First, we study the subskewfield of rational pseudodifferential operators
over a differential field K generated in the skewfield of pseudodifferential
operators over K by the subalgebra of all differential operators.

Second, we show that the Dieudonne' determinant of a matrix
pseudodifferential operator with coefficients in a differential subring A of K
lies in the integral closure of A in K, and we give an example of a 2x2 matrix
differential operator with coefficients in A whose Dieudonne' determiant does
not lie in A.

In this paper, the conjugate-linear anti-involutions and the unitary
irreducible modules of the intermediate series over the twisted
Heisenberg-Virasoro algebra are classified respectively. We prove that any
unitary irreducible module of the intermediate series over the twisted
Heisenberg-Virasoro algebra is of the form $\mathcal{A}_{a,b,c}$ for $a\in
\mathbb{R}, b\in 1/2+\sqrt{-1}\mathbb{R}, c\in \mathbb{C}.$

We show that the support of a simple weight module over the Neveu-Schwarz
algebra, which has an infinite-dimensional weight space, coincides with the
weight lattice and that all non-trivial weight spaces of such module are
infinite-dimensional. As a corollary we obtain that every simple weight module
over the Neveu-Schwarz algebra, having a non-trivial finite-dimensional weight
space, is a Harish-Chandra module (and hence is either a highest or lowest
weight module, or else a module of the intermediate series).

We introduce on the abstract level in real Clifford algebras \cl_{p,q} of a
non-degenerate quadratic space (V,Q), where Q has signature \epsilon=(p,q), a
transposition anti-involution \tp. In a spinor representation, the
anti-involution \tp gives transposition, complex Hermitian conjugation or
quaternionic Hermitian conjugation when the spinor space \check{S} is viewed as
a \cl_{p,q}-left and \check{K}-right module with \check{K} isomorphic to R or
R^2, C, or, H or H^2.

By elementary and direct calculations the vanishing of the (algebraic) second
Lie algebra cohomology of the Witt and the Virasoro algebra with values in the
adjoint module is shown. This yields infinitesimal and formal rigidity or these
algebras. The first (and up to now only) proof of this important result was
given 1989 by Fialowski in an unpublished note. It is based on cumbersome
calculations. Compared to the original proof the presented one is quite elegant
and considerably simpler.

The PostLie algebra is an enriched structure of the Lie algebra that has
recently arisen from operadic study. It is closely related to pre-Lie algebra,
Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter
equations and integrable systems. We give a complete classification of PostLie
algebra structures on the Lie algebra sl(2,C) up to isomorphism. We first
reduce the classification problem to solving an equation of 3 x 3 matrices.

Given a ring R, we define its right injective profile as the collection of
injectivity domains of right R-modules. We show that the injective profile of R
is in bijective correspondence with a cofinal interval of the lattice of linear
filters of R, so we apply torsion-theoretic techniques in the study of relative
injectivity. Similarly, we define the right projective profile of a ring, and
prove some of its properties when R is a right perfect ring. In the final
section, we apply our results in the study of a special class of QF rings.

We develop a graded version of the theory of cyclotomic q-Schur algebras, in
the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on
q-Schur algebras. As an application, we identify the coefficients of the
canonical basis on a higher level Fock space with q-analogues of the
decomposition numbers of cyclotomic q-Schur algebras.

Let K be an arbitrary field, and a,b,c,d be elements of K such that the
polynomials t^2-at-b and t^2-ct-d are split in K[t]. Given a square matrix M
with entries in K, we give necessary and sufficient conditions for the
existence of two matrices A and B such that M=A+B, A^2=a A+bI_n and B^2=c
B+dI_n. Prior to this paper, such conditions were known in the case b=d=0, a<>0
and c<>0, and also in the case a=b=c=d=0. Here, we complete the study, which
essentially amounts to determining when a matrix is the sum of an idempotent
and a square-zero matrix.

A valuation theoretic approach is presented that directly leads to division
algebras that are noncrossed products (instead of, e.g., describing Brauer
classes of noncrossed products in an abstract manner). While this feature is
shared by Amitsur's original construction, the new approach works over small
fields. It is further demonstrated how it can be used to obtain very explicit
examples of noncrossed products in the form of iterated twisted function fields
over division algebras over global fields. The examples allow even to write
down structure constants of noncrossed products.

It is shown that if a bilinear map f: A x B --> C of modules over a
commutative ring k is nondegenerate (i.e., if no nonzero element of A
annihilates all of B, and vice versa), and A and B are Artinian, then A and B
are of finite length.

Some consequences are noted. Counterexamples are given to some attempts to
generalize the above statement to balanced bilinear maps of bimodules over
noncommutative rings, while the question is raised whether other such
generalizations are true.

Let G be any group and F an algebraically closed field of characteristic
zero. We show that any two G-graded finite dimensional G-simple algebras over F
are G-graded isomorphic if and only if the satisfy the same G-graded polynomial
identities. This result was proved by Koshlukov and Zaicev in case G is
abelian.

Let K be a skew field and (G,<) an ordered group. We show that the skew field
generated by the group ring K[G] inside the Malcev-Neumann series ring K((G;<))
contains noncommutative free group algebras.

We study ruled orders. These arise naturally in the Mori program for orders
on projective surfaces and morally speaking are orders on a ruled surface
ramified on a bisection and possibly some fibres. We describe fibres of a ruled
order and show they are in some sense rational. We also determine the Hilbert
scheme of rational curves and hence the corresponding non-commutative Mori
contraction. This gives strong evidence that ruled orders are examples of the
non-commutative ruled surfaces introduced by Van den Bergh.

