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In the present paper we prove a weak form of sandwich classification for the overgroups of the subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ where $\Phi$ is a simply laced root sysetem and $\Delta$ is its sufficiently large subsystem. Namely we show that for any such an overgroup $H$ there exists a unique net of ideals $\sigma$ of the ring $R$ such that $E(\Phi,\Delta,R,\sigma)\le H\le {\mathop{\mathrm{Stab}}\nolimits}_{G(\Phi,R)}(L(\sigma))$ where $E(\Phi,\Delta,R,\sigma)$ is an elementary subgroup associated with the net and $L(\sigma)$ is a corresponding subalgebra of the Chevalley Lie algebra., Comment: 25 pages, to appear in St.Petersburg Math Journal

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(\Delta_V^{it})_{t \in R}$ of $V$ and the corresponding modular conjugation~$J_V$. We study the semigroup \[ S_V = \{ g\in G \cap U(H) : gV \subseteq V\} \] and determine its Lie wedge $L(S_V) = \{ x \in L(G) : exp(R_+ x) \subseteq S_V\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $L(G)$ of~$G$. The semigroup $S_V$ is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form $G exp(iC)$, where $C \subseteq L(G)$ is an $Ad(G)$-invariant closed convex cone. Our main results assert that the Lie wedge $L(S_V)$ spans a $3$-graded Lie subalgebra in which it can be described explicitly in terms of the involution $\tau$ of $L(G)$ induced by $J_V$, the generator $h \in L(G)^\tau$ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup $S_V$ itself, Comment: This version has been completely rewritten. The results are now stronger and the proofs more direct. We also corrected some minor inaccuracies

In this paper, we study basic properties of global $\jmath$-crystal bases for integrable modules over a quantum symmetric pair coideal subalgebra $\mathbf{U}^{\jmath}$ associated to the Satake diagram of type AIII with even white nodes and no black nodes. Also, we obtain an intrinsic characterization of the $\jmath$-crystal bases, whose original definition is artificial., Comment: 34 pages, minor corrections

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations., v3: 18 pages. Minor revision. To appear in Trasactions of the AMS

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_\Delta$ denote the diagonal subalgebra of $S$ corresponding to $\Delta = \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_\Delta$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_\Delta$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul, and Cohen-Macaulay property of the diagonal subalgebras of their Rees algebras., Comment: Typos and errors have been fixed in the current version. Scheduled to appear in Proc. Amer. Math. Soc

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

The first author was partially supported by the National Research, Development and Innovation Office of Hungary (NKFIH) K119934.

For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$., A subtle mistake in the proof of Theorem 4.4 has been corrected (will appear in a corrigendum et addendum, TAMS). In the latter paper we also strengthen some of the results by removing the "square free'' condition in Section 5 and in this paper we also prove new homological equivalences in Theorem 4.4

Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras., Comment: 35 pages, 2 figures. v2: include suggestions and minor corrections from the referee. Final version, to appear in Transactions of the AMS. v3: minor changes

We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a non-unitary isometry, and so it is infinite., Comment: 5 pages

We describe the structure of projective indecomposable modules for the subalgebra consisting of the elements of degree 0 in the hyperalgebra of the r r -th Frobenius kernel for the algebraic group SL 2 ( k ) \textrm {SL}_2(k) , using the primitive idempotents which were constructed before by the author.

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with gl2(Q). Finally, we introduce and compute examples of graded subobject zeta functions.

For any reductive Lie algebra $\mathfrak{g}$ and commutative, associative, unital algebra $S$, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S $ with finite weight multiplicities. In particular, any such module is parabolically induced from a simple admissible module for a Levi subalgebra. Conversely, all modules obtained in this way have finite weight multiplicities. These modules are isomorphic to tensor products of evaluation modules at distinct maximal ideals of $S$. Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras., 15 pages

Given − 1 > q > 1 -1>q>1 and a separable real Hilbert space H R \mathcal {H}_{\mathbb {R}} with dimension no less than 2, we prove that the generator subalgebra in the q q -Gaussian algebra Γ q ( H R ) \Gamma _q(\mathcal {H}_{\mathbb {R}}) is singular.

