On actions of Drinfel'd doubles on finite dimensional algebras

Let q be an nth root of unity for $n>2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel $u_q({\mathfrak{sl}}_2)$.     

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the ${\mathbb{Z}}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.     

The nil Temperley–Lieb algebra of type affine C

We introduce a type affine C analogue of the nil Temperley–Lieb algebra, in terms of generators and relations. We show that this algebra $T(n)$, which is a quotient of the positive part of a Kac–Moody algebra of type $D_{n+1}^{(2)}$, has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of $T(n)$ consists of polynomials in a certain element Q, and $T(n)$ is a free module of finite rank over its centre. We show how to localize $T(n)$ by adjoining an inverse of Q, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although $T(n)$ has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of $T(n)$ in which Q acts invertibly.     

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the 2-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.     

Multiplicative preprojective algebras of Dynkin quivers

For a commutative ring R and an ADE Dynkin quiver Q, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as R-algebras if and only if the bad primes for Q – 2 in type D, 2 and 3 for $Q=E_6$, $E_7$ and 2, 3 and 5 for $Q=E_8$ – are invertible in R. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2,1/3]$ for $Q=E_6$, $E_7$ and over $\mathbb{Z}[1/2,1/3,1/5]$ for $Q=E_8$. Conversely, if some bad prime is not invertible in R, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest.In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q\ne A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q=D_{2n}$, $n\ge 2$ and $Q=E_7$, $E_8$.     

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units $R^{\times }$, we study the algebra $A_R(G,\sigma )$ consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra $A_R(G;\mathrm{\Sigma })$, and we show that it coincides with $A_R(G,{\sigma }^{?1})$. Given any discrete field ${\mathbb{F}}_d$, we prove a graded uniqueness theorem for $A_{{\mathbb{F}}_d}(G,\sigma )$, and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of $A_{{\mathbb{F}}_d}(G,\sigma )$ is equivalent to minimality of G.     

On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type $G/N$ where $N\subseteq G$ is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of $G/N$ when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.     

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.     

The classification of 3-dimensional noetherian cubic Calabi–Yau algebras

It is known that every 3-dimensional noetherian Calabi–Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S.P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi–Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi–Yau algebras. As an application, we show that if S is a 3-dimensional noetherian cubic Calabi–Yau algebra and σ is a graded algebra automorphism of S, then the homological determinant of σ can be calculated by the formula $\mathrm{hdet}\sigma ={(\det \sigma )}^2$ with one exception.     

Vertex F-algebra structures on the complex oriented homology of H-spaces

We give a topological construction of graded vertex F-algebras by generalizing Joyce's vertex algebra construction to complex-oriented homology. Given an H-space X with a $B\mathrm{U}(1)$-action, a choice of K-theory class, and a complex oriented homology theory E, we build a graded vertex F-algebra structure on $E_{?}(X)$ where F is the formal group law associated with E.