We classify, up to isomorphism, all gradings by an arbitrary abelian group on
simple finitary Lie algebras of linear transformations (special linear,
orthogonal and symplectic) on infinite-dimensional vector spaces over an
algebraically closed field of characteristic different from 2.

Let $r,s$ be two parameters chosen from $\C^{\ast}$ such that $r^{m}s^{n}=1$
implies $m=n=0$. We compute the derivations of the two-parameter quantized
enveloping algebra $\U$ and calculate its first degree Hochschild cohomology
group. We further determine the group of algebra automorphisms for the
two-parameter Hopf algebra $\V$. As a result, we determine the group of Hopf
algebra automorphisms for $\V$.

It is shown that the Gelfand--Kirillov dimension for modules over quantum
Laurent polynomials is tensor-minimal. The Brookes--Groves invariant associated
with a tensor product of modules is determined. It is also shown that there can
be nonholonmic simple modules.

In this paper we study the $p$-adic equation $x^q=a$ over the field of
$p$-adic numbers. We construct an algorithm of calculation of criteria of
solvability in the case of $q=p^m$ and present a computer program to compute
the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability
criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz
algebras.

In the present paper by Frobenius algebra Y we mean a finite dimensional
algebra possessing an associative and invertible (nondegenerate) form a scalar
product, referred to as the Frobenius structure. The nondegenerate form has an
inverse. We drop the extra conditions of associativity and unitality of Y.
Frobenius algebra is formulated within the monoidal abelian category of operad
of graphs cat(m,n). Operad of graphs, i.e. diagrammatic language, is used both
to illustrate the construction as well as a method of proof for the main
Theorem.

A ring is clean (resp. almost clean) if each of its elements is the sum of a
unit (resp. regular element) and an idempotent. In this paper we define the
analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its
every element is the sum of a unit (resp. regular element) and a projection.
Although *-clean is a stronger notion than clean, for some *-rings we
demonstrate that it is more natural to use.

In this paper, we study typical ranks of 3-tensors and show that there are
plural typical ranks for m x n x p tensors over the real number field in the
following cases. (1) 2<m<5, 4|n and (m-1)(n-1)<p<(m-1)n+1. (2) 4<m<9, 8|n and
(m-1)(n-1)<p<(m-1)n+1. (3) m=9, 16|n and $8n-8<p<8n+1. (4) For some integer s
with s>4, 9<m<2s+1, 2^s|n and (m-1)(n-1)<p<(m-1)n+1. (5) m=3, 4|(n-3) and
p=2n-1. (6) m=4, 4|(n-2), n>5 and p=3n-2. (7) m=6, 8|(n-4), n>11 and p=5n-4.

Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed
field of characteristic 0. In this paper, we obtain the structure theorems for
semisimple Hopf algebras of dimension $pq^3$.

We show the characterization analogous to dimension groups of partially
ordered real vector spaces with interpolation works, but sequential direct
limits of simplicial vector spaces only under strong assumptions. We also
provide and generalize a proof of a result of Fuchs asserting that the real
polynomial algebra with pointwise ordering coming from an interval satisfies
Riesz interpolation

Artin solved Hilbert's $17^{th}$ problem by showing that every positive
semidefinite polynomial can be realized as a sum of squares of rational
functions. Pfister gave a bound on the number of squares of rational functions:
if $p$ is a positive semi-definite polynomial in $n$ variables, then there is a
polynomial $q$ so that $q^2p$ is a sum of at most $2^n$ squares.

In this paper we consider a multicriteria aggregation model where local
utility functions of different sorts are aggregated using Sugeno integrals, and
which we refer to as Sugeno utility functions. We propose a general approach to
study such functions via the notion of pseudo-Sugeno integral (or,
equivalently, pseudo-polynomial function), which naturally generalizes that of
Sugeno integral, and provide several axiomatizations for this class of
functions.

In this paper we study contravariant functors from the category of rings to
the category of sets whose restriction to the full subcategory of commutative
rings is isomorphic to the prime spectrum functor Spec. The main result of this
paper reveals a common characteristic of these functors: every such functor
assigns the empty set to M_n(C) for n > 2. The proof relies, in part, on the
Kochen-Specker Theorem of quantum mechanics. The analogous result for
noncommutative extensions of the Gelfand spectrum functor for C^*-algebras is
also proved.

Kinyon showed that the tangent space of a Lie Rack at the neutral element has
a Leibniz algebra structure. This provided a promising lead towards solving the
Coquecigrue problem for Leibniz algebras. In this paper, we introduce the
category of Lie $n$-racks and generalize several results known on racks. In
particular, we generalize Kinyon's result to Leibniz $n$-algebras.

Some lower bounds of GK-dimension of Hopf algebras are given.

Using the Luthar--Passi method, we investigate the possible orders and
partial augmentations of torsion units of the normalized unit group of integral
group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.

Associating to each pre-order on the indices 1,...,n the corresponding
structural matrix ring, or incidence algebra, embeds the lattice of n-element
pre-orders into the lattice of n x n matrix rings. Rings within the
order-convex hull of the embedding, i.e. matrix rings that contain the ring of
diagonal matrices, can be viewed as incidence algebras of ideal-valued,
generalized pre-order relations.

I give a simpler proof of the generalisation of Engel's Theorem to Leibniz
algebras.

We define a new class of algebras, cyclotomic Temperley-Lieb algebras of type
D, in a diagrammatic way, which is a generalization of Temperley-Lieb algebras
of type D. We prove that the cyclotomic Temperley-Lieb algebras of type D are
cellular. In fact, an explicit cellular basis is given by means of
combinatorial methods. After determining all the irreducible representations of
these algebras, we give a necessary and sufficient condition for a cyclotomic
Temperley-Lieb algebra of type D to be quasi-hereditary.

