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Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by $\mathfrak {R}(A)=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}$ . In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, $b\in \mathbb {N}$ , we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group $G\times \mathbb {Z}_{2}$ . Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural $\mathbb {Z}_{2}$ -grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group $G\times \mathbb {Z}_{2}$ , so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is $\mathbb {Z}_{3}\times \mathbb {Z}_{2}$ (resp. $\mathbb {Z}\times \mathbb {Z}_{2}$ ), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1).

Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show that, for fixed $R$ and $\sigma$, there is a natural algebra map $A(R,\sigma,tt')\to A(R,\sigma,t)\otimes_R A(R,\sigma,t')$. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all $t$) of the Grothendieck groups of the categories of weight modules for $A(R,\sigma,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]$. As a consequence, for $n=1$, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over $\mathfrak{sl}_2$ is a tensor product of two Weyl algebra modules., Comment: 31 pages, 3 figures

Let $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

Let F be an infinite field of characteristic different from two and E be the unitary Grassmann algebra of an infinite dimensional F-vector space L. Denote by Egr an arbitrary $\mathbb {Z}_{2}$ -grading on E such that the subspace L is homogeneous. We consider Egr ⊗ E⊗n as a $(\mathbb {Z}_{2}\times {\mathbb {Z}_{2}^{n}})$ -graded algebra, where the grading on E is supposed to be the canonical one, and we find its graded ideal of identities.

We give a realization of the Kirillov–Reshetikhin crystal B1, s using Nakajima monomials for $\widehat {\mathfrak {s}\mathfrak {l}}_{n}$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes _{i=1}^{N} B^{1,s_{i}}$ in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals Br,1 using Nakajima monomials.

In this paper we study irreducible tensor products of representations of alternating groups and classify such products in characteristic 5.

We introduce a procedure to fold the structure of a crystal B of simply-laced Cartan type ${\mathscr{C}}$ by the action of an automorphism σ. This produces a crystal Bσ for the folded Langlands dual datum ${\mathscr{C}}^{\sigma \vee }$ which properly contains the well-studied ${\mathscr{C}}^{\sigma \vee }$ crystal of σ-invariant points. Our construction preserves normality and the Weyl group action, and is compatible with Kashiwara’s tensor product rule. Combinatorial properties of $B(\infty )_{\sigma }$ reflect the structure of a subalgebra of $U_{q}^{-}({\mathscr{C}})$ , which is naturally a module over the graded-σ-fixed-point subalgebra of $U_{q}^{-}({\mathscr{C}})$ via Berenstein and Greenstein’s quantum folding procedure. We find that $B(\infty )_{\sigma }$ is generated by a set of highest-weight elements over the monoid of root operators. Through the Kashiwara-Nakashima-Zelevinsky polyhedral realization, the highest-weight set identifies with a commutative monoid which admits a Hilbert basis in finite type. A subset of the Weyl group called the balanced parabolic quotient is in one-to-one correspondence with the Hilbert basis for the pair $\left ({\mathscr{C}}, {\mathscr{C}}^{\sigma \vee } \right ) = \left (D_{r}, C_{r-1} \right )$ , and identifies with a proper subset of the Hilbert basis in other finite types. We obtain an explicit combinatorial description of the highest-weight set of $B(\infty )_{\sigma }$ by establishing a connection between the action of root operators on $B(\infty )$ and the semigroup structure in the polyhedral realization.

In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple modules with a nice property over the infinite-dimensional Heisenberg algebra ${\H}$, and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra $\LL$ from generalized oscillator representations of $\LL$ by extending these $\H$-modules. We give the necessary and sufficient conditions for Whittaker modules over $\LL$ (in the more general setting) to be simple. We use the "shifting technique" to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over $\LL$ to be simple. At last we establish the "embedding trick" to obtain a lot more simple $\LL$-modules., 28 pages

For n ≥ a ≥ b, the tensor product $V=\bigwedge ^{a}(\mathbb {C}^{n})\otimes \bigwedge ^{b}(\mathbb {C}^{n})$ has a natural filtration 0 = Vm+ 1 ⊆ Vm ⊆⋯ ⊆ V2 ⊆ V1 ⊆ V0 = V of Gln submodules where m = min(n − a,b) and V/V1 is the Cartan product. For each 1 ≤ u ≤ m, we construct a basis for Vu and a basis for the quotient V/Vu. The elements in the basis for Vu can be regarded as a generalization of the quadratic relations, and the elements in the basis for V/Vu are parametrized by a set of skew tableaux satisfying a condition that cleanly extends the well known semistandardness condition defining a basis for V/V1.

