We study the representation theory of the affine nilCoxeter algebra $A$ of type $\tilde A_{n-1}$, over a field $k$ of any characteristic. Our main theorem states that this is a Noetherian prime affine PI algebra of PI degree $n!$. As a consequence, the simple $A$-modules are all finite dimensional, and the maximum dimension of a simple module is $n!$ over a suitable finite extension of $k$. To achieve this, we investigate a large commutative subalgebra $C$ which is finitely generated as an algebra and over which $A$ is finitely generated as a module. We show that the associated primes of $C$ are minimal primes, and there are $n!$ of them, regularly permuted by $\mathfrak{S}_n$. The algebra $R=C^{\mathfrak{S}_n}$ is equal to the centre of $A$, and isomorphic to $C/\mathfrak{p}$ for each of the minimal primes $\mathfrak{p}$. We prove that the ring $R$ is isomorphic to $k[X_1,\dots,X_{n-1}]^{\mu_n}$, where $\mu_n$ is the finite group scheme of $n$th roots of unity, acting so that $X_i$ has degree $i$ modulo $n$. The ring $R$ is Cohen--Macaulay, and is Gorenstein if and only if $n$ is odd or $n=2$. It is a toric ring, with divisor class group $\mathsf{Cl}(R)\cong\mathbb{Z}/n$, and every projective $R$-module is free.
We investigate the structure of central extensions for algebras in a congruence modular variety. We use a multisorted algebraic object called a clonoid to understand the term clone of such a central extension. We develop the difference clonoid of such a central extension and use it to show that the number of $2$-step n...
Given a finite dimensional Lie algebra $L$ on a finite field $\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\mathcal{L}(L)$ and introduce the notion of ``subalgebra commutativity degree''of $L$....
Seid Kassaw Muhie, Daniele Ettore Otera, F. Russo· 0 citations
Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$....
Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $\tau(T)=0$ for all $\tau\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds:...
In this paper, we introduce a family of infinite-dimensional Lie algebras $B(q)$ of Takiff Block type, containing Heisenberg-Virasoro algebra, $W$-algebra $W(2,2)$, and BMS-Kac-Moody algebra as subalgebras. For $q\in\mathbb C^*$, we classify the $ B(q)$-modules that are free of rank one over $U(\mathfrak h)$, where $\m...
We exhibit a simple unital infinite-dimensional C$^*$-algebra $A$ which is not $K_1$-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator $u$ of $\pi_4(U(2))\cong\mathbb{Z}/2$, and we us...
A. Toms· 0 citations
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