We generalize to the case of Lie superalgebras the classical symplectic
double extension of symplectic Lie algebras introduced in [2]. We use this
concept to give an inductive description of nilpotent homogeneous-symplectic
Lie superalgebras. Several examples are included to show the existence of
homogeneous quadratic symplectic Lie superalgebras other than even-quadratic
even-symplectic considered in [6]. We study the structures of even (resp.
odd)-quadratic odd (resp.

We consider a Weitzenb\"och derivation $\Delta$ acting on a polynomial ring
$R=K[\xi_1,\xi_2,...,\xi_m]$ over a field $K$ of characteristic 0. The
$K$-algebra $R^\Delta = \{h \in R \mid \Delta(h) = 0\}$ is called the algebra
of constants. Nowicki considered the case where the Jordan matrix for $\Delta$
acting on $R_1$, the degree 1 component of $R$, has only Jordan blocks of size
2. He conjectured (\cite{N}) that a certain set generates $R^{\Delta}$ in that
case.

We consider the $\delta$-derivations of classical Lie superalgebras and prove
that these superalgebras admit nonzero $\delta$-derivations only when $\delta =
0,1/2,1$. The structure of $1/2$-derivations for classical Lie superalgebras is
completely determined.

We provide a construction of minimal injective resolutions of simple
comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau
condition of algebras, we introduce the Calabi-Yau condition to coalgebras.
Then we give some descriptions of Calabi-Yau coalgebras with lower global
dimensions. An appendix is included for listing some properties of cohom
functors.

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially
nilpotent if there is some nilpotent real matrix with nonzero entries in
precisely the entries indicated by the pattern. We develop ways to construct
some potentially nilpotent patterns, including some balanced tree patterns. We
explore the index of some of the nilpotent matrices constructed,and observe
that some of the balanced trees are spectrally arbitrary using the
Nilpotent-Jacobian method. Inspired by an argument in [R. Pereira, Nilpotent
matrices and spectrally arbitrary sign patterns. Electron. J.

In this document we consider a way of localizing an MV-algebra. Given any
prime filter $F$ we find a local MV-algebra which has the same poset of prime
filters as the poset of prime filters comparable to $F$.

Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra
$A$ by an abelian group $G$, we have two subgroups of the group of
automorphisms of $A$: the automorphisms that stabilize each homogeneous
component $A_g$ (as a subspace) and the automorphisms that permute the
components. By the Weyl group of $\Gamma$ we mean the quotient of the latter
subgroup by the former. In the case of a Cartan decomposition of a semisimple
complex Lie algebra, this is the automorphism group of the root system, i.e.,
the so-called extended Weyl group.

It is shown that over an arbitrary countable field, there exists a finitely
generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov
dimension at most three.

We prove that there exists an exponent beyond which all continuous
conventional powers of n-by-n doubly nonnegative matrices are doubly
nonnegative. We show that this critical exponent cannot be less than $n-2$ and
we conjecture that it is always $n-2$ (as it is with Hadamard powering). We
prove this conjecture when $n<6$ and in certain other special cases. We
establish a quadratic bound for the critical exponent in general.

We first discuss the construction by Perez-Izquierdo and Shestakov of
universal nonassociative enveloping algebras of Malcev algebras. We then
describe recent results on explicit structure constants for the universal
enveloping algebras (both nonassociative and alternative) of the 4-dimensional
solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We
include a proof (due to Shestakov) that the universal alternative enveloping
algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the
8-dimensional division algebra of real octonions.

For n even, we prove Pozhidaev's conjecture on the existence of associative
enveloping algebras for simple n-Lie algebras. More generally, for n even and
any (n+1)-dimensional n-Lie algebra L, we construct a universal associative
enveloping algebra U(L) and show that the natural map from L to U(L) is
injective. We use noncommutative Grobner bases to present U(L) as a quotient of
the free associative algebra on a basis of L and to obtain a monomial basis of
U(L). In the last section, we provide computational evidence that the
construction of U(L) is much more difficult for n odd.

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$
be the corresponding companion matrices. We find necessary and sufficient
conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in
the full integral lattice $M_n(Z)$, in which case we compute the exact value of
this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative
ring with identity we determine when $R< C,D>=M_n(R)$, in which case a
presentation for $M_n(R)$ in terms of $C$ and $D$ is given.

In this paper, we present the construction of a geometric object, called a
generalized flag geometry, $(X^+;X^-)$, corresponding to a (2k +1)-graded Lie
algebra $g=g_k\oplus\dots\oplus g_{-k}$. We prove that $(X^+;X^-) can be
realized inside the space of inner filtrations of g and we use this realization
to construct "algebraic bundles" on $X^+$ and $X^-$ and some sections of these
bundles.

Using a recent result of Bogdanov and Guterman on the linear preservers of
pairs of simultaneously diagonalizable matrices, we determine all the
automorphisms of the vector space M_n(R) which stabilize the set of
diagonalizable matrices. To do so, we investigate the structure of linear
subspaces of diagonalizable matrices of M_n(R) with maximal dimension.

We prove that, if A is a strongly simply connected algebra of polynomial
growth, then A is torsionless-finite. In particular, its representation
dimension is at most three.

The notion of a $\mathcal{K}_2$-algebra was recently introduced by Cassidy
and Shelton as a generalization of the notion of a Koszul algebra. The Yoneda
algebra of any connected graded algebra admits a canonical $A_{\infty}$-algebra
structure. This structure is trivial if the algebra is Koszul. We study the
$A_{\infty}$-structure on the Yoneda algebra of a $\mathcal{K}_2$-algebra.

