Clifford algebra analogue of the Cartan theorem

30.04.2024 15:30 – 16:30

Let $\mathfrak{g}$ be a complex simple Lie algebra. The Hopf-Koszul-Samelson theorem asserts that the algebra of $\mathfrak{g}$-invariants in the exterior algebra of $\mathfrak{g}$ is the exterior algebra over the space of primitive invariants $P$. Kostant proved the analogous result for the Clifford algebra of $\mathfrak{g}$. Namely, the algebra of $\mathfrak{g}$-invariants in $Cl(\mathfrak{g})$ is the Clifford algebra over the space of primitive invariants.

Let $\mathfrak{k}$ be a symmetric subalgebra of $\mathfrak{g}$ and $\mathfrak{p}$ be the corresponding isotropy representation. The Cartan theorem states that the algebra of $\mathfrak{k}$-invariants in the
exterior algebra of $\mathfrak{p}$ is isomorphic to the tensor product of the exterior algebra of the Samelson subspace of $P$, corresponding to the pair $(\mathfrak{g},\mathfrak{k})$, and a certain commutative algebra $A$. We prove a Clifford algebra analogue of the Cartan theorem. Namely, we show that the algebra of $\mathfrak{k}$-invariants in $Cl(\mathfrak{p})$ is the tensor product of the Clifford algebra over the space of primitive invariant of $\mathfrak{p}$ with a certain filtered deformation of $A$.

This is joint work with K. Calvert, K. Grizelj, and P. Pandžić.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Groupes de Lie et espaces de modules"

entrée libre