Let g be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form ((.),(.)), sigma an elliptic automorphism of g leaving the form ((.),(.)) invariant, and a a sigma-invariant subalgebra of g, such that the restriction of the form ((.),(.)) to a non-degenerate. Let (L) over cap (g, sigma) and (L) over cap (a, sigma) be the associated twisted affine Lie algebras and F(sigma)(p) the sigma-twisted Clifford module over (L) over cap (a, sigma), associated to the orthocomplement p of a in g. Under suitable hypotheses on sigma and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight (L) over cap (g, sigma)-module and F(sigma)(p), into irreducible (L) over cap (a, sigma)-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.
ON THE KERNEL OF THE AFFINE DIRAC OPERATOR / Papi, Paolo; V. G., Kac; P., Moseneder Frajria. - In: MOSCOW MATHEMATICAL JOURNAL. - ISSN 1609-3321. - STAMPA. - 8:4(2008), pp. 759-788.
ON THE KERNEL OF THE AFFINE DIRAC OPERATOR
PAPI, Paolo;
2008
Abstract
Let g be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form ((.),(.)), sigma an elliptic automorphism of g leaving the form ((.),(.)) invariant, and a a sigma-invariant subalgebra of g, such that the restriction of the form ((.),(.)) to a non-degenerate. Let (L) over cap (g, sigma) and (L) over cap (a, sigma) be the associated twisted affine Lie algebras and F(sigma)(p) the sigma-twisted Clifford module over (L) over cap (a, sigma), associated to the orthocomplement p of a in g. Under suitable hypotheses on sigma and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight (L) over cap (g, sigma)-module and F(sigma)(p), into irreducible (L) over cap (a, sigma)-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.