Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S₀(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S₀(G) with an operator space structure. With this structure S₀(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L¹(G). We show that this operator space structure is consistent with the major functorial properties: (i) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map is completely surjective, if H is a closed subgroup; and (iii) is completely surjective, where N is a normal subgroup and . We also show that S₀(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.      

Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients ak and ak+n of the product _∞(q,q)/_∞(qn,qn)=k_∑akqk have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487-501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.     

A Family of Irreducible Representations of the Witt Lie Algebra with Infinite-Dimensional Weight Spaces

We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S ¹, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.     

On Restricted Leibniz Algebras

ABSTRACT

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ?/n? with n = 4 or n = p ^r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.     

The stabilizer of immanants

Immanants are homogeneous polynomials of degree n in n² variables associated to the irreducible representations of the symmetric group S_n of n elements. We describe immanants as trivial S_n modules and show that any homogeneous polynomial of degree n on the space of n×n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. Building on works of Duffner [5] and Purificação [3], we prove that for n?6 the identity component of the stabilizer of any immanant (except determinant, permanent, and π=(4,1,1,1)) is Δ(S_n)?T(GL_n×GL_n)?Z₂, where T(GL_n×GL_n) is the group consisting of pairs of n×n diagonal matrices with the product of determinants 1, acting by left and right matrix multiplication, Δ(S_n) is the diagonal of S_n×S_n, acting by conjugation (S_n is the group of symmetric group) and Z₂ acts by sending a matrix to its transpose. Based on the work of Purificação and Duffner [4], we also prove that for n?5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is Δ(S_n)?T(GL_n×GL_n)?Z₂.     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

Cohomology of the Schrdinger Algebra S(1)

We explicitly compute the first and second cohomology groups of the Schrdinger algebra S(1) with coefficients in the trivial module and the finite-dimensional irreducible modules.We also show that the first and second cohomology groups of S(1) with coefficients in the universal enveloping algebras U(S(1))(under the adjoint action) are infinite dimensional.     

Graded automorphism group of TKK Lie algebra over semilattice

Every extended affine Lie algebra of type A ₁ and nullity ν with extended affine root system R(A ₁, S), where S is a semilattice in ℝ ^ν , can be constructed from a TKK Lie algebra T (J (S)) which is obtained from the Jordan algebra J (S) by the so-called Tits-Kantor-Koecher construction. In this article we consider the ℤ ⁿ -graded automorphism group of the TKK Lie algebra T (J (S)), where S is the “smallest” semilattice in Euclidean space ℝ ⁿ .     

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.     

Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.     

Counterexamples to isosystolic inequalities

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifoldS ² ×S ² admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case ofS ⁿ ×S ⁿ ,n ≥ 3. Our point of departure is M. Gromov's metric onS ¹ ×S ³, and more general examples, due to C. Pittet, of metrics onS ¹ ×S ⁿ with ‘voluminous’ homology. We take the metric product of these metrics with a sphereS ^n?1 of a suitable volume, and perform surgery to obtain the desired metrics onS ⁿ ×S ⁿ .     

Higher dimensional manifolds with S 1-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁, W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹. We show that for n?>?3, if ${{\rm cat}_{S^1}(M^n )=2}$ then M ⁿ is homeomorphic to S ⁿ or S ^n–1 × S ¹ or the non-orientable S ^n–1-bundle over S ¹. We also obtain an unknotting theorem for locally flat knots of S ^n–2 in S ⁿ and a characterization of S ¹?× S ^n–1.     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

The Banach algebra l1(S)

Let S be the free semigroup with a finite or countably infinite set of generators plus an identity. It is shown that there is a natural involution 1 on the convolution Banach algebra l¹(S) such that (l¹(S), 1) has a separating family of finite-dimensional star representations. The star representations of the l¹-algebra of some other semigroups are also considered. The spectrum of every element of l¹(S) which is not a scalar multiple of the identity is shown to be a connected set with interior.