Abstract :
Given any commutative field k, denote R=Oq(Mn(k)) the coordinate ring of quantum n×n matrices over k and assume q is a nonzero element in k which is not a root of unity. Recall that R is generated by n2 variables Xi,α ((i,α) 1,n 2) subject (only) to the following relations:
If is any 2×2 sub-matrix of then: (a) yx=q−1xy, zx=q−1xz, tz=q−1zt, ty=q−1yt, zy=yz; (b) tx=xt−(q−q−1)yz.
Denote the k-algebra generated by the same variables Xi,α subject to the same relations, except relations (b) which are replaced by: (c) tx=xt; so that is just the algebra of regular functions on some quantum affine space of dimension n2 over k.
The theory of “derivative elimination” defines a natural embedding and asserts that:
• The “canonical image” (Spec(R)) is a union of strata (in the sense of [Goodearl, Letzter, in: CMS Conf. Proc., Vol. 22 (1998) 39–58]), where w describes some subset W of .
• The sets (w W) define the Goodearl–Letzter H-stratification of Spec(R) in the sense of [Goodearl, Letzter, Trans. Amer. Math. Soc. 352 (2000) 1381–1403].
In this paper, we give the precise description of the set W and we compute its cardinality. Using that description and the derivative elimination algorithm, we can verify (Theorems 6.3.1, 6.3.2) that H-Spec(R) has an H-normal separation (in the sense of [Goodearl, in: Lecture Notes in Pure and Appl. Math. 210 (2000) 205–237]), so that Spec(R) has normal separation (in the sense of [Brown, Goodearl, Trans. Amer. Math. Soc. 348 (1996) 2465–2502]). This property was conjectured by K. Brown and K. Goodearl. Since R is Auslander–Regular and Cohen–Macaulay, this implies (by [Goodearl, Lenagan, J. Pure Appl. Algebra 111 (1996) 123–142]) that R is catenary and satisfies the Tauvelʹs height formula.