The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M \mathcal M obtained by successive refinements of the identity x 0 + y 0 = x 0 + z 0 x_0+y_0=x_0+z_0 . This monoid is known to be realizable by the algebra A = K [ F ] A= K[\mathcal F] of the monogenic free inverse monoid F \mathcal F , for any choice of field K K , but A A is not an exchange ring. We show that, for any uncountable field K K , M \mathcal M is not realizable by a regular K K -algebra, but that a suitable universal localization Σ − 1 A \Sigma ^{-1}A of A A provides an exchange, non-regular, K K -algebra realizing M \mathcal M . For any countable field F F , we show that a skew version of the above construction gives a regular F F -algebra realizing M \mathcal M . Finally, we develop some connections with the Atiyah Problem for the lamplighter group. We prove that the algebra A A can be naturally seen as a ∗ * -subalgebra of the group algebra k G kG over the lamplighter group G = Z 2 ≀ Z G= \mathbb {Z}_2\wr \mathbb {Z} , for any subfield k k of C \mathbb {C} closed under conjugation, and we determine the structure of the ∗ * -regular closure of A A in U ( G ) \mathcal U(G) . Using this, we show that the subgroup of R \mathbb R generated by the von Neumann dimensions of matrices over k G kG contains Q \mathbb {Q} .

Let G be a connected, semisimple algebraic group over a field k whose characteristic is very good for G. In a canonical manner, one associates to a nilpotent element X is an element of Lie(G) a parabolic subgroup P - in characteristic zero, P may be described using an sl(2)-triple containing X; in general, P is the "instability parabolic" for X as in geometric invariant theory. In this setting, we are concerned with the center Z(C) of the centralizer C of X in G. Choose a Levi factor L of P, and write d for the dimension of the center Z(L). Finally, assume that the nilpotent element X is even. In this case, we can deform Lie(L) to Lie(C), and our deformation produces a d-dimensional subalgebra of Lie(Z(C)). Since Z(C) is a smooth group scheme, it follows that dim Z(C) >= d = dim Z(L). In fact, Lawther and Testerman have proved that dim Z(C) = dim Z(L). Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case-checking carried out by Lawther and Testerman.

In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called symplectic. Here we show that a formal power series is symplectic if and only if it is a formal composite with the formal power series x 2 / ( 1 − x ) x^2/(1-x) . Hence the set of symplectic power series forms a subalgebra of the algebra of formal power series. The subalgebra property is translated into an identity for the coefficients of the even Euler polynomials, which can be interpreted as a cubic identity for the Bernoulli numbers. Furthermore we show that a rational power series is symplectic if and only if it is invariant under the idempotent Möbius transformation x ↦ x / ( x − 1 ) x\mapsto x/(x-1) . It follows that the Hilbert series of a graded Cohen-Macaulay algebra A A is symplectic if and only if A A is Gorenstein with its a-invariant and its Krull dimension adding up to zero. It is shown that this is the case for algebras of regular functions on symplectic quotients of unitary representations of tori.

ANTHONY TO-MING LAU AND YONG ZHANGAbstract. Motivated by a result of Ky Fan in 1965, we establish a charac-terization of a left amenable F-algebra (which includes the group algebra andthe Fourier algebra of a locally compact group and quantum group algebras,or more generally the predual algebra of a Hopf von Neumann algebra) interms of a nite dimensional invariant subspace property. This is done by rstrevealing a xed point property for the semigroup of norm one positive lin-ear functionals in the algebra. Our result answers an open question posted inTokyo in 1993 by the rst author (see [25, Problem 5]). We also show that theleft amenability of an ideal in an F-algebra may determine the left amenabilityof the algebra.

We construct new irreducible weight modules over quantum affine algebras of type I with all weight spaces infinite-dimensional. These modules are obtained by parabolic induction from irreducible modules over the Heisenberg subalgebra.

In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.

We call a subalgebra $U$ of a Lie algebra $L$ a $CAP$-subalgebra of $L$ if for any chief factor $H/K$ of $L$, we have $H \cap U = K \cap U$ or $H+U = K+U$. In this paper we investigate some properties of such subalgebras and obtain some characterizations for a finite-dimensional Lie algebra $L$ to be solvable under the assumption that some of its maximal subalgebras or $2$-maximal subalgebras be $CAP$-subalgebras.

