17,375 research outputs found
From conformal embeddings to quantum symmetries: an exceptional SU(4) example
We briefly discuss several algebraic tools that are used to describe the
quantum symmetries of Boundary Conformal Field Theories on a torus. The
starting point is a fusion category, together with an action on another
category described by a quantum graph. For known examples, the corresponding
modular invariant partition function, which is sometimes associated with a
conformal embedding, provides enough information to recover the whole
structure. We illustrate these notions with the example of the conformal
embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the
exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures
(ps files) in such a way that they are still understood when converted to
gray levels. Version 3: Several references have been adde
Exceptional quantum subgroups for the rank two Lie algebras B2 and G2
Exceptional modular invariants for the Lie algebras B2 (at levels 2,3,7,12)
and G2 (at levels 3,4) can be obtained from conformal embeddings. We determine
the associated alge bras of quantum symmetries and discover or recover, as a
by-product, the graphs describing exceptional quantum subgroups of type B2 or
G2 which encode their module structure over the associated fusion category.
Global dimensions are given.Comment: 33 pages, 27 color figure
Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups
We obtain formulae giving global dimensions for fusion categories defined by
Lie groups G at level k and for the associated module-categories obtained via
conformal embeddings. The results can be expressed in terms of Lie quantum
superfactorials of type G. The later are related, for the type Ar, to the
quantum Barnes function.Comment: 20 pages, talk given at: Coloquio de Algebras de Hopf, Grupos
Cuanticos y Categorias Tensoriales, Cordoba, Argentina, 200
Recognition of finite exceptional groups of Lie type
Let be a prime power and let be an absolutely irreducible subgroup of
, where is a finite field of the same characteristic as \F_q,
the field of elements. Assume that , a quasisimple group of
exceptional Lie type over \F_q which is neither a Suzuki nor a Ree group. We
present a Las Vegas algorithm that constructs an isomorphism from to the
standard copy of . If with even, then the
algorithm runs in polynomial time, subject to the existence of a discrete log
oracle
The adjoint representation inside the exterior algebra of a simple Lie algebra
For a simple complex Lie algebra we study the space of
invariants , (which describes the isotypic component of type
in ) as a module over the algebra of
invariants . As main result
we prove that is a free module, of rank twice the rank of ,
over the exterior algebra generated by all primitive invariants in , with the exception of the one of highest degree.Comment: Final version. More misprints corrected. To appear in Advances in
Mathematic
Roots of Unity: Representations of Quantum Groups
Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra
of rank n, are constructed from arbitrary representations of rank n-1 quantum
groups for q a root of unity. Representations which have the maximal dimension
and number of free parameters for irreducible representations arise as special
cases.Comment: 23 page
Embeddings of Sz(32) in E_8(5)
We show that the Suzuki group Sz(32) is a subgroup of E_8(5), and so is its automorphism group. Both are unique up to conjugacy in E_8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E_8(5)
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