N ov 2 00 4 CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES
نویسنده
چکیده
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.
منابع مشابه
2 00 4 Cluster Algebras of Finite Type and Positive Symmetrizable Matrices
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...
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The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...
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The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...
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We name an indecomposable symmetrizable generalized Cartan matrix A and the corresponding Kac–Moody Lie algebra g(A) of the arithmetic type if for any β ∈ Q with (β|β) < 0 there exist n(β) ∈ N and an imaginary root α ∈ ∆ such that n(β)β ≡ α mod Ker (.|.) on Q. Here Q is the root lattice. This generalizes ”symmetrizable hyperbolic” type of Kac and Moody. We show that generalized Cartan matrices ...
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