We classify connected Hopf algebras of Gelfand-Kirillov dimension 4 over an algebraic closed field of characteristic zero.

The (n + 1)-ary derivations of simple non-Lie n-ary Maltsev algebras are described in the case of the binary algebra M7 and the ternary algebra M8 .A s a consequence, a description is obtained for the 3-ary derivations of simple non-Lie Maltsev algebras and simple finite-dimensional non-Lie binary-Lie algebras. Exam- ples of semisimple Maltsev algebras having nontrivial 3-ary derivations are given.

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property., 5 pages; comments welcome; v2 only minor changes, most suggested by referee

In this article we develop a unifying theory of many years of work by a number of researchers. Especially, we establish general sufficient conditions for maps between algebras of bounded continuous functions on locally compact Hausdorff spaces to be almost composition or almost weighted composition operators, which extend the main results of many previous works on this subject. The following are some typical results. Let T : A → B T\colon A \to B be a surjective map between two function algebras on locally compact Hausdorff spaces X X and Y Y with Choquet boundaries δ A ⊂ X \delta A\subset X and δ B ⊂ Y \delta B\subset Y . If ‖ T f T g ‖ = ‖ f g ‖ \|Tf\,Tg\|=\|fg\| and there is an ε , 0 ≤ ε > 2 / 3 \varepsilon ,\,0\leq \varepsilon >2/3 , so that the peripheral spectrum σ π ( T f T g ) \sigma _\pi (Tf\,Tg) is contained in an ε ‖ f g ‖ \varepsilon \,\|fg\| -neighborhood of σ π ( f g ) \sigma _\pi (fg) for all f ∈ A f\in A and all g ∈ A g\in A with ‖ g ‖ = 1 \|g\|=1 , then there is a continuous function α : δ B → { ± 1 } \alpha \colon \delta B\to \{\pm 1\} and a homeomorphism ψ : δ B → δ A \psi \colon \delta B\to \delta A so that | ( T f ) ( y ) − α ( y ) f ( ψ ( y ) ) | ≤ 2 ε | f ( ψ ( y ) ) | |(Tf)(y)-\alpha (y)\,f(\psi (y))|\leq 2\varepsilon \,|f(\psi (y))| for each f ∈ A f\in A and every y ∈ δ B y\in \delta B . Therefore, T T is an almost weighted composition operator on δ B \delta B . If ‖ T f T g ‖ = ‖ f g ‖ \|Tf\,Tg\|=\|fg\| , there are ε , 0 ≤ ε > 1 \varepsilon ,\,0\leq \varepsilon >1 , and η , 0 ≤ η > 1 \eta ,\ 0\leq \eta >1 , so that d ( σ π ( T f T g ) , σ π ( f g ) ) ≤ ε ‖ f g ‖ d(\sigma _\pi (Tf\,Tg),\sigma _\pi (fg))\leq \varepsilon \,\|fg\| , while σ π ( T f ) \sigma _\pi (Tf) is contained in an η \eta -neighborhood of σ π ( f ) \sigma _\pi (f) for all f ∈ A f\in A and all g ∈ A g\in A with ‖ g ‖ = 1 \|g\|=1 . Then | ( T f ) ( y ) − f ( ψ ( y ) ) | ≤ ( ε + η ) | f ( ψ ( y ) ) | |(Tf)(y)-f(\psi (y))|\leq (\varepsilon +\eta )\,|f(\psi (y))| for each y ∈ δ B y\in \delta B and every f ∈ A f\in A ; i.e. T T is an almost composition operator on δ B \delta B . If σ π ( T f T g ) ⊂ σ π ( f g ) \sigma _\pi (Tf\,Tg)\subset \sigma _\pi (fg) ( ( or σ π ( f g ) ⊂ σ π ( T f T g ) ) \sigma _\pi (fg)\subset \sigma _\pi (Tf\,Tg)) , and d ( σ π ( T g ) , σ π ( g ) ) ≤ η d(\sigma _\pi (Tg),\sigma _\pi (g))\leq \eta for some η , 0 ≤ η > 1 \eta ,\,0\leq \eta >1 , for all f ∈ A f\in A and all g ∈ A g\in A with ‖ g ‖ = 1 \|g\|=1 , then ( T f ) ( y ) = f ( ψ ( y ) ) (Tf)(y)=f(\psi (y)) for each f ∈ A f\in A and every y ∈ δ B y\in \delta B . Consequently, T T is a composition operator on δ B \delta B and, therefore, an algebra isomorphism.

