).
a(28) = 3 and not 4 since, because of triality for Spin(8), the semi-spin group HSpin(8) is isomorphic to SO(8). Thus, the only groups are Spin(8), SO(8), PSO(8). See the nLab link.
The unusually large value a(78) = 6 is due to Spin(13), SO(13), Sp(6), PSp(6), E_6, E_6/Z3 all of dimension 78.

Andrea Aveni, Table of n, a(n) for n = 1..1000
Math Overflow, Number of compact connected Lie groups of given dimension
nLab, semi-spin group

Number.Divisors=function(n){
  out=c()
  for(j in 1:n){if(n%%j==0){out=c(out,j)}}
  return(length(out))
}
a178176=function(n){
  kSU=sqrt(n+1)
  kSO=(sqrt(8*n+1)+1)/2
  kSp=(sqrt(8*n+1)-1)/4
  a=0
  if(n %in% c(14,52,248)){a=a+1} # G2, F4, E8 with center Z1
  if(n %in% c(78,133)){a=a+2} # E7 with center Z2, E6 with center Z3
  if(kSp%%1==0 & kSp>=2){a=a+2} # Sp(k), PSp(k)
  if(kSU%%1==0 & kSU>=2){a=a+Number.Divisors(kSU)} # SU(n)/Zd
  if(kSO%%1==0 & kSO>=7 & kSO!=8){
    if(kSO%%2!=0){a=a+2} # Spin(k), SO(k)
    if(kSO%%2==0 & kSO%%4!=0){a=a+3} # Spin(k), SO(k), PSO(k)
    if(kSO%%4==0){a=a+4} # Spin(k), SO(k), HSpin(k), PSO(k)
  }
  if(n==28){a=3} # Because of Triality: Spin(8), HSpin(8)=SO(8), PSO(8)
  return(a)
} # Andrea Aveni, Mar 23 2025

a(28) corrected by Andrea Aveni, Mar 23 2025

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

Original entry on oeis.org

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, 442, 517, 619, 716, 833, 980, 1132, 1308, 1533, 1758, 2027, 2370, 2703, 3095, 3594, 4081, 4668, 5397, 6125, 6970, 8007, 9065, 10281, 11753, 13289, 15036, 17120, 19305, 21788, 24690, 27768, 31294, 35381, 39690, 44591, 50261, 56267, 63047

Offset: 0

Andrew Rupinski, Dec 18 2010

nonn

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.

			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.

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

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.

a(28) and following corrected by Andrea Aveni, Mar 22 2025

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User: Andrew Rupinski

A178930 Number of semisimple Lie algebras of dimension n.

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A178176 a(n) is the number of central quotients of simple compact Lie groups of dimension n.

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

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