Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Hall algebra of a cyclic quiver at $q=0$
136
0
0.0
(
0
)
Authors
Stefan Wolf
Ask ChatGPT about the research
No Arabic abstract
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
rate research
Read More
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$ and $widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
Let $Lambda$ be the set of partitions of length $geq 0$. We introduce an $mathbb{N}$-graded algebra $mathbb{A}_q^d(Lambda)$ associated to $Lambda$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of $mathbb{A}^d_q(Lambda)$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra $mathcal{H}^d_q$ of the $d$-loop quiver. It turns out that $mathbb{A}^d_q(Lambda)$ is isomorphic to $mathcal{H}^d_q$, thus can be viewed as a combinatorial realization for $mathcal{H}^d_q$.
We show that the $imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $imath$Hall algebra to the ring of symmetric functions in two parameters $t, theta$, which maps the $imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($imath$HL) functions. The (modified) $imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $imath$HL functions. The modified $imath$HL functions specialize at $theta=0$ to the modified HL functions; they specialize at $theta=1$ to the deformed universal characters of type C, which further specialize at $(t=0, theta =1)$ to the universal characters of type C.
Let $L$ be a Lie algebra of Block type over $C$ with basis ${L_{alpha,i},|,alpha,iinZ}$ and brackets $[L_{alpha,i},L_{beta,j}]=(beta(i+1)-alpha(j+1))L_{alpha+beta,i+j}$. In this paper, we shall construct a formal distribution Lie algebra of $L$. Then we decide its conformal algebra $B$ with $C[partial]$-basis ${L_alpha(w),|,alphainZ}$ and $lambda$-brackets $[L_alpha(w)_lambda L_beta(w)]=(alphapartial+(alpha+beta)lambda)L_{alpha+beta}(w)$. Finally, we give a classification of free intermediate series $B$-modules.
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
