“Classical” Flag Varieties for Quantum Groups: The Standard Quantum SL(n,C)

Abstract

We suggest a possible programme to associate geometric ''flag-like'' data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SLðnÞ of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety. # 2002 Elsevier Science (USA) INTRODUCTIONThe study of quantum analogues of flag varieties, first suggested by Manin [31], has been undertaken during the past decade by several authors, from various points of view; see e.g. [1,8,13,16,17,23,26,29,35,38,39,40]. Around the same time, an approach to noncommutative projective algebraic geometry was initiated by Artin et al. [4,5] and Artin and Van den Bergh [6], and considerably developed since (see e.g. [3, 7, 9, 24, 28, 37, 41, 42, 44]). One attractive feature of their approach is the association of actual geometric data to certain classes of graded noncommutative algebras.The present work is an attempt to study quantum flag varieties from this point of view. As a consequence, our ''quantum flag varieties'' will be actual varieties (with some bells and whistles).Recall the original idea of [4][5][6]. If A is the homogeneous coordinate ring of a projective scheme E; then the points of E are in one-to-one correspondence with the isomorphism classes of the so-called point modules of A; i.e. N-graded cyclic A-modules P such that dim P n ¼ 1 for all n: Now if A is an N-graded noncommutative algebra, one may still try to parametrize its point modules by the points of some projective scheme E: Of course, one cannot hope to reconstruct A from E alone, but there is now an additional ingredient: the shift operation s : P/P½1; where P½1 is the N-graded A-module defined by P½1 n :¼ P nþ1 : (When A is commutative, this shift is trivial: P½1 ' P for every point module P:) Assume that s may be viewed as an automorphism of E: one may then hope, at least in ''good'' cases, to recover A from the triple ðE; s; LÞ; where L is the line bundle over E defined by its embedding into a projective space. The first step of this recovery is the construction of the twisted homogeneous coordinate ring BðE; s; LÞ of a triple ðE; s; LÞ; defined in [6] as follows:(where L s denotes the pullback of L along s), the multiplication being given by ab :¼ a b s m for all a 2 B m ; b 2 B n : (When s is the identity, this algebra coincides, in high degree, with the (commutative) homogeneous coordinate ring of E w.r.t. the polarization L:) If the triple ðE; s; LÞ comes from an algebra A as above, the second step then consists in analysing the canonical morphism A ! BðE; s; LÞ: The initial success of this method has been a complete study of all regular algebras of dimension three [4] (where the kernel of A ! BðE; s; LÞ turns out to be generated by a single element of degree three).The present paper is organized as follows.In Part I, we give a general outline of a possible theory of flag varieties for quantum groups, usi...

Quantum- and quasi-Plücker coordinates

“…To this end we use quasideterminants. Other means of attaching geometric data may be found in [19], where Ohn follows the Artin-Tate-van den Bergh approach to noncommutative projective geometry, and in [21], where Škoda uses quasideterminant-theory to provide localizations of the quantum algebras in question.…”