Hechun Zhang

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.

A class of quadratic matrix algebras arising from the quantized enveloping algebra Uq(A2n−1)

A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside Uq(A2n−1), it consists of quadratic algebras with the same Hilbert series as polynomials in n2 variables. We discuss their general properties and investigate some members of the family in great detail with respect to associated varieties, degrees, centers, and symplectic leaves. Finally, the space of rank r matrices becomes a Poisson submanifold, and there is an associated tensor category of rank ⩽r matrices.

Dual canonical bases for the quantum special linear group and invariant subalgebras

AbstractA string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a “canonical basis” for every finite dimensional irreducible

Cyclic representations of the quantum matrix algebras

U_q({\mathfrak{sl}}(n))

The center of the quantized matrix algebra

-module. It is also shown that the algebra of functions on any quantum homogeneous space is generated by quantum minors.

Dual canonical bases for the quantum general linear supergroup

We investigate the algebra Fq(N) introduced by Faddeev, Reshetikhin and Takhadjian. In the case where q is a primitive root of unity, the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini–Kac–Procesi Conjecture is proved for this case.

The Exponential Nature and Positivity

In this paper we give a complete classification of the minimal cyclic M q(n)-modules and construct them explicitly. Also, we give a complete classifica-tion of the minimal cyclic modules of the so-called Dipper-Donkin quantum matrix algebra as well as of two other natural quantized matrix algebras. In the last part of the paper we relate the results to the De Concini - Procesi conjecture.