The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$.
This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBW-type basis whose $\bz[q^{-1}]$-span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.
We give some discussions to the relations between canonical left cells and the lowest two-sided cell of an affine Weyl group. In particular, we use the relations to construct irreducible modules attached to the lowest two-sided cell and some one dimensional representations of an affine Hecke algebra.
We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.
Let Q be an affine quiver of type A. Let C be the associated generalized Cartan matrix. Let U^- be the negative part of the quantized enveloping algebra attached to C. In terms of perverse sheaves on the moduli space of representations of a quiver, Lusztig constructed U geometrically and gave a canonical basis B of U^- at the same time. The simple perverse sheaves which enter B are defined abstractly in general. In Lusztig's paper [L2], by using McKay's correspondence, he gave a description of these canonical basis elements in affine cases, i.e. specifying the corresponding supports and local …
We generalize the theory of the second invariant cohomology group $H^2_{\rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that for connected affine algebraic groups $G$ over an algebraically closed field of characteristic $0$, the map $Θ$ from [GK] is bijective (unlike for some finite groups, as shown in [GK]). This allows us to compute $H^2_{\rm inv}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown …
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $σ$ on ${\mathfrak g}$. Let ${\mathbf B}^σ$ be the set of $σ$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^σ \simeq \underline{\mathbf B}$ based on …
Let ${\mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $σ$ the algebra automorphism on ${\mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{\mathbf U}_q^-$ be the quantum algebra obtained from $σ$, and $\widetilde{\mathbf B}$ (resp. $\widetilde{\underline{\mathbf B}}$) the canonical signed basis of ${\mathbf U}_q^-$ (resp. $\underline{\mathbf U}_q^-$). Assume that ${\mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection $\widetilde{\math…
We define canonical bases of the higher-level q-deformed Fock space modules of the affine Lie algebra sl(n)^. This generalizes the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of affine Kazhdan-Lusztig polynomials.
We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When $\mathfrak{g}=\mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $\hat{\mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of …
Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis using the data of the corresponding semistandard Young tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.
We use \cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $\dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U(\g)$ as algebras, where $\g={\rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function …
In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases. Suggested by these formulas, we define natural $q$-deformation of arithmetical functions such as (multi-)partition function and Ramanujan $τ$-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider $q$-deformation of Kostant's function and study certain $q$-polynomials whose special values are weight multiplicities.
We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.
A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein-Zelevinsky ring.
For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave th…
In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.
In this article, we define an algebraic version of the Knizhnik--Zamolodchikov functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo--Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.
Given an affine algebraic variety V and a quantization A of its coordinate ring, it is conjectured that the primitive ideal space of A can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological quotient maps are given in the case of quantum toric varieties.
The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${\mathbf U}^+$ of ${\mathbf U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf U}$ of ${\mathbf U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf U}(\widehat{\frak{sl}}_n)$ coincide …
It is shown that the algebraic--geometrical (or quasiperiodic) solutions of the Conformal Affine $sl(2)$ Toda model are generated from the vacuum via dressing transformations. This result generalizes the result of Babelon and Bernard which states that the $N$--soliton solutions are generated from the vacuum by dressing transformations.
Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(kΛ_0)$ called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor product of the twisted module $V_L^T$ for the lattice vertex operator algebra $V_L$. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in arXiv:math/0102113.
We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.
In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If $G$ is an affine supergroup and $H$ is its normal supersubgroup, then we prove that a dur $K$-sheaf $\tilde{\tilde{G/H}}$ is again affine supergroup. Additionally, if $G$ is algebraic, then a $K$-sheaf $\tilde{G/H}$ is also algebraic supergroup and it coincides with $\tilde{\tilde{G/H}}$. In particular, any normal supersubgroup of an affine supergroup is faithfully exact.
We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra ${\frak H}_Δ(n)$ of a cyclic quiver $Δ(n)$ given in \cite[Thm~4.5]{DuFu2015quantum}. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established in \cite{Guo1995hallpoly} and \cite{Ringel1993composition}, and derive a recursive formula to compute them. We will further use the formula and the construction of certain monomial base for ${\mathfrak H}_Δ(n)$ given in \cite{DengDuXiao2007generic}, together with the double Ringel--Hall algebra…
Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions (q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.
Let $\textbf{U}^+$ be the positive part of the quantum group $\textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $\textbf{B}$ of $\textbf{U}^+$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $\textbf{U}^+$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give severa…
We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We show that ${\mathbf K}^{\mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${\bf U}(\hat{\mathfrak{gl}}_n…
In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and …