In this paper, by using Gr\"obner-Shirshov bases for Rota-Baxter algebras, we
prove that every dendriform dialgebra over a field of characteristic 0 can be
embedded into its universal enveloping Rota-Baxter algebra of weight 0.

This is a brief and informal introduction to cluster algebras. It roughly
follows the historical path of their discovery, made jointly with A.Zelevinsky.
Total positivity serves as the main motivation.

We introduce the notion of characteristic function of a quaternionic matrix,
whose roots are the left eigenvalues. We prove that for all $2\times 2$
matrices and for $3\times 3$ matrices having some zero entry outside the
diagonal there is a characteristic function which satisfies Hamilton-Cayley
theorem.

We associate two linear categories with two objects to a module over the
subalgebra of coinvariants of a Hopf-Galois extension, and prove that they are
isomorphic. The structure Theorem for cleft extensions, and the Militaru
\cStefan lifting Theorem can be obtained using these isomorphisms.

We report on some computations with reachable elements in simple Lie algebras
of exceptional type within the SLA package of GAP4. These computations confirm
the classification of such elements by Elashvili and Grelaud. Secondly they
answer a question from Panyushev. Thirdly they show in what way a recent result
of Yakimova for the Lie algebras of classical type extends to the exceptional
types.

Given an arbitrary field K, let V be a linear subspace of M_n(K) consisting
of matrices of rank lesser or equal to some r<n. A theorem of Atkinson and
Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the matrices of V
vanish on some common (n-r)-dimensional subspace of K^n, or it is true of the
matrices of its transpose V^t. Following some arguments of our recent proof of
the Flanders theorem for an arbitrary field, we show that this result holds for
any field.

When K is an arbitrary field, we study the affine automorphisms of M_n(K)
that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine
subspaces of singular matrices, this is easily reduced to the known case of
linear preservers when n>2 or #K>2. We include a short new proof of the more
general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank.
We also find that the group of affine transformations of M_2(F_2) that
stabilize GL_2(F_2) does not consist solely of linear maps.

We consider semigroups of transformations (partial mappings defined on a set
$A$) closed under the set-theoretic intersection of mappings treated as subsets
of $A\times A$. On such semigroups we define two relations: the relation of
semicompatibility which identifies two transformations at the intersection of
their domains and the relation of semiadjacency when the image of one
transformation is contained in the domain of the second. Abstract
characterizations of such semigroups are presented.

In this paper we present two algorithms for the computation of a diagonal
form of a matrix over non-commutative Euclidean domain over a field with the
help of Gr\"obner bases. This can be viewed as the pre-processing for the
computation of Jacobson normal form and also used for the computation of Smith
normal form in the commutative case. We propose a general framework for
handling, among other, operator algebras with rational coefficients. We employ
special "polynomial" strategy in Ore localizations of non-commutative
$G$-algebras and show its merits.

We generalize the well-known construction of dendriform dialgebras and
trialgebras from Rota-Baxter algebras to a construction from O-operators. We
then show that this construction from O-operators gives all dendriform
dialgebras and trialgebras. Furthermore there are bijections between certain
equivalence classes of invertible O-operators and certain equivalence classes
of dendriform dialgebras and trialgebras.

Signatures of quadratic forms have been generalized to hermitian forms over
algebras with involution. In the literature this is done via Morita theory,
which causes sign ambiguities in certain cases. The main result of this paper
consists of a method for resolving this problem, using properties of the
underlying algebra with involution.

The paper describes interrelations between: (1) algebraic structure on sets
of scalars, (2) properties of monads associated with such sets of scalars, and
(3) structure in categories (esp. Lawvere theories) associated with these
monads. These interrelations will be expressed in terms of "triangles of
adjunctions", involving for instance various kinds of monoids (non-commutative,
commutative, involutive) and semirings as scalars. It will be shown to which
kind of monads and categories these algebraic structures correspond via
adjunctions.

In this paper, we study certain algebras related to a graded selfinjective
algebra with Nakayama translation $\tau$. We prove that a graded selfinjective
algebra is a regular covering of its orbit algebra, as a consequence, the McKay
quiver of a finite subgroup of a general linear group is a covering of the
McKay quiver of its intersection with the special linear group. We describe the
bound quiver of its Beilinson algebra.

We define a transcendence degree for division algebras, by modifying the
lower transcendence degree construction of Zhang. We show that this invariant
has many of the desirable properties one would expect a noncommutative analogue
of the ordinary transcendence degree for fields to have. Using this invariant,
we prove the following conjecture of Small. Let $k$ be a field, let $A$ be a
finitely generated $k$-algebra that is an Ore domain, and let $D$ denote the
quotient division algebra of $A$.

In this paper we construct ternary $q$-Virasoro-Witt algebras which
$q$-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie
and Zachos using $su(1,1)$ enveloping algebra techniques. The ternary
Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a
parameter and are not Nambu-Lie algebras for all but finitely many values of
this parameter.

Hom-Maltsev(-admissible) algebras are defined, and it is shown that
Hom-alternative algebras are Hom-Maltsev-admissible. With a new definition of a
Hom-Jordan algebra, it is shown that Hom-alternative algebras are
Hom-Jordan-admissible. Hom-type generalizations of some well-known identities
in alternative algebras, including the Moufang identities, are obtained.

The problem of construction of Barabanov norms for analysis of properties of
the joint (generalized) spectral radius of matrix sets has been discussed in a
number of publications. The method of Barabanov norms was the key instrument in
disproving the Lagarias-Wang Finiteness Conjecture. The related constructions
were essentially based on the study of the geometrical properties of the unit
balls of some specific Barabanov norms.

