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The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafe asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

If the enveloping central simple algebra of an Okubo algebra comes with a valuation, then we can compute the residue of that Okubo algebra, at least as a vector subspace of the central simple algebra. We give a criterion for the residue of an Okubo algebra without nonzero idempotents to be an Okubo algebra. We also prove that Okubo algebras without nonzero idempotents over a field of characteristic 3 are always the residue of an Okubo algebra over a field of characteristic 0.

Various classes of simple torsion modules are classified over the quantum spatial ageing algebra (this is a Noetherian algebra of Gelfand-Kirillov dimension 4). Explicit constructions of these modules are given and for each module its annihilator is found.\ud \ud

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.

Danz computes the depth of certain twisted group algebra extensions in Comm. Alg. (2011), which are less than the values of the depths of the corresponding untwisted group algebra extensions in Burciu et al, I.E.J.A. (2011). In this paper, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right $H$-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A tensor C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R < H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that subgroup depth behaves exactly like combinatorial depth with respect to the core of a subgroup, and extend results in Kadison J.Pure Appl.Alg. (2014) to coideal subalgebras of finite dimension., 25 pp. A proposition 1.3 is added connecting relative separability with finite depth, and an example 1.6 computing Q for H* in D(H). The introduction and abstract are rewritten

Any (associative) left Noetherian algebra over a field, which is finitely generated as an algebra over a central subring, is representable. As a special case, we give a short proof of the well-known theorem that any algebra over a field, which also is finite (as a module) over a central affine algebra, is representable. We give a counterexample to some natural generalizations, as well as a conjectured generic counterexample, together with specific positive results for irreducible algebras and finitely presented algebras. Also, based on joint work of the second author with Amitsur (posthumous), we show that any semiprimary PI-algebra with radical squared 0 is weakly representable.

In this paper we describe the blocks of the partition algebra over a field of positive characteristic.

In 1993 I. P. Shestakov came up with the question whether there exists a central simple finite-dimensional algebra over a field of characteristic 0, whose identities are not given by a finite set (Dniester Notebook, Question 3.103). In 2012 I. M. Isaev and the author constructed a desired example, giving an affirmative answer to the question posed. Here research into Shestakov’s question is continued for the case of commutative algebras. A seven-dimensional central simple commutative algebra over a field of characteristics 0 having no finite basis of identities is exemplified.

This paper fills in the relationship between the standard and dual representations of irreducible Brauer algebra modules in all cases, semisimple and non-semisimple. This parallels the correspondence given in earlier works for the constructions of the Specht modules. In addition, due to the close relationship between Brauer algebra irreducible modules in the non-semisimple case and certain multiplications, some general work on polytabloids is discussed. This gives an extension to previous results about the non-semisimple case.

Extending two classical embedding theorems of Albert and Jacobson and Jacobson for Albert (exceptional simple Jordan) algebra over fields of characteristic not two to base fields of arbitrary characteristic, we show that any element of a reduced Albert algebra can be embedded into a reduced absolutely simple subalgebra of degree 3 and dimension 9 which may be chosen to be split if the Albert algebra was split to begin with.

We determine decomposition matrices of Brauer algebras over a field of characteristic 0 and for fields of prime characteristic p with p larger than the degree of the Brauer algebra. We reprove and extend work of Martin as well as by Cox and De Visscher over the complex field by using new methods which uniformly treat the characteristic 0 and prime characteristic cases. These methods are inspired by the theory of Jucys–Murphy elements and the use of refined induction and restriction functors which are generalizations of similar functors for symmetric groups.

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.

A classification of the simple weight modules is given for the (6-dimensional) Euclidean Lie algebra 𝔢(3) = 𝔰𝔩2⋉V3. As an intermediate step, a classification of all simple modules is given for the centralizer C of the Cartan element H (in the universal enveloping algebra 𝒰 = U(𝔢(3))). Generators and defining relations for the algebra C are found (there are three quadratic relations and one cubic relation). The algebra C is a Noetherian domain of Gelfand-Kirillov dimension 5. Classifications of prime, primitive, completely prime, and maximal ideals are given for the algebra U .

The Brauer algebra was introduced by R. Brauer in 1937 as a tool in invariant theory. The problem of determining the Cartan decomposition matrix of the Brauer algebra over the complex field has remained open since then. Here we determine this fundamental invariant.

Let u k (2, r) be a little q-Schur algebra over k, where k is a field containing an l′-th primitive root ɛ of 1 with l′ ≥ 4 even, the author constructs a certain monomial base for little q-Schur algebra u k (2, r).