We consider the Belkale-Kumar cup product ⊙ t \odot _t on H ∗ ( G / P ) H^*(G/P) for a generalized flag variety G / P G/P with parameter t ∈ C m t \in \mathbb {C}^m , where m = dim ⁡ ( H 2 ( G / P ) ) m=\dim (H^2(G/P)) . For each t ∈ C m t\in \mathbb {C}^m , we define an associated parabolic subgroup P K ⊃ P P_K \supset P . We show that the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) contains a graded subalgebra A A isomorphic to H ∗ ( P K / P ) H^*(P_K/P) with the usual cup product, where P K P_K is a parabolic subgroup associated to the parameter t t . Further, we prove that ( H ∗ ( G / P K ) , ⊙ 0 ) (H^*(G/P_K), \odot _0) is the quotient of the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) with respect to the ideal generated by elements of positive degree of A A . We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.

Sufficient conditions are obtained for the existence of a vector with a one-dimensional or simple three-dimensional stationary subalgebra for an irreducible compact linear Lie algebra., Comment: in Russian

We introduce and analyse the structure of C*-algebras arising from ideals in right tensor C*-precategories, which naturally generalize both relative Cuntz-Pimsner and Doplicher-Roberts algebras. We establish an explicit intrinsic construction of the considered algebras and a number of key results such as structure theorem, gauge-invariant uniqueness theorem or description of the gauge-invariant ideal structure. In particular, these statements give a new insight into the corresponding results for relative Cuntz-Pimsner algebras and are applied to Doplicher-Roberts associated to C*-correspondences.

The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C*-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C*-algebra D without bounded 2-quasitraces or unital quotients has the property that M_n(D) is stable., Comment: 19 pages

Let g \mathfrak g be a finite-dimensional complex semisimple Lie algebra and b \mathfrak b a Borel subalgebra. Then g \mathfrak g acts on its exterior algebra ∧ g \wedge \mathfrak g naturally. We prove that the maximal eigenvalue of the Casimir operator on ∧ g \wedge \mathfrak g is one third of the dimension of g \mathfrak g , that the maximal eigenvalue m i m_i of the Casimir operator on ∧ i g \wedge ^i\mathfrak g is increasing for 0 ≤ i ≤ r 0\le i\le r , where r r is the number of positive roots, and that the corresponding eigenspace M i M_i is a multiplicity-free g \mathfrak g -module whose highest weight vectors correspond to certain ad-nilpotent ideals of b \mathfrak {b} . We also obtain a result describing the set of weights of the irreducible representation of g \mathfrak g with highest weight a multiple of ρ \rho , where ρ \rho is one half the sum of positive roots.

We denote by A A the ring of Laurent polynomials in the indeterminate v v and by K K its field of fractions. In this paper, we are interested in representation theory of the “generic” q q -Schur algebra S q ( n , r ) \mathcal {S}_q(n,r) over A A . We will associate to every symmetrising trace form τ \tau on K S q ( n , r ) K\mathcal {S}_q(n,r) a subalgebra J τ \mathcal {J}_{\tau } of K S q ( n , r ) K\mathcal {S}_q(n,r) which is isomorphic to the “asymptotic” algebra J ( n , r ) A \mathcal {J}(n,r)_A defined by J. Du. As a consequence, we give a new hypothesis which implies James’ conjecture.

Consider the following three properties for countable discrete groups G: (1) G has an infinite subgroup with relative property (T), (2) the group von Neumann algebra L(G) has a diffuse von Neumann subalgebra with relative property (T) and (3) G does not have Haagerup’s property. It is clear that (1) implies (2) and (2) implies (3). I will prove that both of the converses are false. This is joint work with Ionut Chifan.