The problem of computation of the joint (generalized) spectral radius of
matrix sets has been discussed in a number of publications. In the paper an
iteration procedure is considered that allows to build numerically Barabanov
norms for the irreducible matrix sets and simultaneously to compute the joint
spectral radius of these sets.

An associative ring $R$ with identity is called left morphic if
$\frac{R}{Ra}\cong l_{R}(a)$ for every $a\in R$ and $R$ is called morphic if it
is both left and right morphic. Given a commutative B\'{e}zout domain $R$ with
classical quotient field $Q$, Diesl, Dorsey, and McGovern [A characterization
of certain morphic trivial extensions, {\it arXiv:0907.1141v1[math.RA]}, 7 Jul
2009 ] asked whether or not $R\ltimes M$ is morphic iff $M\cong \frac{Q}{R}$.
Here we affirmatively answer this question.

For a matrix coalgebra $C$ over some field, we determine all small
subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a
bjective antipode on $C$, and the free Hopf algebra with antipode $S$
satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information
to find the endomorphisms of these free Hopf algebras, and to determine the
centers of the categories of Hopf algebras, Hopf algebras with bijective
antipode, and Hopf algebras with antipode of order dividing 2d.

C.H. Yang discovered a polynomial version of the classical Lagrange identity
expressing the product of two sums of four squares as another sum of four
squares. He used it to give short proofs of some important theorems on
composition of delta-codes (now known as T-sequences). We investigate the
possible new versions of his polynomial Lagrange identity. Our main result
shows that all such identities are equivalent to each other.

By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an
exhaustive and not repetitive description of all 4-dimensional absolute-valued
algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}.
For such an algebras the number N(p,q,r) of isomorphism classes is 2 or 3, or
is infinite. Concretely 1. N(1,1,1)=N(1,1,2)=N(1,2,1)=N(2,1,1)=2, 2.
N(1,2,2)=N(2,2,1)=3, 3. N(2,1,2)=N(2,2,2)=\infty. Besides, each one of the
above algebras contains 2-dimensional subalgebras. However, the problem in
dimension 8 is far from being completely solved.

Let A be a connected-graded algebra with trivial module k, and let B be a
graded Ore extension of A. We relate the structure of the Yoneda algebra E(A)
:= Ext_A(k,k) to E(B). Cassidy and Shelton have shown that when A satisfies
their K_2 property, B will also be K_2. We prove the converse of this result.

Let W be an associative PI-algebra over a field F of characteristic zero.
Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote
the codimension growth of W and of the identity component W_e, respectively.
The following inequality had been conjectured by Bahturin and Zaicev:
exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is
affine (i.e. finitely generated). Here we prove the conjecture in general.

Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop
$\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this
paper, all the finite-dimensional commutative Hopf algebras over the sub
coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the
commutative infinitesimal groups $\mathcal{G}$ with
dim$_{k}$Lie$(\mathcal{G})=1$ are classified.

In this work we are interested in the general problem of the determination of
the normed division algebras. Our fundamental results are obtained in the
particular subclass of those 8-dimensional quadratic flexible real division
algebras.

In these notes we provide the foundation for the deformation theoretic parts
of arXiv:0807.375 and arXiv:math/0102005.

We apply the full centre construction, defined in arXiv:0908.1250, to
algebras in and module categories over categories of representations of Hopf
algebras. We obtain a compact formula for the full centre of a module algebra
over a Hopf algebra.

This paper unifies several generalizations of coherent rings in one notion.
Namely, we introduce $n$-$\mathscr{X}$-coherent rings, where $\mathscr{X}$ is a
class of modules and $n$ is a positive integer, as those rings for which the
subclass $\mathscr{X}_n$ of $n$-presented modules of $\mathscr{X}$ is not
empty, and every module in $\mathscr{X}_n$ is $n+1$-presented. Then, for each
particular class $\mathscr{X}$ of modules, we find correspondent relative
coherent rings.

We show that the discrete complex, and numerous hypercomplex, Fourier
transforms defined and used so far by a number of different researchers can be
unified into a single theoretical framework based on a matrix exponential
version of Euler's formula $e^{j\theta}=\cos\theta+j\sin\theta$, and a matrix
root of -1 in place of the imaginary root $j$.

The aim of this paper is to give a survey of nonassociative Hom-algebra and
Hom-superalgebra structures. The main feature of these algebras is that the
identities defining the structures are twisted by homomorphisms. We discuss
Hom-associative algebras, Hom-Flexible algebras, Hom-Lie algebras,
$G$-hom-associative algebras, Hom-Poisson algebras, Hom-alternative algebras
and Hom-Jordan algebras and $\mathbb{Z}_2$-graded versions. We give an overview
of the development of Hom-algebras structures which have been intensively
investigated recently.

Given an integer n greater of equal to 3, we investigate the minimal
dimension for a subalgebra of square matrices of order n with a trivial
centralizer. It is shown that this dimension is 5 when n is even and 4 when it
is odd. In this latter case, all 4-dimensional subalgebras with a trivial
centralizer are explicitely computed.

We consider the two-sided eigenproblem Ax= lBx over max algebra. It is shown
that any finite system of real intervals and points can be represented as
spectrum of this eigenproblem.

We show how to use Groebner bases for operads to prove various freeness
theorems: freeness of certain operads as nonsymmetric operads, freeness of an
operad Q as a P-module for an inclusion P into Q, freeness of a suboperad. This
gives new proofs of many known results of this type and helps to prove some new
results.

