In this paper, we investigate the structure and representations of the quantum group ${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$. We will present a realization for $\mathbf{U}(\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism $ζ_r$ from $\mathbf{U}(\infty)$ to the infinite $q$-Schur algebra ${\boldsymbol{\mathcal S}}(\infty,r)$ is not surjective for any $r\geq 1$. We will give a BLM type realization for the image $\mathbf{U}(\infty,r):=ζ_r(\mathbf{U}(\infty))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra $\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$ so that $ζ_r$ can be extended to an algebra epimorphism $\tildeζ_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will investigate the representation theory of ${\bf U}(\infty)$, especially the polynomial representations of ${\bf U}(\infty)$.
Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from $\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct…
The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine $\frak{gl}_n$. Finite dimensional irreducible modules for the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ were classified in \cite{DDF}, where $v\in{\mathbb C}^*$ is not a root of unity. We will classify blocks of the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ in this paper.
Let $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ be the modified quantum affine $\frak{sl}_n$ and let ${\bf U}(\widehat{\frak{sl}}_N)^+$ be the positive part of quantum affine $\frak{sl}_N$. Let $\dot{\mathbf{B}}(n)$ be the canonical basis of $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ and let $\mathbf{B}(N)^{\mathrm{ap}}$ be the canonical basis of ${\bf U}(\widehat{\frak{sl}}_N)^+$. It is proved in \cite{FS} that each structure constant for the multiplication with respect to $\dot{\mathbf{B}}(n)$ coincide with a certain structure constant for the multiplication with respect to $\mathbf{B}(N)^{\mat…
Let ${\mathbf U}(n)$ be the quantum enveloping algebra of ${\frak {gl}}_n$ over $\mathbb Q(v)$, where $v$ is an indeterminate. We will use $q$-Schur algebras to realize the integral form of ${\mathbf U}(n)$. Furthermore we will use this result to realize quantum $\frak{gl}_n$ over $k$, where $k$ is a field containing an l-th primitive root $\varepsilon$ of 1 with $l\geq 1$ odd.
We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that this algebra is isomorphic to the quantum enveloping algebra of the loop algebra of $\mathfrak {gl}_n$. Though this construction is motivated by the work \cite{BLM} by Beilinson--Lusztig--MacPherson for quantum $\frak{gl}_n$, our approach is purely algebraic and combinatorial, independent of the geometric method which seems to work only for quantum $\mathfr…
In this note we present an explicit realization of the affine vertex algebra $V^{cri}(\frak{gl}(1 \vert 1)) $ inside of the tensor product $F\otimes M$ where $F$ is a fermionic verex algebra and $M$ is a commutative vertex algebra. This immediately gives an alternative description of the center of $V^{cri}(\frak{gl}(1 \vert 1) ) )$ as a subalgebra $M _ 0$ of $M$. We reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the center of $V^ {cri}(\frak{gl}(1 \vert 1) )$. Moreover, we construct a family of irreducible $V^{cri}(\frak{gl}(1 \vert 1))$ -modules realized on $F$ and pa…
In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\…
Let $M$ be an oriented manifold and let $\frak N$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\frak N, M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\frak N$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\frak N$. We consider the oriented bordism group $Ω_*(G_{\frak N, M})=\oplus_{i=0}^{\infty} Ω_i(G_{\frak N, M}).$ We introduce the operations $\star$ and $[\cdot, \cdot]$ on $Ω_*(G_{\frak N,M})\otimes \mathbb Q,$ that make $Ω…
We construct an efficient quantum algorithm to compute the quantum Schur-Weyl transform for any value of the quantum parameter $q \in [0,\infty]$. Our algorithm is a $q$-deformation of the Bacon-Chuang-Harrow algorithm, in the sense that it has the same structure and is identically equal when $q=1$. When $q=0$, our algorithm is the unitary realization of the Robinson-Schensted-Knuth (or RSK) algorithm, while when $q=\infty$ it is the dual RSK algorithm together with phase signs. Thus, we interpret a well-motivated quantum algorithm as a generalization of a well-known classical algorithm.
We construct a Fock space representation and the action of the two-parameter quantum algebra $U_{r,s}(\frak{gl}_{\infty})$ using extended Young diagrams. In particular, we obtain an integrable representation of the two-parameter quantum affine algebra of type $C_n^{(1)}$ which is a two-parameter generalization of Kang-Misra-Miwa's realization.
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for $TL_{n}$ that generalizes to give extensions of $TL_{\infty}$ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.
In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the sa…
The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They have well-known representations on the Fock space of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$). Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of …
