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A340223
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Number of n-dimensional semisimple algebras over the reals.
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1
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1, 1, 2, 2, 5, 5, 8, 8, 15, 16, 23, 24, 37, 40, 53, 56, 80, 87, 113, 120, 162, 175, 221, 234, 304, 329, 407, 434, 544, 589, 711, 760, 929, 1006, 1198, 1283, 1542, 1665, 1958, 2093, 2480, 2673, 3112, 3329, 3894, 4194, 4829, 5165, 5971, 6426, 7339, 7850, 8996, 9667, 10965
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OFFSET
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0,3
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COMMENTS
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The Artin-Wedderburn theorem states that every finite-dimensional semisimple algebra A over a field K can be uniquely (up to the permutation of direct factors) written as A = M_(m_1)(D_1) X M_(m_2)(D_2) X ... X M_(m_n)(D_n), where D_i is a division algebra over K, M_m(D) is the algebra of m X m matrices over D. Here each D_i is necessarily associative since semisimple algebras are defined to be so. By the Frobenius theorem, the only finite-dimensional associative division algebras over the reals are isomorphic to R (the real numbers), C (the complex numbers) or H (the quaternions). (Note that there are other non-associative division algebras over R, the octonions, for example.) As a consequence, every finite-dimensional semisimple algebra over R is built up by M_k(R) (dimension k^2), M_k(C) (dimension 2*k^2) and M_k(H) (dimension 4*k^2).
Also, since the complex numbers C is an algebraically closed field, the only finite-dimensional associative division algebras over C is C itself, so every finite-dimensional semisimple algebra A over C must have C = M_(m_1)(C) X M_(m_2)(C) X ... X M_(m_n)(C). Hence the number of n-dimensional semisimple algebras over C is given by A001156.
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LINKS
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FORMULA
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G.f.: Product_{m>=1} 1/((1-x^(m^2))*(1-x^(2*m^2))*(1-x^(4*m^2))) = f(x)*f(x^2)*f(x^4), where f(x) = Product_{m>=1} 1/(1-x^(m^2)) is the g.f. of A001156.
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EXAMPLE
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List of n-dimensional semisimple algebras over R for n <= 9:
n = 0: {0} (1 in total);
n = 1: R (1 in total);
n = 2: R^2 (the split-complex numbers), C (2 in total);
n = 3: R^3, R x C (2 in total);
n = 4: H, M_2(R) (the split-quaternions), C^2, R^2 x C, R^4 (5 in total);
n = 5: R x H, R x M_2(R), R x C^2, R^3 x C, R^5 (5 in total);
n = 6: C x H, C x M_2(R), R^2 x H, R^2 x M_2(R), C^3, R^2 x C^2, R^4 x C, R^6 (8 in total);
n = 7: R x C x H, R x C x M_2(R), R^3 x H, R^3 x M_2(R), R x C^3, R^3 x C^2, R^5 x C, R^7 (8 in total);
n = 8: M_2(C), H^2, H x M_2(R), (M_2(R))^2, C^2 x H, C^2 x M_2(R), R^2 x C x H, R^2 x C x M_2(R), R^4 x H, R^4 x M_2(R), C^4, R^2 x C^3, R^4 x C^2, R^6 x C, R^8 (15 in total);
n = 9: M_3(R), R x M_2(C), R x H^2, R x H x M_2(R), R x (M_2(R))^2, R x C^2 x H, R x C^2 x M_2(R), R^3 x C x H, R^3 x C x M_2(R), R^5 x H, R^5 x M_2(R), R x C^4, R^3 x C^3, R^5 x C^2, R^7 x C, R^9 (16 in total).
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MATHEMATICA
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CoefficientList[ Series[Product[1/((1 - x^(m^2))*(1 - x^(2 m^2))*(1 - x^(4 m^2))), {m, 70}], {x, 0, 68}], x]
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PROG
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(PARI) seq(n) = Vec(1/prod(k=1, sqrtint(n+1), (1-x^(k^2)+x*O(x^n)) * (1-x^(2*k^2)+x*O(x^n)) * (1-x^(4*k^2)+x*O(x^n)))) \\ Jianing Song, Apr 10 2021, after Paul D. Hanna's first PARI program for A001156.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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