The center of Uq(nω){\mathcal U}_q({\mathfrak n}_\omega)

We determine the center of a localization of ${\mathcal U}_q({\mathfrak n}_\omega)\subseteq {\mathcal U}^+_q({\mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up, ${\mathfrak g}$ is any finite-dimensional complex Lie algebra and $\omega$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+\omega$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $\omega^{\mathfrak a}+\omega^{\mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.Comment: 28 pages LaTeX. Relevant references as well as a new section relating to the root-of-unity case have been added. Now in print with minor change

Quantized Dirac Operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.Comment: To appear in the Proceedings of the Quantum Groups And Integrable Systems meeting in Prag, June 22-24 2000. To be published with the Czechoslovak Journal of Physi

Algebras of Variable Coefficient Quantized Differential Operators

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients should be. It is an emediate point that even $0$th order operators, given as multiplications by polynomials, have to be specified as e.g. left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings which allows us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings. The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra ${\mathcal W}eyl_q(n,n)$ introduced by T. Hyashi (Comm. Math. Phys. 1990) plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over ${\mathcal W}eyl_q(n,n)$. We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.Comment: 37 pages LaTe

Quantized Heisenberg Space

We investigate the algebra $F_q(N)$ introduced by Faddeev, Reshetikhin and Takhadjian. In case $q$ 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. In the case of $q$ generic, the primitive ideals are described. A related algebra studied by Oh is also treated.Comment: 20 pages LaTeX documen

Special classes of homomorphisms between generalized Verma modules for Uq(su(n,n)){\mathcal U}_q(su(n,n))

We study homomorphisms between quantized generalized Verma modules $M(V_{\Lambda})\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}M(V_{\Lambda_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $\phi^k_{\Lambda,\Lambda_1}$. We examine when one can have a series of such homomorphisms $\phi^1_{\Lambda_{n-1},\Lambda_{n}} \circ \phi^1_{\Lambda_{n-2}, \Lambda_{n-1}} \circ\cdots\circ \phi^1_{\Lambda,\Lambda_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_{\Lambda})\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{\Lambda_n})$. If, classically, $su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb C})\oplus {\mathfrak p}^+$, then $\Lambda = (\Lambda_L,\Lambda_R,\lambda)$ and $\Lambda_n =(\Lambda_L,\Lambda_R,\lambda+2)$. The answer is then that $\Lambda$ must be one-sided in the sense that either $\Lambda_L=0$ or $\Lambda_R=0$ (non-exclusively). There are further demands on $\lambda$ if we insist on ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${\mathcal U}^-_q({\mathfrak g}^{\mathbb C})$ homomorphisms, in which case the conditions on $\lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms $\phi^1_{\Lambda,\Lambda_1}$.Comment: 10 pages proceedings of Group 32, Prague 201

The exponential nature and positivity

In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of $U_q(n^-)$, the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type $A$, which has been conjectured by Lusztig, are proved