We mainly discuss the representation dimension of the 2×2 triangular matrix algebra over an artin algebra. Let Λ be an artin algebra, and let T2(Λ) be the 2×2 triangular matrix algebra over Λ. We will show that the representation dimension of T2(Λ) is upper bounded by the maximum of the representation dimension of Λ plus 1 and the global dimension of Λ plus 2. In particular, we will show that if Λ is a hereditary algebra or a tilted algebra, then the representation dimension of T2(Λ) is at most 4.
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