Download e-book for iPad: Recent Developments in Quantum Affine Algebras and Related by Naihuan Jing, Kailash C. Misra
By Naihuan Jing, Kailash C. Misra
This quantity displays the complaints of the foreign convention on Representations of Affine and Quantum Affine Algebras and Their purposes held at North Carolina country college (Raleigh). in recent times, the speculation of affine and quantum affine Lie algebras has turn into a major quarter of mathematical study with a number of purposes in different components of arithmetic and physics. 3 parts of modern growth are the point of interest of this quantity: affine and quantum affine algebras and their generalizations, vertex operator algebras and their representations, and functions in combinatorics and statistical mechanics. Talks given by way of best foreign specialists on the convention provided either overviews at the topics and present study effects. The e-book properly offers the interaction of those subject matters lately occupying 'center degree' within the concept of limitless dimensional Lie conception
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Extra info for Recent Developments in Quantum Affine Algebras and Related Topics: Representations of Affine and Quantum Affine Algebras and Their Applications, North ... May 21-24, 1998
1 the linear operator Q ψ is compact and self-adjoint on Vπ . It is not difficult to show that Q ψ is non-zero if ψ = 0, and we assume this from now on. It follows that Q ψ has a real non-zero eigenvalue and the corresponding eigenspace Wπ is finite-dimensional. 1 that Q ψ is intertwining, hence Wπ is invariant. Since π is irreducible, Wπ = Vπ and we are done. 2 Every finite-dimensional representation of a compact Hausdorff group is either irreducible or completely reducible. Proof Assume that π is a representation acting on a finite-dimensional space Vπ and that it is not irreducible.
A character of G is a continuous homomorphism χ from G to the one-torus T1 . So if g, h ∈ G, χ(gh) = χ(g)χ(h) = χ(h)χ(g); χ(g −1 ) = χ(g); χ(e) = 1. It is clear that a character defines a one-dimensional representation of G. Conversely we have 3 Of course, in a generic quantum group, there is no underlying group G, only the Hopf algebra structure. 5 If G is a compact Hausdorff abelian group, then every irreducible representation of G is one-dimensional and is given by a character of G. Proof Suppose that π : G → Vπ is an irreducible representation of G.
Right invariant vector fields are defined similarly. The linear space of all left invariant vector fields forms a Lie algebra L(G) under the usual vector field bracket operations. We can similarly form the Lie algebra of all right invariant vector fields on G, and the linear mapping d I is a Lie algebra In some books, both instances of C ∞ in the definition of a Lie group are replaced by the stronger condition of “real analyticity”. In fact, these two seemingly different ways of defining a Lie group give rise to exactly the same objects.
Recent Developments in Quantum Affine Algebras and Related Topics: Representations of Affine and Quantum Affine Algebras and Their Applications, North ... May 21-24, 1998 by Naihuan Jing, Kailash C. Misra