let $k$ be a number field and let $a$ be an abelian variety over $k$. in an effort of proper setting of the sato-tate conjecture concerning the equidistribution of frobenius elements in the representation of the galois group $g_k$ on the tate module of $a$, one of attempts is the introduction of the algebraic sato-tate group $ast_{k}(a)$. maximal compact subgroups of $ast_{k}(a)(\mathbb{c})$ are expected to be the key tool for the statement of the sato-tate conjecture for $a$. at the lecture, following an idea of j-p. serre, an explicit construction of $ast_{k}(a)$ will be presented based on p. deligne's motivic category for absolute hodge cycles. i will discuss the arithmetic properties of $ast_{k}(a)$ along with explicit computations of $ast_{k}(a)$ for some families of abelian varieties. i will also explain how this construction extends to absolute hodge cycles motives in the deligne's motivic category for absolute hodge cycles. this is joint work with kiran kedlaya.