Jonathan Brundan, Alexander Kleshchev
We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of $p$-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.
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We define an analogue of Shapovalov forms for Q-type Lie superalgebras and factorize the corresponding Shapovalov determinants which are responsible for simplicity of highest weight modules. We apply the factorization to obtain a description of the centres of Q-type Lie superalgebras.
Let $U$ be either classical or quantized universal enveloping algebra of $\sł(n+1)$ extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in $U$ over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse of the form is explicitly constructed.
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig's small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation
Shapovalov determinant for loop superalgebras
本文研究李超代数对应的Kac-Moody超代数的Casimir元，借助Wick正规形式由有限维单李超代数的二次Casimir元恢复loop超代数相关Kac-Moody超代数的二次Casimir元，还计算了奇不变型情形的三次Casimir元，并拓展到poi(0|n)等情形。
Shapovalov elements $θ_{β,m}$ of the classical or quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root $β$ and a positive integer $m$. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of $θ_{β,m}$ to a product of $θ_{β,1}$ and calculate $θ_{β,1}$ as a residue of a matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm. This way we explicitly express $θ_{β,m}$ of a classical simple Lie algebra through the Cartan-Weyl basis i…
The Cartan determinant conjecture for representation-finite Artin algebras
该研究将Enomoto关于有限表示型代数Cartan行列式的结果，从代数闭域推广到完美域，证明有限整体维数的此类代数其整Cartan行列式为1。
We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring $\mathbb Z[v,v^{-1}]$. This proves and generalizes a conjecture of Ando-Suzuki-Yamada …
We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to Külshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.
A counterexample to a conjecture on Cartan determinants of monoid algebras
本文通过构造有限幺半群的实例，证明了其复数域代数的Cartan矩阵非奇异而正特征域代数的Cartan矩阵奇异，从而反驳了Steinberg关于幺半群代数Cartan行列式的猜想。
The Cartan determinant conjecture for representation-finite algebras
本文针对代数闭域上的有限表示型代数，证明了满足特定条件的自同态代数的嘉当行列式为1，从而验证了该类代数的嘉当行列式猜想成立。
We study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For the natural grading by path lengths, any locally gentle algebra is a Koszul algebra. Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver.…
We explore some interesting features of the characteristic polynomial of the Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely related with the Chebyshev polynomials of first and second kind. In addition, we give explicit formulas for the characteristic polynomial of the Coxeter adjacency matrix, we compute the associated polynomials and use them to derive the Coxeter polynomial of the underlying graph. We determine the expression of the Coxeter and associated polynomials as a product of cyclotomic factors. We use this data to propose an algorithm for factoring Ch…
A $q$-analogue of combinatorics concerning the Cartan matrix for the Iwahori-Hecke algebra of type $A$ is investigated. We give several descriptions for the determinant of the graded Cartan matrix, which imply some combinatorial identities. A conjectural expression for the elementary divisors is also presented.
Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $Γ:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $Γ$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $Γ$. We also investigate the situation where both $A,Γ$…
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors…
n-recollements of triangulated categories and n-derived-simple algebras are introduced. The relations between the n-recollements of derived categories of algebras and the Cartan determinants, homological smoothness and Gorensteinness of algebras respectively are clarified. As applications, the Cartan determinant conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry conjecture is reduced to 2-derived-simple algebras.
In this paper we consider the category $C (\tilde k, \tilde H)$ of the $(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$ are the corresponding affine Lie algebra and the affine Cartan group, respectively. To this category we apply the Zuckerman functor and its derivatives. By using the determinant bundle structure, we prove the natural duality of the Zuckerman derived functors, and deduce a Borel-Weil-Bott type theorem on decomposition of the nilpotent part cohomology.
Inverses of Cartan matrices of Lie algebras and Lie superalgebras
本文计算了多类李（超）代数不可分解Cartan矩阵的逆，发现三个新现象，同时纠正了前人对相关结果的误解并补充了更早期的文献依据。
Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the $\ell$-Cartan matrix for $S_n$ (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra $\mathcal{H}_n(q)$, where $q$ is a primitive $\ell$th root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when $\ell=p^r$, $p$ prime, and $r\leq p$ and went on to conjecture that the formulae should hold for all $r$. Another result was obtained, which is surprising and counteri…
We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S_n and their double covers. In particular, we give a simple computation, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table of S_n with respect to an integer r>1. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas' Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke alg…
Singular equivalences and homological conjectures
通过中心化子矩阵代数的奇异范畴和奇异等价，验证了Auslander-Reiten和Cartan行列式猜想，进而证明所有经典同调猜想对域上中心化子矩阵代数成立。
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