Generalized spin representations

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• Guntram Hainke, University of Birmingham
• Thursday 24 February 2011, 16:00-17:00
• Watson Building, Lecture Room A.

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

The special orthogonal group SO_n(R) is a maximal compact subgroup of SL_n(R). Its Lie algebra so_n(R) is therefore called a maximal compact subalgebra of sl_n(R), and it can be characterized as the fixed point set of the Cartan–Chevalley involution ω: A→ −A^T of sl_n(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.

This talk is included in these lists:

• Algebra Seminar
• School of Mathematics Events
• Watson Building, Lecture Room A

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