357 research outputs found
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction
Analytic Loops and Gauge Fields
In this paper the linear representations of analytic Moufang loops are
investigated. We prove that every representation of semisimple analytic Moufang
loop is completely reducible and find all nonassociative irreducible
representations. We show that such representations are closely associated with
the (anti-)self-dual Yang-Mills equations in Comment: 10 pages, LaTeX, no figure
Alternating quaternary algebra structures on irreducible representations of sl(2,C)
We determine the multiplicity of the irreducible representation V(n) of the
simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power
. The multiplicity is 1 (resp. 2) if and only if n = 4, 6
(resp. n = 8, 10). For these n we determine the multilinear polynomial
identities of degree satisfied by the sl(2,C)-invariant alternating
quaternary algebra structures obtained from the projections . We represent the polynomial identities as the nullspace of a large
integer matrix and use computational linear algebra to find the canonical basis
of the nullspace.Comment: 26 pages, 13 table
A commutant realization of Odake's algebra
The bc\beta\gamma-system W of rank 3 has an action of the affine vertex
algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W)
contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an
extension of the N=2 superconformal algebra with c=9, and is generated by eight
fields which close nonlinearly under operator product expansions. Our main
result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual
commutants) inside C. More generally, any finite-dimensional representation of
a Lie algebra g gives rise to a similar Howe pair, and this example corresponds
to the adjoint representation of sl_2.Comment: Minor corrections, discussion of Odake's algebra in Section 2
expanded, final versio
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