In this paper, we consider the simple non-associative algebra WN(F(e x r ;0;1)(@;@2)). There are many papers on nding

In this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

Rota-Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota-Baxter operators of weight zero on a 4-dimensional complex simple associative algebra, which is isomorphic to the algebra of all 2x2 matrices over the field of complex numbers. Such operators satisfy (the operator form of) the classical Yang-Baxter equation on the general linear Lie algebra gl"2(C). We provide two different solving methods: one by hand to show some tricks for solving a large nonlinear system, and another one by Computer Algebra to show the possibility of solving higher dimensional cases.

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D_n is shown to be semisimple and free of rank (2^n + 1)n!! − (2^n−1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D_n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ℤ[δ^(±1)]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D_n is a subalgebra of the BMW algebra of the same type.

Brauer algebras form a tower of cellular algebras. There is a well-defined notion of limiting blocks for these algebras. In this paper we give a complete description of these limiting blocks over any field of positive characteristic. We also prove the existence of a class of homomorphisms between cell modules.

We investigate the Brauer group BRM(k, H) of a cocommutative Hopf algebra H (H can be nonfinitely generated). BRM(k, H) is the bigger Brauer group of H-module Taylor–Azumaya algebras (which do not necessarily contain a unit). An exact sequence is obtained. Furthermore, we prove the surjectivity of under certain circumstances. This generalizes Beattie's exact sequence to the “infinite” case.

Under study are the centralizers of 3-dimensional simple Lie subalgebras in the universal enveloping algebra of a 7-dimensional simple Malcev algebra. We find some sets of generators for these centralizers in characteristic not 2 nor 3 and for the subalgebra generated by the centralizer in the central closure of the universal enveloping algebra in characteristic 3. As a corollary of the main theorem we obtain the available description of the center of universal enveloping algebra of a 7-dimensional simple Malcev algebra.

We prove that each exceptional differentiably simple Jordan algebra over a field of characteristic 0 is an Albert ring whose elements satisfy a cubic equation with the coefficients in the center of the algebra. If the characteristic of the field is greater than 2 then such an algebra is the tensor product of its center and a central exceptional simple 27-dimensional Jordan algebra. Some remarks made on special algebras.

We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Brou\'e, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group $G_{32}$, of order 155,520.

By Merkurjev's Theorem every central simple algebra of exponent two is Brauer equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over a given field is split, then there exists no central simple algebra of exponent two over this field. This note provides an independent elementary proof for the latter fact. (C) 2016 Elsevier B.V. All rights reserved.

Let A be a finite dimensional central simple algebra. By the Skolem–Noether theorem, every automorphism of A is inner. We will give a short proof of a somewhat more general result. The concept behind this proof is the fact that every linear map on A belongs to the multiplication algebra of A. As an application we will describe linear maps α, β : A → A such that α(x)β(y) = 0 whenever xy = 0.

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) 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 L → 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.

The Jacobian algebra An is obtained from the Weyl algebra An by inverting (not in the sense of Ore) of certain elements (An is neither a Noetherian algebra nor a domain, An contains the algebra K〈x1,…,xn,∂∂x1,…,∂∂xn,∫1,…,∫n〉 of polynomial integro-differential operators). The group of automorphisms Gn of the Jacobian algebra An is found (Gn is a huge group): Gn=Sn⋉(Tn×Ξn)⋉Inn(An)⊇Sn⋉(Tn×(Zn)(Z))⋉GL∞(K)⋉⋯⋉GL∞(K)︸2n−1times,G1≃(T1×Z(Z))⋉GL∞(K), where Sn is the symmetric group, Tn is the n-dimensional algebraic torus, Ξn≃Zn is a group given explicitly, Inn(An) is the group of inner automorphisms of An (which is huge), GL∞(K) is the group of invertible infinite dimensional matrices, and (Zn)(Z) is a direct sum of Z copies of the free abelian group Zn. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group G1. The stabilizers in Gn of all the ideals of An are found, they are subgroups of finite index in Gn. It is shown that the group Gn has trivial center. An explicit inversion formula is given for the elements of Gn. Defining relations are found for the algebra An.

Using the determination of conjugacy classes in an earlier paper, we study the center of the Brauer algebra. In the case of finite groups, conjugacy class sums determine the center of the group algebra. In the case of the Brauer algebra the corresponding class sums only yield a basis of the centralizer of the symmetric group in the Brauer algebra. However, we exhibit an explicit algorithm to determine conditions for a centralizer element to be central and show how to compute a basis for the center using these methods. We will outline how this can be used to compute blocks over fields of arbitrary characteristic. We will also show that similar methods can be applied for computing a basis of the center of the walled Brauer algebra.