Let $\mathcal{D}$ be the Drinfeld double of $\mathcal{FK}_3\#\Bbbk{\mathbb S}_3$. The simple $\mathcal{D}$-modules were described in arXiv:1409.0438. In the present work, we describe the indecomposable summands of the tensor product between them. We classify the extensions of the simple modules and show that $\mathcal{D}$ is of wild representation type. We also investigate the projective modules and their tensor products.

Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$ is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where $A=\Bbbk[t, t^{-1}]$), and current superalgebras (where $A=\Bbbk[t]$). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where $\mathfrak{g}$ is either $\mathfrak{sl} (n,n)$ with $n \ge 2$, or a finite-dimensional simple Lie superalgebra not of type $\mathfrak{q}(n)$. Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated)., 31 pages

Let n be a positive integer, and $\mathfrak {A}(n)=\mathbb {F}[x]/(x^{p^{n}})$ , the divided power algebra over an algebraically closed field $\mathbb {F}$ of prime characteristic p > 2. Let π(n) be the tensor product of $\mathfrak {A}(n)$ and the Grassmann superalgebra $\bigwedge (1)$ in one variable. The Zassenhaus superalgebra $\mathcal {Z}(n)$ is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra $\mathcal {Z}(n)$ with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained.

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of “trivial” modules. The nodes of the fundamental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.

We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities., Comment: To apear in Algebras and Representation Theory. 17 pages

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by a cocycle. We develop a theory of support varieties for these Hopf algebras to use as a tool in our investigations., We have condensed the preliminaries and we have corrected some typos. 20 pages

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field k. Let H=kG(χ,a,δ) be a Hopf-Ore extension of kG and H′ a rank one quotient Hopf algebra of H, where k is a field, G is a group, a is a central element of G and χ is a k-valued character for G with χ(a)≠1. We first show that the simple weight modules over H and H′ are finite dimensional. Then we describe the structures of all simple weight modules over H and H′, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over H′ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over H and H′, and classify them. Finally, when χ(a) is a primitive n-th root of unity for some n≥2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over H′.

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz–Krieger algebras associated with a certain non-cocommutative comultiplication in our previous work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor products of type III factor representations of Cuntz–Krieger algebras which are different from results of the tensor product of factors of type III.

The restriction of a Verma module of \({\bf U}(\mathfrak{sl}_3)\) to \({\bf U}(\mathfrak{sl}_2)\) is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of \({\bf U}(\mathfrak{sl}_2)\). The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category \(\mathcal O_{\rm{int}}\) of quantum \(\mathfrak{sl}_2\) is given.

In this text, we compare several invariants of the reduced Whitehead group SK1 of a central simple algebra. For biquaternion algebras, we compare a generalised invariant of Suslin as constructed by the author in a previous article to an invariant introduced by Knus-Merkurjev-Rost-Tignol. Using explicit computations, we prove these invariants are essentially the same. We also prove the non-triviality of an invariant introduced by Kahn. To obtain this result, we compare Kahn's invariant to an invariant introduced by Suslin in 1991 which is non-trivial for Platonov's examples of non-trivial SK1. We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras., Comment: 26 pages

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V μ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V μ . A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}'_e$ coming from a right action. It is $\mathcal{F}'_e$ which may coincide with $\mathcal{F}$ . In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}'_e$ . An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$ . This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.

