In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A.
The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer. [1] However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as coherent algebras. [2] [3] [4]
Let be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also be an -algebra.
A cell datum for is a tuple consisting of
This definition was originally given by Graham and Lehrer who invented cellular algebras. [1]
Let be an anti-automorphism of -algebras with (just called "involution" from now on).
A cell ideal of w.r.t. is a two-sided ideal such that the following conditions hold:
A cell chain for w.r.t. is defined as a direct decomposition
into free -submodules such that
Now is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent. [5] Every basis gives rise to cell chains (one for each topological ordering of ) and choosing a basis of every left ideal one can construct a corresponding cell basis for .
is cellular. A cell datum is given by and
A cell-chain in the sense of the second, abstract definition is given by
is cellular. A cell datum is given by and
A cell-chain (and in fact the only cell chain) is given by
In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset .
Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as . [6] This includes for example the integral group algebra of the symmetric groups as well as all other finite Weyl groups.
A basic Brauer tree algebra over a field is cellular if and only if the Brauer tree is a straight line (with arbitrary number of exceptional vertices). [5]
Further examples include q-Schur algebras, the Brauer algebra, the Temperley–Lieb algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category of a semisimple Lie algebra. [5]
Assume is cellular and is a cell datum for . Then one defines the cell module as the free -module with basis and multiplication
where the coefficients are the same as above. Then becomes an -left module.
These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A.
There is a canonical bilinear form which satisfies
for all indices .
One can check that is symmetric in the sense that
for all and also -invariant in the sense that
for all ,.
Assume for the rest of this section that the ring is a field. With the information contained in the invariant bilinear forms one can easily list all simple -modules:
Let and define for all . Then all are absolute simple -modules and every simple -module is one of these.
These theorems appear already in the original paper by Graham and Lehrer. [1]
If is an integral domain then there is a converse to this last point:
If one further assumes to be a local domain, then additionally the following holds:
Assuming that is a field (though a lot of this can be generalized to arbitrary rings, integral domains, local rings or at least discrete valuation rings) and is cellular w.r.t. to the involution . Then the following hold
In mathematics, specifically in functional analysis, a C∗-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator on M. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras..
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure, which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures.
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects.
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center of the universal enveloping algebra of a reductive Lie algebra to the elements of the symmetric algebra of a Cartan subalgebra that are invariant under the Weyl group .
In mathematics, a finitely generated algebra is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K.
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
In the representation theory of semisimple Lie algebras, Category O is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations. Let be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over is semisimple as a module
In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extensione is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.
In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.