We introduce the notions of a commutative square ring $R$ and of a quadratic
map between modules over $R$, called $R$-quadratic map. This notion generalizes
various notions of quadratic maps between algebraic objects in the literature.
We construct a category of quadratic maps between $R$-modules and show that it
is a right-quadratic category and has an internal Hom-functor. Along our way,
we recall the notions of a general square ring $R$ and of a module over $R$,
and discuss their elementary properties in some detail, adopting an operadic
point of view.

Let A be a finite-dimensional associative algebra and $\phi$ a symmetric
linear function on $A$. In this note, we will show that the pseudotrace maps
are obtained as special cases of well-known symmetric linear functions on the
endomorphism rings of projective modules. We also prove that modules are
interlocked with $\phi$ if and only if they are projective. As an application
of our approach, we will give proofs of several propositions and theorems for
pseudotrace maps for an arbitrary finite-dimensional associative algebra.

The question of whether or not any zero torsion linear map on a non abelian
real Lie algebra g is necessarily an extension of some CR-structure is
considered and answered in the negative. Two examples are provided, one in the
negative and one in the positive.In both cases, the computation up to
equivalence of all zero torsion linear maps on g is used for an explicit
description of the equivalence classes of integrable complex structures on the
direct product g x g.

Graded skew-commutative rings occur often in practice. Here are two examples:
1) The cohomology ring of a compact three-dimensional manifold. 2) The
cohomology ring of the complement of a hyperplane arrangement (the
Orlik-Solomon algebra). We present some applications of the homological theory
of these graded skew-commutative rings. In particular we find compact oriented
3-manifolds without boundary for which the Hilbert series of the Yoneda
Ext-algebra of the cohomology ring of the fundamental group is an explicit
transcendental function.

In this article, we describe the right ideals of $A_1:=k[t,\partial]$, the
first Weyl agebra, over any field $k$ of characteristic zero. For this, we
define the notion of primary decomposable subspaces of $k[t]$. This description
generalizes a result of Cannings and Holland obtained for an algebraically
closed field $k$. Dans cet article, on d\'ecrit les id\'eaux \`a droite de
$A_1$ sur un corps quelconque de caract\'eristique nulle. Pour cela on
d\'efinit la notion de sous-espaces d\'ecomposables primaires de $k[t]$.

We continue the study of the lower central series of a free associative
algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410).

In this paper, we generalize the concepts of level and sublevels of a
composition algebra to algebras obtained by the Cayley-Dickson process. In
1967, R. B. Brown constructed, for every $t\in \Bbb{N},$ a division algebra
$A_{t}$ of dimension $2^{t}$ over the power-series field
$K\{X_{1},X_{2},...,X_{t}\}.$ This gives us the possibility to construct a
division algebra of dimension 2$^{t}$ and prescribed level 2$^{k}$ $ k, t\in
\Bbb{N}^{*}.$

The inner automorphisms of a group G can be characterized within the category
of groups without reference to group elements: they are precisely those
automorphisms of G that can be extended, in a functorial manner, to all groups
H given with homomorphisms G --> H. Unlike the group of inner automorphisms of
G itself, the group of such extended systems of automorphisms is always
isomorphic to G.

We study finiteness conditions on essential extensions of simple modules over
the quantum plane and over some Noetherian down-up algebras. The results
achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.

We characterize prelie algebras in words of left ideals of the enveloping
algebras and in words of modules, and use this result to prove that a simple
complex finite-dimensional Lie algebra is not prelie, with the possible
exception of f4.

The principal filtration of the infinite-dimensional odd Contact Lie
superalgebra over a field of characteristic $p>2$ is proved to be invariant
under the automorphism group by investigating ad-nilpotent elements and
determining certain invariants such as subalgebras generated by some
ad-nilpotent elements. Then, it is proved that two automorphisms coincide if
and only if they coincide on the -1 component with respect to the principal
grading. Finally, all the odd Contact superalgebras are classified up to
isomorphisms.

For a given abelian group G, we classify the isomorphism classes of
G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1)
(m>=2), in terms of numerical and group-theoretical invariants. Our main tool
is automorphism group schemes, which we determine for the simple restricted Lie
algebras of types S(m;1) and H(m;1). The ground field is assumed to be
algebraically closed of characteristic p>3.

Let $\mathbb K$ be a field and suppose $p, q\in\mathbb K^*$ are not roots of
unity. We prove that the two quantum groups $U_q(\mathfrak {sl}_{n+1})$ and
$U_p(\mathfrak{sl}_{n+1})$ are isomorphic as $\mathbb K$-algebras implies that
$p=\pm q^{\pm 1}$ when $n$ is even. This new result answers a classical
question of Jimbo.

The fundamental properties of biquaternions (complexified quaternions) are
presented including several different representations, some of them new, and
definitions of fundamental operations such as the scalar and vector parts,
conjugates, semi-norms, polar forms, and inner and outer products. The notation
is consistent throughout, even between representations, providing a clear
account of the many ways in which the component parts of a biquaternion may be
manipulated algebraically.

When $\mathbb{K}$ is a field, and $\mathcal{A}$ and $\mathcal{B}$ denote
commuting subspaces of $\text{M}_n(\K)$ each of which contains a non-scalar
matrix, we prove that $\dim \mathcal{A} +\dim \mathcal{B} \leq (n-1)^2+3$. We
also give a complete description of the cases when equality holds.

We use techniques from both real and complex algebraic geometry to study
K-theoretic and related invariants of the algebra C(X) of continuous
complex-valued functions on a compact Hausdorff topological space X. For
example, we prove a parametrized version of a theorem of Joseph Gubeladze; we
show that if M is a countable, abelian, cancellative, torsion-free, seminormal
monoid, and X is contractible, then every finitely generated projective module
over C(X)[M] is free.

