arXiv subjects

V. Ostrik

Publications and source records attributed to V. Ostrik.

3 recordsLinked to original sources

arXiv1998

Support varieties for quantum groups

For any module $M$ over small quantum group one defines the support variety using construction from the theory of restricted Lie algebras. It is a closed conical subset of nilpotent cone of the corresponding Lie algebra. If module $M$ is a module over the quantum group $U_ξ$ with divided powers then its support variety is invariant under the action of the corresponding algebraic group. In this case we relate codimension $2a$ of the support variety of $M$ in nilpotent cone and dimension of $M$. Namely, we prove that $\dim M$ is `almost' divisible by $l^a$. Further, we give an a priori estimate for support variety of a module in a given linkage class. We compute the support varieties for Weyl modules. Also we compute the support varieties for tilting modules over quantum $SL_n$ and verify in this case Humphreys' Conjecture which relates support varieties of tilting modules with Lusztig's bijection between nilpotent orbits and two-sided cells in the affine Weyl group.

q-alg↗

arXiv1997

Tensor ideals in the category of tilting modules

We study the tensor category $\cQ$ of tilting modules over a quantum group $U_q$ with divided powers. The set $X_+$ of dominant weights is a union of closed alcoves $\oC_w$ numbered by the elements $w\in W^f$ of a certain subset of affine Weyl group $W$. G.Lusztig and N.Xi defined a partition of $W^f$ into canonical right cells and the right order $\le_R$ on the set of cells. For a cell $A\subset W^f$ we consider a full subcategory $\cQ_{<A}$ formed by direct sums of tilting modules $Q(λ)$ with highest weights $λ\in \bigcup_{w\in B<_RA} \oC_w$. We prove that $\cQ_{<A}$ is a tensor ideal in $\cQ$, generalizing H.Andersen's Theorem about the ideal of negligible modules which in our notations is nothing else then $\cQ_{<\{ e\}}$. The proof is an application of a recent result by W.Soergel who has computed the characters of tilting modules.

q-alg↗

arXiv1995

Decomposition of the adjoint representation of the small quantum $sl_2$

Given a finite type root datum and a primitive root of unity $q=\sqrt[l]{1}$, G.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra $\fu$ over the cyclotomic field ${\Bbb Q}(\sqrt[l]{1})$. In this note we study the adjoint representation $\ad$ of $\fu$ in the simplest case of the root datum $sl_2$. The semisimple part of this representation is of big importance in the study of local systems of conformal blocks in WZW model for $\hat{sl}_2$ at level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple part is closely related to the problem of integral representation of conformal blocks (see [BFS]). We find all the indecomposable direct summands of $\ad$ with multiplicities. It appears that $\ad$ is isomorphic to a direct sum of simple and projective modules. It can be lifted to a module over the (infinite dimensional) quantum universal enveloping algebra with divided powers $U_q(sl_2)$ which is also a direct sum of simples and projectives.

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