We prove that every finite-dimensional homogeneously simple associative algebra over an algebraically closed field is representable as the product of a full matrix algebra and a graded field.

In this paper we construct an algebra associated to a cubic curve C defined over a field F of characteristic not two or three. We prove that this algebra is an Azumaya algebra of rank nine. Its center is the affine coordinate ring of an elliptic curve, the Jacobian of the cubic curve C . The induced function from the group of F -rational points on the Jacobian into the Brauer group of F is a group homomorphism with image precisely the relative Brauer group of classes of central simple F -algebras split by the function field of C . We also prove that this algebra is split if and only if the cubic curve C has an F -rational point. These results generalize Haile's work on the Clifford algebra of a binary cubic form.

In this paper, by using Gröbner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We give another proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie algebra).

In this paper, we describe finite-dimensional homogeneously simple algebras of associative type whose 1-component is a full matrix algebra. In addition, we prove that a finite-dimensional division ring of associative type over an algebraically closed field is isomorphic to a group algebra.

Let $r \in \BC$ be a complex number, and $d \in \BZ_{\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\Vam$ explicitly., 30 pages, no figure

The simple non-associative algebra and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra and its antisymmetrized algebra . We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations of te antisymmetrized algebra and every derivation of the algebra is outer in this paper.

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A_(n − 1) on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A_(n − 1), D_n , E_6, E_7, E_8). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.

We study properties of n-tuple algebras of associative type. We show that the nilpotency of an n-tuple algebra of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an n-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple n-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple n-tuple algebra of associative type.

In the algebraic study of deep matrices ℳ X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ℳ X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.

We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.

This paper continues the study of the lower central series quotients of an associative algebra A, regarded as a Lie algebra, which was started in math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second quotient in the case when A is the free algebra in n generators (note that the Hilbert series of this quotient was determined earlier in math/0610410). Further, it uses this basis to determine the structure of the second quotient in the case when A is the free algebra modulo the relations saying that the generators have given nilpotency orders. Finally, it determines the structure of the third and fourth quotient in the case of 2 generators, confirming an answer conjectured in math/0610410. Finally, in the appendix, the results of math/0610410 are generalized to the case when A is an arbitrary associative algebra (under certain conditions on $A$)., Comment: 23 pages, latex, corrected some more typos

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

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras.

In the paper, it is proved that the radical of a relatively free associative algebra of countable rank over an infinite field of characteristic p > 0 is a nil ideal of bounded index if the complexity of the corresponding variety is less than p. Moreover, a description of a basis for trace identities for the matrix algebra M n over an infinite field of characteristic p > 0, n < p, is obtained in the paper.

G ∞ -structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is the extension of a B ∞ -algebra by an associative algebra. Actions of B ∞ -algebras on associative and B ∞ -algebras are analyzed; extensions of B ∞ -algebras by associative and B ∞ -algebras that they act upon are constructed. The resulting G ∞ -algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the G ∞ -algebra on the deformation complex of the corresponding diagram algebra.

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra [Formula: see text] and its restricted subalgebra [Formula: see text] are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra [Formula: see text] and the semi-Lie algebra [Formula: see text], where [Formula: see text]. We prove that the algebra is simple and find all its non-associative algebra derivations.

This paper presents some results on the simple exceptional Jordan algebra over an algebraically closed field Φ of characteristic not 2. Namely an example of simple decomposition of H(O3) into the sumof two subalgebras of the typeH(Q3) is produced, and it is shown that this decomposition is the only one possible in terms of simple subalgebras.

Let k be a field of characteristic not two. To each irreducible quartic f ( x ) over k we associate a certain algebra A f , given by explicit generators and relations. We prove A f is an Azumaya algebra of rank four over its center and that its center is the coordinate ring of an affine piece of an elliptic curve, the Jacobian of the curve C : y 2 = f ( x ) . Its simple images are quaternion algebras and the resulting function from the group of k-rational points on the Jacobian to the Brauer group of k is a group homomorphism whose image is the relative Brauer group of central simple k-algebra split by the function field of C. We also show that the algebra A f is split if and only if C has a k-rational point.

We give an algorithm for finding an isomorphism of a given central simple algebra of degree 4 over the rationals and the full matrix algebra, provided it exists. It reduces the task to classical problems in number theory, for which there are already known algorithms.