We study formal deformations of a crossed product $S(V)#_f G$, of a
polynomial algebra with a group, induced from a universal deformation formula
introduced by Witherspoon. These deformations arise from braided actions of
Hopf algebras generated by automorphisms and skew derivations. We show that
they are nontrivial in the characteristic free context, even if $G$ is
infinite, by showing that their infinitesimals are not coboundaries. For this
we construct a new complex which computes the Hochschild cohomology of $S(V)#_f
G$.

We study the behavior of max-algebraic powers of a reducible nonnegative n by
n matrix A. We show that for t>3n^2, the powers A^t can be expanded in
max-algebraic powers of the form CS^tR, where C and R are extracted from
columns and rows of certain Kleene stars and S is diadonally similar to a
Boolean matrix. We study the properties of individual terms and show that all
terms, for a given t>3n^2, can be found in O(n^4 log n) operations.

In this paper we explore graded algebras of quotients of Lie algebras with
special emphasis on the 3-graded case and answer some natural questions
concerning its relation to maximal Jordan systems of quotients.

It is shown that the problem of reduction can be formulated in a uniform way
using the theory of invariants. This provides a powerful tool of analysis and
it opens the road to new applications of these algebras, beyond the context of
integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras
associated to the icosahedral group I, the octahedral group O, the tetrahedral
group T, and the dihedral group Dn are isomorphic.

Consider the $n$th degree polynomial equation,
$X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices.
If this equation has more than ${2n \choose 2}$ solutions, then it has
infinitely many solutions. We show here that for any $n,m \in\N$ such that
$m\leq{2n \choose 2}$, there exists an $n$th degree polynomial equation with
exactly $m$ solutions.

Let $F$ be any field of characteristic $p$. It is well-known that there are
exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of
$C_p$ defined over $F$. Thus if $V$ is any finite dimensional
$C_p$-representation there are non-negative integers $0\leq n_1,n_2,..., n_k
\leq p-1$ such that $V \cong \oplus_{i=1}^k V_{n_i+1}$. It is also well-known
there is a unique (up to equivalence) $d+1$ dimensional irreducible complex
representation of $\SL_2(\C)$ given by its action on the space $R_d$ of $d$
forms. Here we prove a conjecture, made by R.J.

We prove that the algebra $\mI_n:=K\langle x_1, \ldots , x_n,
\frac{\der}{\der x_1}, \ldots ,\frac{\der}{\der x_n}, \int_1, \ldots ,
\int_n\rangle $ of integro-differential operators on a polynomial algebra is a
prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull
dimension $n$ and of Gelfand-Kirillov dimension $2n$. Its weak homological
dimension is $n$, and $n\leq \gldim (\mI_n)\leq 2n$.

We provide sufficient conditions for a lattice polynomial function to be
self-commuting. We explicitly describe self-commuting polynomial functions over
chains.

It is shown that any finite-dimensional homomorphic image of an inverse limit
of nilpotent not-necessarily-associative algebras over a field is nilpotent.
More generally, this is true of algebras over a general commutative ring k,
with "finite-dimensional" replaced by "of finite length as a k-module".

This is an old paper put here for archeological purposes. It is proved that a
finite-dimensional Lie algebra over a field of characteristic p>5, that can be
written as a vector space (not necessarily direct) sum of two nilpotent
subalgebras, is solvable. The same result (but covering also the cases of low
characteristics) was established independently by V. Panyukov (Russ. Math.
Surv. 45 (1990), No.4, 181-182), and the homological methods utilized in the
proof were developed later in arXiv:math/0204004.

Two similar embedding theorems for algebras and groups are presented, basing
on a certain old ultrafilter construction. As an application, we outline
alternative proofs of some results from the theory of PI algebras, and
establish some interesting properties of Tarski's monsters.

In this note we study morphisms from P2 to Gr(2,C4) from the point of view of
the cohomology class they represent in the Grassmannian. This leads to some new
result about projection of d-uple imbedding of P2 to P5.

We show that there are exactly three types of Hilbert series of
Artin-Schelter regular algebras of dimension five with two generators. One of
these cases (the most extreme) may not be realized by an enveloping algebra of
a graded Lie algebra. This is a new phenomenon compared to lower dimensions,
where all resolution types may be realized by such enveloping algebras.

An infinite filiform Lie algebra L is residually nilpotent and its graded
associated with respect to the lower central series has smallest possible
dimension in each degree but is still infinite. This means that gr(L) is of
dimension two in degree one and of dimension one in all higher degrees. We
prove that if L is solvable, then already [L,L] is abelian. The isomorphism
classes in this case are given in a paper by Bratzlavsky, but the proof there
is incomplete. We make the necessary additional computations and restate
Bratzlavskys result when the ground field is the complex numbers.

We give a new and elementary proof that simultaneous similarity and
simultaneous equivalence of families of matrices are invariant under extension
of the ground field, a result which is non-trivial for finite fields and first
appeared in a paper of Klinger and Levy.

In their "Cluster Algebras IV" paper, Fomin and Zelevinsky defined
F-polynomials and g-vectors, and they showed that the cluster variables in any
cluster algebra can be expressed in a formula involving the appropriate
F-polynomial and g-vector.

A square matrix is called {\it Hessenberg} whenever each entry below the
subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$
denote a nonzero finite-dimensional vector space over a field $\fld$. We
consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to
V$ which satisfy both (i), (ii) below. \begin{enumerate} \item There exists a
basis for $V$ with respect to which the matrix representing $A$ is Hessenberg
and the matrix representing $A^*$ is diagonal.

