[Math.RT] 2 Feb 2005 Ler for Algebra Eti Nt Ru Hc Cso H Peeado Matrices
CERTAIN EXAMPLES OF DEFORMED PREPROJECTIVE ALGEBRAS AND GEOMETRY OF THEIR *-REPRESENTATIONS ANTON MELLIT λ Abstract. We consider algebras eiΠ (Q)ei obtained from deformed preprojective algebra λ of affine type Π (Q) and an idempotent ei for certain concrete value of the vector λ which corresponds to the traces of −1 ∈ SU(2, C) in irreducible representations of finite subgroups of SU(2, C). We give a certain realization of these algebras which allows us to construct the C∗-enveloping algebras for them. Some well-known results, including description of four projections with sum 2 happen to be a particular case of this picture. 1. Introduction Let us take the *-algebra, generated by self-adjoint projections a1, a2, a3, a4 with relation a1 + a2 + a3 + a4 = 2: ∗ A := C a1, a2, a3, a4 ai = a ,σ(ai) 0, 1 , a1 + a2 + a3 + a4 = 2 . D4 h | i ⊂ { } i Here and belowf σ(a) denotes the spectrum of self-adjoint element a and writing σ(a) ⊂ x1,x2,...xk for real numbers x1,x2,...,xk we mean that there is a relation { } (a x1)(a x2) . (a xk) = 0. − − − Classification of irreducible representations of A is well known (see [OS]). All irreducible D4 representations, except some finite number of exceptional cases, form a two dimensional family f of irreducible representations in dimension 2. The paper [M1] was devoted to the classification of irreducible representations of the algebra ∗ A := C a1, a2, a3 ai = a ,σ(ai) 0, 1, 2 , a1 + a2 + a3 = 3 . E6 h | i ⊂ { } i After that, the classificationf problem for irreducible representations of the *-algebra ∗ A := C a1, a2, a3 ai = a ,σ(a1) 0, 1, 2, 3 ,σ(a2 ) 0, 1, 2, 3 ,σ(a3 ) 0, 2 , E7 h | i ⊂ { } ⊂ { } ⊂ { } f a1 + a2 + a3 = 4 i was solved in [O].
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