The Radon-Nikodym property for the Banach algebras A p r ( G ) = A p ∩ L r ( G ) A_p^r(G)=A_p\cap L^r(G) , where A 2 ( G ) A_2(G) is the Fourier algebra, is investigated. A complete solution is given for amenable groups G G if 1 > p > ∞ 1>p>\infty and for arbitrary G G if p = 2 p=2 and A 2 ( G ) A_2(G) has a multiplier bounded approximate identity. The results are new even for G = R n G=\mathbb {R}^n .

Let g \mathfrak {g} be a finite dimensional Lie algebra over a field k k of characteristic zero. An element x x of g \mathfrak {g} is said to have an abstract Jordan-Chevalley decomposition if there exist unique s , n ∈ g s,n\in \mathfrak {g} such that x = s + n x=s+n , [ s , n ] = 0 [s,n]=0 and given any finite dimensional representation π : g → g l ( V ) \pi :\mathfrak {g}\to \mathfrak {gl}(V) the Jordan-Chevalley decomposition of π ( x ) \pi (x) in g l ( V ) \mathfrak {gl}(V) is π ( x ) = π ( s ) + π ( n ) \pi (x)=\pi (s)+\pi (n) . In this paper we prove that x ∈ g x\in \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if x ∈ [ g , g ] x\in [\mathfrak {g},\mathfrak {g}] , in which case its semisimple and nilpotent parts are also in [ g , g ] [\mathfrak {g},\mathfrak {g}] and are explicitly determined. We derive two immediate consequences: (1) every element of g \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if g \mathfrak {g} is perfect; (2) if g \mathfrak {g} is a Lie subalgebra of g l ( n , k ) \mathfrak {gl}(n,k) , then [ g , g ] [\mathfrak {g},\mathfrak {g}] contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie’s Theorem, in addition to the fundamental theorems of Ado and Levi.

We prove that we have an isomorphism of type A a u t ( C σ [ G ] ) ≃ A a u t ( C [ G ] ) σ A_{aut}(\mathbb C_\sigma [G])\simeq A_{aut}(\mathbb C[G])^\sigma , for any finite group G G , and any 2 2 -cocycle σ \sigma on G G . In the particular case G = Z n 2 G=\mathbb Z_n^2 , this leads to a Haar measure-preserving identification between the subalgebra of A o ( n ) A_o(n) generated by the variables u i j 2 u_{ij}^2 and the subalgebra of A s ( n 2 ) A_s(n^2) generated by the variables X i j = ∑ a , b = 1 n p i a , j b X_{ij}=\sum _{a,b=1}^np_{ia,jb} . Since u i j u_{ij} is “free hyperspherical” and X i j X_{ij} is “free hypergeometric”, we obtain in this way a new free probability formula, which at n = ∞ n=\infty corresponds to the well-known relation between the semicircle law and the free Poisson law.

We prove that integral blocks of parabolic category O \mathcal {O} associated to the subalgebra g l m ( C ) ⊕ g l n ( C ) \mathfrak {gl}_m(\mathbb {C}) \oplus \mathfrak {gl}_n(\mathbb {C}) of g l m + n ( C ) \mathfrak {gl}_{m+n}(\mathbb {C}) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2 2 -Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A A .

The dominant dimension of an algebra A A provides information about the connection between A -mod A\textrm {-mod} and B -mod B\textrm {-mod} for B = e A e B=eAe , a certain centralizer subalgebra of A A . Well-known examples of such a situation are the connection (given by Schur-Weyl duality) between Schur algebras and group algebras of symmetric groups, and the connection (given by Soergel’s ’Struktursatz’) between blocks of the category O \mathcal O of a complex semisimple Lie algebra and the coinvariant algebra. We study cohomological aspects of such connections, in the framework of highest weight categories. In this setup we characterize the dominant dimension of A A by the vanishing of certain extension groups over A A , we determine the range of degrees, for which certain cohomology groups over A A and over e A e eAe get identified, we show that Ringel duality does not change dominant dimensions and we determine the dominant dimension of Schur algebras.

We give a formula for the universal R R -matrix of the quantized universal enveloping algebra U q ( g ) . U_q(\mathfrak {g}). This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the braid group element T w 0 T_{w_0} on each representation V V , we show that one can instead use a system of weight preserving endomorphisms. One advantage of our construction is that it is well defined for all symmetrizable Kac-Moody algebras. However, we have only established that the result is equal to the universal R R -matrix in finite type.

