Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z\({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)^H, where Z(A) is the center of algebra A, H⁰ is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)^H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A^H for a pointed Hopf algebra over a field of non-zero characteristic.     

Super Jack-Laurent Polynomials

Let \(\mathcal {D}_{n,m}\) be the algebra of quantum integrals of the deformed Calogero-Moser-Sutherland problem corresponding to the root system of the Lie superalgebra \(\frak {gl}(n,m)\). The algebra \(\mathcal {D}_{n,m}\) acts naturally on the quasi-invariant Laurent polynomials and we investigate the corresponding spectral decomposition. Even for general value of the parameter k the spectral decomposition is not multiplicity free and we prove that the image of the algebra \(\mathcal {D}_{n,m}\) in the algebra of endomorphisms of the generalised eigenspace is k[ε]^?r where k[ε] is the algebra of dual numbers. The corresponding representation is the regular representation of the algebra k[ε]^?r.     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

The inclusion of the Schur algebra in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">ℓ</Emphasis><Superscript>2</Superscript>) is not inverse-closed

The Schur algebra is the algebra of operators which are bounded on ? ¹ and on ? ^∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ? ², and whose inverse is not bounded on ? ¹ nor on ? ^∞. In particular, this shows that the Schur algebra is not inverse-closed.     

Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra

An element a of a complex Banach algebra with unit \(1I\) and with standard conditions on the norm (‖ab‖ ? ‖a‖ · ‖b‖ and ‖\(1I\)‖ = 1) is said to be Hermitian if ‖e ^ita ‖ = 1 for any real number t. An element is said to be decomposable if it admits a representation of the form a + ib in which a and b are Hermitian. The decomposable elements form a Banach Lie algebra (with respect to the commutator). The Hermitian components are determined uniquely, and hence this Lie algebra has the natural involution a + ib = x → x* = a ? ib. One can readily see that ‖x*‖ ? 2‖x‖. Among other things, we prove that ‖ x*‖ ? γ‖x‖, where γ < 2. In fact, the situation is treated in more detail: the original problem is included in a continuous family parametrized by the numerical radius of the element. Finding the exact value of the constant γ is reduced to a variational problem in the theory of entire functions of exponential type. Approximately, γ is equal to 1.92 ± 0.04.     

On the radical for some class of Banach algebras

Let A be a complex Banach algebra. It is well known that the second dual A** of A can be equipped with a multiplication that extends the original multiplication on A and makes A** a Banach algebra. We show that Rad(A) = ^⊥(A ^* · A) and Rad(A ^**) = (A ^* · A)^⊥ for some classes of Banach algebras A with scattered structure space. Some applications of these results are given.     

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UT _n , compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UT _n , for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UT_n^(?), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UT_n^(?) and for any grading on the Jordan algebra UJ _n .It turns out that the graded exponent for UT _n , considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UT _n coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UT_n^(?).     

Nonmatricial version of the Arveson-Wittstock extension principle,and its generalization

We consider the algebra ? = ?(H) of bounded operators in a Hilbert space H, ?-bimodules, and morphisms of these bimodules into the algebra ?(L ? H), where L is a Hilbert space. We study the problem of extension of a morphism defined on a sub-?-bimodule Y ? Z to Z. This problem is solved for Ruan bimodules.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra

An n ×nω-circulant matrix which has a specific structure is a type of important matrix. Several norm equalities and inequalities are proved for ω-circulant operator matrices with ω = e^iθ (0 ≤ θ < 2π) in this paper. We give the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Meanwhile, we present that the set of ω-circulant matrices with complex entries has an idempotent basis. Based on this basis, we introduce an automorphism on the ω-circulant algebra and then show different operators on linear vector space that are isomorphic to the ω-circulant algebra. The function properties, other idempotent bases and a linear involution are discussed for ω-circulant algebra. These results are closely related to the special structure of ω-circulant matrices.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.     

Invariants and rings of quotients of <Emphasis Type="Italic">H</Emphasis>-semiprime <Emphasis Type="Italic">H</Emphasis>-module algebras satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra H on a PI-algebra. We prove that an H-semiprime H-module algebra A has a Frobenius artinian classical ring of quotients Q, provided that A has a finite set of H-prime ideals with zero intersection. The ring of quotients Q is an H-semisimple H-module algebra and a finitely generated module over the subalgebra of central invariants. Moreover, if algebra A is a projective module of constant rank over its center, then A is integral over its subalgebra of central invariants.     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

Endomorphism Algebras of 2-term Silting Complexes

We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D ^b (A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D ^b (A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite.     

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D~b(A)and the subcategory K~b(P) of perfect complexes in D~b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K~b(P), and finding an example such that D_(hf)~b(A)≠K~b(P). We realize the bounded derived category D~b(A) as a Verdier quotient of the relative derived category D_C~b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT ∞ such that ~⊥T is finite, then D~b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.     

Stochastic Hamiltonian flows with singular coefficients

In this paper, we study the following stochastic Hamiltonian system in ?^2d (a second order stochastic differential equation):

$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$

where b(x; v) : ?^2d → ?^d and σ(x; v): ?^2d → ?^d ? ?^d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H _p ^2/3,0 and ?σ ∈ L^p for some p > 2(2d+1), where H _p ^{α, β} is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that (x, v) ? Z_t(x, v) := (X_t, ?_t)(x, v) forms a stochastic homeomorphism flow, and (x, v) ? Z_t(x, v) is weakly differentiable with ess.sup_{x, v} E(sup_{t∈[0, T]} |?Z_t(x, v)|^q) < ∞ for all q ? 1 and T ? 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).