On endomorphisms of quantum tensor space

On tensor products of path algebras of type A

We derive a formula for the Coxeter polynomial of the s-fold tensor product F[A_{n_1-1}] x ... x F[A_{n_s-1}] of path algebras of linearly oriented quivers of Dynkin type A_{n_i-1}, in terms of the weights n_1, ..., n_s > 1, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.

Topological tensor representations of gl(V) for a space V of countable dimension

The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space $V$. We define a tensor category of topological representations of the Lie algebra $gl(V)$, so that $V$, its dual and the adjoint representation $gl(V)$ are objects of this category. This makes it an analogue of the category of finite-dimensional modules over the finite-dimensional Lie algebra $gl(n)$. Our main result is that this category is antiequivalent as a symmetric monoidal category to the category of tensor representations of the Lie algebra of finitary infinite matrices.

线性代数与伽罗瓦理论

Linear Algebra and Galois Theory

本文基于伽罗瓦扩张模型，利用张量代数语言重新阐述并简化了线性代数中自同态结构、迹及线性无关判据等结果的证明。

On derived equivalences of lines, rectangles and triangles

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are "componentwise" tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizing the ADE-chain related to singularity theory, incidence algebras of posets and certain Auslander algebras or more generally endomorphism algebras of initial preprojective modules over path algebras of …

The braid group surjects onto $G_2$ tensor space

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this linearly to a map on the braid group algebra. Lehrer and Zhang (MR2271576) prove this map is surjective, as a special case of a more general result. Using Kuperberg's spider for G_2 from arXiv:math.QA/9201302, we give an elementary diagrammatic proof of this result.

Universal tensor categories generated by dual pairs

Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $φ$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $φ^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that …

An in-Depth Look at Quotient Modules

The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that $Q$ has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of $Q$ and its tensor pow…

通过重构方法研究Brieskorn-Pham奇点

A recollement approach to Brieskorn-Pham singularities

本文通过构造扩展倾斜$n$-立方体，研究Brieskorn-Pham奇点的奇点范畴，并将其自同态代数与复制代数导出等价。

Lie wedges of endomorphism semigroups of standard subspaces in admissible Lie algebras

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ on $\mathfrak{g}$ such that the $(\pm 1)$-eigenspaces $\mathfrak{g}_{\pm 1}$ of $D$ are generated by the intersections with generating cones of the form $W = \mathcal{O}_f^*$, where $\mathcal{O}_f$ is the coadjoint orbit of a linear functional $f \in \mathfrak{z}(\mathfrak{g})^*$ and $\mathcal{O}_f^*$ is the …

Trace-like invariant for representations of nilpotent liftings of quantum planes

We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.

Ozsvath-Szabo bordered algebras and subquotients of category O

We show that Ozsváth-Szabó's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a $\mathfrak{q}$-presentable quotient of parabolic category $\mathcal{O}$. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsváth-Szabó algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation $V^{\otimes n}$ for $U_q(\mathfrak{gl}(1|1))$. Our isomorphism al…

Schur-Weyl duality for certain infinite dimensional $\rm{U}_q(\mathfrak{sl}_2)$-modules

Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endom…

On the affine Schur algebra of type A

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref…

Simple Modules for Temperley-Lieb Algebras and related Algebras

Let $k$ be an arbitrary field and let $q \in k\setminus\{0\}$. In this paper we use the known tilting theory for the quantum group $U_q(sl_2)$ to obtain the dimensions of simple modules for the Temperley-Lieb algebras $TL_n(q+q^{-1})$ and related algebras over $k$. Our main result is an algorithm which calculates the dimensions of simple modules for these algebras. We take advantage of the fact that $TL_n(q+q^{-1})$ is isomorphic to the endomorphism ring of the $n$'th tensor power of the natural $2$-dimensional module for the quantum group for $sl_2$. This algorithm is easy when the characteri…

Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak{gl}^M\left(V,V_*\right)$. Here $\mathfrak{gl}^M\left(V,V_*\right)$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_*\otimes V\to\mathbb{K}$, where $\mathbb{K}$ is the base field. Tensor representations of $\mathfrak{gl}^M\left(V,V_*\right)$ are defined as arbitrary subquotients of finite direct sums of tensor products $(V^*)^{\otimes …