Imbedding of Lie algebras in nonassociative structures
- R. M. Santilli
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Notes on a lecture given at the ICTP, Trieste, June 27, 1967.
- The C.J.A. are nonassociative algebras defined by the relations: i)ab=ba, and ii) (a 2b)a=a 2 (ba). They are subdivided into: i) thespecial C.J.A., which are characterized by the productab=1/2(a·b+b·a)=1/2{a, b} (we calla·b the associative product), and ii) theexceptional C.J.A.,i.e. the algebras which are not special. For an extensive bibliography on C.J.A. seeH. Braun andM. Kocher:Jordan-Algebren (Berlin, 1966).
- Albert, A. A. (1948) Trans. A.M.S. 64: pp. 552-552 CrossRef
- Schafer, R. D. (1955) Proc. A.M.S. 6: pp. 472-472 CrossRef
- Schafer, R. D. (1955) Proc. A.M.S. 6: pp. 472-472 CrossRef
- Albert, A. A. (1949) Proc. N.A.S. 35: pp. 317-317 CrossRef
- However, for λ→∞,ab→[a, b]. The author is indebted to Prof.A. Salam for this remark.
- For the case withA=associative algebra see also:R. M. Santilli andG. Soliani:A statistics and parastatistics formal unification, to appear.
- Given an algebraA with productab and an invertible elementc we can form an algebraA *, called theisotope ofA, with the producta*b=acb. As a particular case we may havec=α1, where α is a free (nonzero) scalar. Then the new multiplication inA * is simply α times the old multiplication inA: a*b=αab.
- The author is indebted to Prof.K. McCrimmon for a very kind letter (of June 12, 1967), where the connections between theA(λ, μ) andA(λ) algebras are explicitly investigated.McCrimmon notes that for ϑ, σ≠0, by putting τ=2σ, λ′=λ/τ and μ′=μ/τ, we have λ′+μ′=1. Then (a, b)=λ′τa·b+μ′τb·a=λ′a *·b+(1−λ′)b *·a. HenceA(λ, м) is just the (λ/2σ)-mutation of the isotopic algebraA *,i.e. A(λ, μ) is isomorphic toA *(λ/2σ). In addition we note that, since σ=λ+μ, the isotopic algebraA * characterized by the producta *·b=(λ+μ)a·b is the zero algebra for λ=−μ. Furthermore for λ=−μA(λ, μ) corresponds to the ∞-mutation of theA * (zero) algebra (see also footnote (6)). Hence theA(λ, μ) andA(λ) algebras are equivalent for every λ≠−μ, while the case λ=−μ corresponds to the explicit Lie content of theA(λ, μ) algebras which occurs when the discriminant has the degenerate value ∞.
- Clearly for conservative systems the Hamiltonian mechanics and Lie algebra are completely satisfactory.
- R. M. Santilli:Some remarks on pseudo-Hamiltonian mechanics, to appear.
- Formànek, J. (1966) Czech. Journ. Phys. 16: pp. 281-281 CrossRef
- For instance, in the decay πarм+ν there in the transition from bosons to fermions, which leaves open the problem of characterization of the decaying region from a statistical viewpoint (7).
- Segal, I. E. (1951) Duke Math. Journ. 18: pp. 221-221 CrossRef
- Title
- Imbedding of Lie algebras in nonassociative structures
- Journal
-
Il Nuovo Cimento A
Volume 51, Issue 2 , pp 570-576
- Cover Date
- 1967-09-01
- DOI
- 10.1007/BF02902200
- Print ISSN
- 0369-3546
- Online ISSN
- 1826-9869
- Publisher
- Società Italiana di Fisica
- Additional Links
- Topics
- Authors
-
- R. M. Santilli (1)
- Author Affiliations
-
- 1. Istituto di Fisica dell’Università, Torino