Let H n , r \mathcal {H}_{n,r} be the Ariki-Koike algebra associated to the complex reflection group S n ⋉ ( Z / r Z ) n \mathfrak {S}_n\ltimes (\mathbb {Z}/r\mathbb {Z})^n , and let S ( Λ ) \mathcal {S}(\varLambda ) be the cyclotomic q q -Schur algebra associated to H n , r \mathcal {H}_{n,r} , introduced by Dipper, James and Mathas. For each p = ( r 1 , … , r g ) ∈ Z > 0 g \mathbf {p} = (r_1, \dots , r_g) \in \mathbb {Z}_{>0}^g such that r 1 + ⋯ + r g = r r_1 +\cdots + r_g = r , we define a subalgebra S p \mathcal {S}^{\mathbf {p}} of S ( Λ ) \mathcal {S}(\varLambda ) and its quotient algebra S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} . It is shown that S p \mathcal {S}^{\mathbf {p}} is a standardly based algebra and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of S ( Λ ) \mathcal {S}(\varLambda ) , which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic q q -Schur algebras associated to Ariki-Koike algebras H n i , r i \mathcal {H}_{n_i,r_i} of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} of type p \mathbf {p} , and prove the Schur-Weyl duality between H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} .

The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M ( G ) M(G) of G G . This has led to the study of the algebra of BSE-functions on the spectrum of an arbitrary commutative Banach algebra and of the concept of a BSE-algebra as introduced by Takahasi and Hatori. Since then BSE-algebras have been studied by several authors. In this paper we investigate BSE-algebras in the general context on the one hand and, on the other hand, we specialize to Fourier and Fourier-Stieltjes algebras of locally compact groups.

To a finite subgroup Γ \Gamma of S L 2 ( C ) SL_2(\mathbb {C}) , we associate a new family of quantum algebras which are related to symplectic reflection algebras for wreath products S l ≀ Γ S_l\wr \Gamma via a functor of Schur-Weyl type. We explain that they are deformations of matrix algebras over rank-one symplectic reflection algebras for Γ \Gamma and construct for them a PBW basis. When Γ \Gamma is a cyclic group, we are able to give more information about their structure and to relate them to Yangians.

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of Mn(C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on Mn(C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼= ⊕g∈GRg is a grading and dimk(Rg) ≤ 1. Thus any component is either 0 or isomorphic to k as a k–vector space. It is easy to show that Supp(R), the subset of elements of G for which Rg is not 0, is a subgroup of G. Moreover

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).

Consider a simple connected graph G embedded in the plane together with a contractible circuit J. For a partition φ of the vertex set of J we denote by p (G,φ) (t) the number of ways of assigning one of t given colours to each vertex of G so that vertices in the same block of φ have the same colour. Tutte showed that this polynomial may be expressed uniquely as a linear combination of p (G,π ) (t) over all planar partitions π of J, with scalars ϑ φ,π (t) that are independent of G. We show that the (chromatic) invariants ϑ φ,π have a natural algebraic setting in terms of the orthogonal projection from the partition algebra ℙ r (t) to the Temperley-Lieb subalgebra TL r (t, 1). We define the genus of a partition and give an extension of the invariants to arbitrary genus g. Finally, we summarise the role of the genus 0 invariants in the algebraic approach of Birkhoff and Lewis to the Four Colour Theorem.

Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest., 26 pages, minor corrections from previous version based on referee report. To appear in Trans. Amer. Math. Soc

We study the coinvariant ring of the complex reflection group G ( r , p , n ) G(r,p,n) as a module for the corresponding rational Cherednik algebra H \mathbb {H} and its generalized graded affine Hecke subalgebra H \mathcal {H} . We construct a basis consisting of non-symmetric Jack polynomials and, using this basis, decompose the coinvariant ring into irreducible modules for H \mathcal {H} . The basis consists of certain non-symmetric Jack polynomials whose leading terms are the “descent monomials” for G ( r , p , n ) G(r,p,n) recently studied by Adin, Brenti, and Roichman as well as Bagno and Biagoli. The irreducible H \mathcal {H} -submodules of the coinvariant ring are their “colored descent representations”.