A382233 Dimensions of the homogeneous component of degree n of the free unital Jordan algebra on 3 generators.

1, 3, 6, 18, 45, 135, 378, 1134, 3324, 9981, 29733, 89280, 267273

Offset: 0

Author

Vladimir Dotsenko, Mar 29 2025

Keywords

• nonn
• hard
• more

Comments

First few terms coincide with A032120 but A032120(8) = 3321. This corresponds to the fact that A032120 gives dimensions of components of the free special Jordan algebra (which follows from Cohn 1959), and 3324 - 3321 = 3 is the dimension of the GL_3-orbit of the so called Glennie identity.

The terms up to a(12) were computed using the Albert nonassociative algebra system.

Examples

			For n = 3, we have a(3)=18 since the following monomials form a basis: x(xx), x(xy), x(xz), x(yy), x(yz), x(zz), y(xx), y(xy), y(xz), y(yy), y(yz), y(zz), z(xx), z(xy), z(xz), z(yy), z(yz), z(zz), these are all commutative nonassociative monomials of degree 3, since the Jordan identity is of degree 4.
		

References

• C. M. Glennie, Identities in Jordan algebras, pp. 307-313 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
• D. P. Jacobs, The Albert nonassociative algebra system: a progress report, pp. 41-44 of Proceedings of the International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, New York, NY, USA, 1994.

Links

• Albert nonassociative algebra system, Homepage
• P. M. Cohn, Two embedding theorems for Jordan algebras, Proceedings of the London Mathematical Society, Volume s3-9, Issue 4, October 1959, pp. 503-524.

Crossrefs

Cf. A001776, A032120.