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many
places in mathematics. In this article we find the multiplicative structure of
their Hochschild cohomology when the coefficients are in an arbitrary bimodule
algebra. When this bimodule algebra is a finite group extension (under a
diagonal action) of a quantum symmetric algebra, we give explicitly the graded
vector space structure. This yields a complete description of the Hochschild
cohomology ring of the corresponding skew group algebra.

A classification of all four-dimensional power-commutative real division
algebras is given.

It is shown that every four-dimensional power-commutative real division
algebra is an isotope of a particular kind of a quadratic division algebra. The
description of such isotopes in dimension four and eight is reduced to the
description of quadratic division algebras. In dimension four this leads to a
complete and irredundant classification. As a special case, the
finite-dimensional power-commutative real division algebras that have a unique
non-zero idempotent are characterised.

We consider two families of finite-dimensional simple Lie superalgebras of
Cartan type, denoted by HO and KO, over an algebraically closed field of
characteristic p>3. Using the weight space decompositions and the principal
gradings we first show that neither HO nor KO possesses a nondegenerate
associative form. Then, by means of computing the superderivations from the Lie
superalgebras in consideration into their dual modules, the second cohomology
groups with coefficients in the trivial modules are proved to be vanishing.

We present the classical Poisson-Lichnerowicz cohomology for the Poisson
algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus.
After presenting some non homogeneous Poisson brackets on this algebra, we
compute Poisson cohomological spaces when the Poisson structure corresponds to
a bracket of a rigid Lie algebra.

Cassidy, Phan and Shelton associate to any regular cell complex X a quadratic
K-algebra R(X). They give a combinatorial solution to the question of when this
algebra is Koszul. The algebra R(X) is a combinatorial invariant but not a
topological invariant. We show that nevertheless, the property that R(x) be
Koszul is a topological invariant.

In the process we establish some conditions on the types of local singular-
ities that can occur in cell complexes X such that R(X) is Koszul, and more
generally in cell complexes that are pure and connected by codimension one
faces.

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan
identity, for a sensible theory of quantum mechanics. All but one of the
algebras that satisfy this condition can be described by Hermitian matrices
over the complexes or quaternions. The remaining, exceptional Jordan algebra
can be described by 3x3 Hermitian matrices over the octonions.

In this paper, as part of a project initiated by A. Mallios consisting of
exploring new horizons for \textit{Abstract Differential Geometry} ($\grave{a}$
la Mallios), \cite{mallios1997, mallios, malliosvolume2, modern}, such as those
related to the \textit{classical symplectic geometry}, we show that results
pertaining to biorthogonality in pairings of vector spaces do hold for
biorthogonality in pairings of $\mathcal A$-modules. However, for the
\textit{dimension formula} the algebra sheaf $\mathcal A$ is assumed to be a
PID.

In this paper we are interested in functionals defined on completely
distributive lattices and which are invariant under mappings preserving
{arbitrary} joins and meets. We prove that the class of nondecreasing invariant
functionals coincides with the class of Sugeno integrals, and that the the
superclass of lattice polynomial functionals can be analogously characterized
by naturally relaxing this invariance condition. Moreover, we show that, in the
case of functionals over complete chains, the nondecreasing condition is
redundant.

When a finite group acts linearly on a complex vector space, the natural
semi-direct product of the group and the polynomial ring over the space forms a
skew group algebra. This algebra plays the role of the coordinate ring of the
resulting orbifold and serves as a substitute for the ring of invariant
polynomials from the viewpoint of geometry and physics. Its Hochschild
cohomology predicts various Hecke algebras and deformations of the orbifold. In
this article, we investigate the ring structure of the Hochschild cohomology of
the skew group algebra.

Let $V$ denote a vector space with finite positive dimension. We consider an
ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that
satisfy (i) and (ii) below:

(i) There exists a basis for $V$ with respect to which the matrix
representing $A$ is irreducible tridiagonal and the matrix representing $A^*$
is diagonal.

(ii) There exists a basis for $V$ with respect to which the matrix
representing $A^*$ is irreducible tridiagonal and the matrix representing $A$
is diagonal.

In this paper, we study the moduli space of $1|2$-dimensional complex
associative algebras, which is also the moduli space of codifferentials on the
tensor coalgebra of a $2|1$-dimensional complex space. We construct the moduli
space by considering extensions of lower dimensional algebras. We also
construct miniversal deformations of these algebras. This gives a complete
description of how the moduli space is glued together via jump deformations.

We describe the classes of operations closed under permutation of variables,
addition of dummy variables and composition in terms of a preservation relation
between operations and certain systems of multisets.

A characterization of right (left) quasi-duo skew polynomial rings of
endomorphism type and skew Laurent polynomial rings are given. In particular,
it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is
commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the
skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These
extend some known results concerning a description of quasi-duo polynomial
rings and give a partial answer to the question posed by Lam and Dugas whether
right quasi-duo rings are left quasi-duo.

Let k be an infinite field, I an infinite set, V a k-vector-space, and
g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under
various hypotheses on card(k) and card(I), if it is finite, respectively
countable, respectively < card(k)), then ker(g) must contain elements
(u_i)_{i\in I} with all but finitely many components u_i nonzero.

Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3
over an uncountable algebraically closed field. Suppose that the graded
quotient ring of R is a skew-Laurent ring over a field; we say that R is a
birationally commutative projective surface. We classify birationally
commutative projective surfaces and show that they fall into four families,
parameterized by geometric data. This generalizes work of Rogalski and Stafford
on birationally commutative projective surfaces generated in degree 1; our
proof techniques are quite different.