1,598 research outputs found
Commutators and squares in free groups
Let F_2 be the free group generated by x and y. In this article, we prove
that the commutator of x^m and y^n is a product of two squares if and only if
mn is even. We also show using topological methods that there are infinitely
many obstructions for an element in F_2 to be a product of two squares.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-27.abs.htm
Global symmetries of Yang-Mills squared in various dimensions
Tensoring two on-shell super Yang-Mills multiplets in dimensions
yields an on-shell supergravity multiplet, possibly with additional matter
multiplets. Associating a (direct sum of) division algebra(s) with
each dimension we obtain formulae for the algebras
and of the U-duality group and its maximal
compact subgroup , respectively, in terms of the internal global symmetry
algebras of each super Yang-Mills theory. We extend our analysis to include
supergravities coupled to an arbitrary number of matter multiplets by allowing
for non-supersymmetric multiplets in the tensor product.Comment: 25 pages, 2 figures, references added, minor typos corrected, further
comments on sec. 2.4 included, updated to match version to appear in JHE
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