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Generalized spin representations

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If you have a question about this talk, please contact David Craven.

The special orthogonal group SOn(R) is a maximal compact subgroup of SLn(R). Its Lie algebra son(R) is therefore called a maximal compact subalgebra of sln(R), and it can be characterized as the fixed point set of the Cartan–Chevalley involution ω: A→ −AT of sln(R).

Kac–Moody algebras were introduced in the 1960s to generalize complex semisimple Lie algebras and have since then found applications in theoretical physics. For a Kac–Moody algebra g one can similarly define its maximal compact subalgebra as k = Fix ω.

In the case of g = E10(R), theoretical physicists have discovered that the spin representation of so10(R) = k(A9) ≤ k(E10) can be extended to a representation of k(E10).

In this talk, we discuss this representation and introduce a general framework which encompasses it. With the help of these so-called generalized spin representations, we derive some algebraic properties of maximal compact subalgebras of simply-laced Kac–Moody algebras.

This is joint work with Ralf Gramlich.

This talk is part of the Algebra Seminar series.

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