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Authors
Hans Plesner Jakobsen
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We set up a framework for discussing `$q$-analogues of the usual covariant differential operators for hermitian symmetric spaces. This turns out to be directly related to the deformation quantization associated to quadratic algebras satisfying certain conditions introduced by Procesi and De Concini.
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Let $mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(widehat{mathfrak{g}})$ of a simple complex Lie algebra ${mathfrak{g}}$. Let $mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $mathbb{A}$ and $mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $mathscr{C}_1$ are factorial for Dynkin types $mathbb{A,D,E}$.
We introduce a factorized difference operator L(u) annihilated by the Frenkel-Reshetikhin screening operator for the quantum affine algebra U_q(C^{(1)}_n). We identify the coefficients of L(u) with the fundamental q-characters, and establish a number of formulas for their higher analogues. They include Jacobi-Trudi and Weyl type formulas, canceling tableau sums, Casorati determinant solution to the T-system, and so forth. Analogous operators for the orthogonal series U_q(B^{(1)}_n) and U_q(D^{(1)}_n) are also presented.
The concept of $lambda$-differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the $lambda$-differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible $lambda$. Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the $lambda$-differential module structures on them.
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.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendiecks notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.
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