We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products., Comment: 30 pages

We show that the principal block \(\mathcal {O}_0\) of the BGG category \(\mathcal {O}\) for a semisimple Lie algebra \(\frak g\) acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category \(\mathcal {O}\). We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on \(\mathcal {O}_0\). Furthermore, all this generalises to parabolic subcategories of \(\mathcal {O}_0\). As an example, we present some explicit computations for the algebra \(\frak{sl}_3\).

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.

The Clebsch–Gordan problem for quiver representations is the problem of decomposing the tensor product of any two representations of a quiver, where the tensor product is defined point-wise and arrow-wise. We introduce the so-called characteristic representations and decompose their tensor product. This is then applied to solve the Clebsch–Gordan problem for quivers of type $\mathbb{A}_n$ and $\mathbb{D}_n$ . In both cases we also provide an explicit description of the representation ring.

Let k be a field of prime characteristic p and E an n-dimensional vector space. We completely describe the tensor space E ⊗r viewed as a module for the Brauer algebra B k (r,δ) with parameter δ=2 and n=2. This description shows that while the tensor space still affords Schur–Weyl duality, it typically is not filtered by cell modules, and thus will not be equal to a direct sum of Young modules as defined in Hartmann and Paget (Math Z 254:333–357, 2006). This is very different from the situation for group algebras of symmetric groups. Other results about the representation theory of these Brauer algebras are obtained, including a new description of a certain class of irreducible modules in the case when the characteristic is two.

The Kostka–Foulkes polynomials $K_{\lambda ,\mu }^{\phi }(q)$ related to a root system $\phi $ can be defined as alternating sums running over the Weyl group associated to $\phi $ . By restricting these sums over the elements of the symmetric group when $\phi $ is of type $B_{n},C_{n}$ or $D_{n}$ , we obtain again a class $\widetilde{K}_{\lambda ,\mu }^{\phi }(q)$ of Kostka–Foulkes polynomials. When $\phi $ is of type $C_{n}$ or $D_{n}$ there exists a duality between these polynomials and some natural $q$ -multiplicities $u_{\lambda ,\mu }(q)$ and $U_{\lambda ,\mu }(q)$ in tensor products [11]. In this paper we first establish identities for the $\widetilde{K}_{\lambda ,\mu }^{\phi }(q)$ which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials $K_{\lambda ,\mu }^{A_{n-1}}(q)$ with nonnegative integer coefficients. Moreover these coefficients are branching coefficients $.$ This allows us to clarify the connection between the $q$ -multiplicities $u_{\lambda ,\mu }(q),U_{\lambda ,\mu }(q)$ and the polynomials $K_{\lambda ,\mu }^{\diamondsuit }(q)$ defined by Shimozono and Zabrocki. Finally we show that $u_{\lambda ,\mu }(q)$ and $U_{\lambda ,\mu }(q)$ coincide up to a power of $q$ with the one dimension sum introduced by Hatayama and co-workers when all the parts of $\mu $ are equal to $1$ , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.

Let \(\hat {\mathfrak{g}}\) be the affine Lie algebra associated to the simple finite-dimensional Lie algebra \({\mathfrak{g}}\). We consider the tensor product of the loop \(\hat {\mathfrak{g}}\)-module \(\overline {V\left( \mu \right)} \) associated to the irreducible finite-dimensional \({\mathfrak{g}}\)-module V(μ) and the irreducible highest weight \(\hat {\mathfrak{g}}\)-module Lk,λ. Then Lk,λ can be viewed as an irreducible module for the vertex operator algebra Mk,0. Let A(Lk,λ) be the corresponding \(A\left( {M_{k,0} } \right)\left( { = U\left( {\mathfrak{g}} \right)} \right)\)-bimodule. We prove that if the \({U\left( {\mathfrak{g}} \right)}\)-module \(A\left( {L_{k,0} } \right) \otimes _{U\left( \mathfrak{g} \right)} V\left( \mu \right)\) is zero, then the \({\hat {\mathfrak{g}}}\)-module \(\left( {L_{k,0} } \right) \otimes _{U\left( {\mathfrak{g}} \right)} V\left( \mu \right)\)is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.